A Survey of Finite Mathematics

Citation preview

a survey

M arvin M arcus The University of California

et Santa Barbara

of finite mathematics

Houghton Mifflin Co mpany • B oston Naw Ytxk Alhnt1 GIMVI. 1//,no,s 01lf11

Palo Airo

To RrnFCCA ELIZABETII

Cort)right

1969 by \larvin Marcu,

Ali nghb rc~r.c:d. 1'0 pdrl of 1h1~ "'or!. may be: rc:producc:d or tran\mlltcd 1n any íorm or b) an) mean\, c:lc,;tronic or mcxhanical, 111clud1n1 pho1ocopymg and recorJma, or b)' any mfonnat1on storagc or rc:m.:H,I S)Stc:m, 1,11thout pc:rnuu1on m 1,1rnm1 from thc pubh!lhc:r.

PnntcJ 1n thc: USA

Preface The primary purpo:,,e of this book is for use as a tcxt in courscs usually enutled ¡.-;nite Mu1hertuuicr that ha\C come into cxistencc O\cr thc last few years in many collcgcs and uni\crs1tics. Suchcourscs are normally one quar1er o r one scmestcr in length and are mtendelmpk linear progrnmm1ng probkms. The ba,ic geometry oí come, scb. mcluC!. "h1eh are marked v.1th a daggcr conta1n re!>ults or ddinitions that are u..c p 11nd q in (10) are somctimcs ~iJ 10 be cqwt· ifq holds. Thus, if p • •q h.is tru1h \aluc T, wc say that p 1s a nt•ussar)' rmd sufficient condiuon for q, or that p holds 1/and only ,Jq holilr)' and suílicient condmon for p. This p:irticular tautology h:is 1hc sorne• \\hJ.t pompous namc of thc /(lw ofd,mMr nl·gatfon, wh1ch "'-C ,erify by mcans oí a truth table p• ·~(~p)

T T

l

In ordmJ.r) language. wc oflcn considcr conJunc11ons or disjunctions of more than '""º statcmcnts. llo\\c,cr, our \ogical S)m· bolum only allo1i1-s l"'-0 statcmcms 81 a tune 10 be pul togcthcr by

11

/

Trvrh Tablfl •Mi App/11 11,ons

9

thesc connttti\t"i. Ne\crtheless, -.-.e w.1111 to assign mcaning to such st.1tcmcntsas (12)

Thcrc are two waysm -.-.hich wc can make (12) intoa conjunction of two st:ucmcnts: (13) (14)

lt is important that we know ( 13) and (14) ha,c thc samc truth ,atucs for ali truth ,alues assign«l to 1he indi,idual st.ucmcnts p 1, pz, and p 1 • To sce 1h1n (13) and (14) ha\c this propcrty, wc can arguc as follows: p 2 /\ p3 has truth \ialue F un1css both p 2 and p 1 ha,c truth \aluc T Thc only way 111 -.-.hich thc conjunction p, /\ (Pi /\ p3) can ha,c truth \alue T is for p 1 and (p, /\ p 1) to both ha\c truth ,aluc T. That is, ( IJ)can ha,e truth ,a\ue T only -.-.hcn cach of p 1, p 2 • and PJ has truth ,a\uc T Similarly one s«s th,11 (14) can ha\C nuth ,aluc T only whcn cach of p., pz, and p 3 has truth \ialue T. Suppos,e now that wc h,nc four statcmcnts p,. p,, p 3 , and P•· -.-.hich JrC connccted in pairs by conjunction, e.g., (15)

Accordmg to our prc,ious argument, p 2 /\ (pa /\ p 4) has truth \aluc F' unless ali thrce of p 2 • p 3 , and p 4 ha,e truth ,aluc T. Thus onc sccs that ( IS) has truth ,alue F unlcss p, has truth ,alue T. i - l .... ,41ngcneral,if-.-.etal can)kpropositionsp1,P2, ,Ple and form a compound proposition using only thc connccti,c /\, it is clear that the compound will ha\C truth ,.1lue T only -.-.hcn c,cry p, has truth ,aluc T, i - l.. . . k. 01hcrwisc thc compound proposition will ha\c truth \ialuc F. Thi§ is thc case for a11)' association of thcsc Slfltcmcnt~ u)ing thc conm."CIÍ\C /\ Theorem 1.1

1// and g a,~ 11''0 comro1md s1atemen1.r obtamed h)' using onf)· the co11nern1,·e /\ 011 ali o/ the k propos11w,u is a tautofngy.

p 1, . .. , p1,,, lh('fl

¡ ..... g

Proo/ By thc aboH: discussion,/ttnd g both ha,c trulh \Blue T ií ;111 ('lf p 1 • •• • p1,, ha,e truth \3lue T and other-.-.isc both ha,e truth 1;1luc F. Thus if onc "ere 10 con~truct a 1ru1h table for ¡ ..... g, the columns for / and g would be ,dentical and /• • g -.-.ould alwa)S ha, c truth ,.1Juc T 1

Ch•¡: r,., 1 /

F'·

Proof Do1h compound !>tatcmcms/::and g wi\l ha,c truth \aluc T unlc!>s C\cry onc oí pi,. , P• has truth \aluc F. Thus / and g h,uc thc !.Jmc truth \i1lucs for ali choiccs of 1ru1h \i1lues for pi, .. ,p, and it fol\o...,;;that/• •gis a t,1utolo@)'- 1

Definltion 1. 1

(16)

Co11j1"' any onc of thc cqui\.1knt compound propo!,itions obt,lined from p 1 , •• ,Pt using 1hc connccmc V.

In discu!,.Sing thcsc nmlt1plc di~Junct1ons and conjunctions it is important w rcahze th,11 it is only thc truth \.llue of a c:omr-ound statcmc:nt th.tt m,1ttcr~. h-.r cx.1mpk. 1ffand g are'""º c:ompound statemcnt~ dcpending on the mdi\idu.11 s1a1cmc:nts P1• . • ,/lt .1nd f • • R is á IJUtOIO@)', thcn/mJ) be \Ubslitutcd for gin an)' [líOPl,'ISÍ· 110n ÍO\Ohingg. Nc,tt kt r, and p 2 be statcmen11, :ind !>Upposc wc consiJer the C(ltnpound !>tatcmcnt

Thc prorx~,.i1ion ,· h.-., the truth \;llm: T ...,h1:n /11 has truth \aluc T .1nJp 2 ha\ truth \J\ue F. For thc remaming threc paÍí$OÍlrUlh \lllucs for p 1 anJ /1:, , hJs truth \alue f. In other \\t1rJi;, thc 1ru1h table for

1.1

f

Truth T•ble11""'1Appl,c~r,on11

11

Pi A ~p2is

P•

P• A ~p2

P2

f

rT

T

T

F

F

F

T

F

F

F

F

T

r--

Nc:11:t considcr thc statcmcnt ru: ~p, A P2 A ~p3.

(Wc ha\c dcnotcd thc statcmcnt by cu for rcasons 10 be discusscd in Thcorcm 1.J.) \\ e know that 1he statcmcnt c 13 has truth \aluc T only v.hcn ~p 1 , p 2 , and ~P'J indi\ídually h¡nc 1ru1h ,aluc T. But ~p 1 and ~Pa ha\C truth \3lue T only v.-hen p 1 and p 3 ha\C truth \liluc F Thus wc scc tha1 ~p 1 A p 2 A ~p3 will ha\e truth \aluc T whcn p 2 has 1ruih valuc T and p 1 and p 2 ha\c truth ,aluc f'. f'or ali othcr truth \ttlucs assigncd to p 1, pz, and p 3 , thc statcmcnt rs bet11t>en I and k. let c, 1 . ... be the C'Ompound statement

P1 A Pz A··· A ~p, A· • A ~p, A'·· A ~p,. A··· A pa. ThM e,,_ .. /un 1rr11h value T only when p,. p 1 , •• • , p .,. ali hove truth 1,"Q/uc f' and al/ 1hc remammg s/uttments hai;e truth volue T. Proof Wc first illustratc cxamplcs of {18) in thc case of four stalcmcnts. Thus, supposc i - 2,J - 4 . Thcn c,i is l'24:

P1 A ~P2 A Pa A ~p4

Again, if 1 - l, J • 2, m - 4, 1hcn ( 18) becomcs l'1 2 4:

~p1 A ~P2 A Ps A ~p4.

To procced to the proof of the theorern v.1: first recall that the s1ate• mcnt (18) has truth \aluc T only "'-hcn c:ich of thc indi\idual stalC• mcnts appcaring in thc conJunction has truih \aluc T. This rcquircs that ,111 oí ~p,. ~p,, ... , ~p,., as v.ell .1s thc rcmainings1:uemcnts, ha\c truth \aluc T Thus p,. p 1 , ,p,. must h:nc truth \aluc F and 1he rcmaining st.itcmcnts must ha\c 1ruth \aluc T. 1

\\e \et r 11 denote the st,ncmcnt

1n "'hi.:h nonc nf thc st,llcment,; i,; preceded by a ncgatlon

Example 1.5

'iÍ(!n.

Consuuct ,1 compound st.itement usinp p 1, p 1• and p,. ...,h,,h has 1rmh ,alue T onl) "'hcn p 1 has truth ,.1\uc I and p 2 and PI ha,c truth \,1lue F. In 1hilitdie~. (d) The flow tal.ies plaL-e from .110 B íor all rnnfi¡urntion, of~,,_.,tches.

7

í hree 1.-11\es, 11, si, and , _,, nre to be ust'd to control thc flo,w of,,_.,'.ller from po1n1s A lo O man 1rngauon S)~tem. Des1gn a net""orl. us,ng thoc ,al,es (und 1he1r nc¡¡a11on)) ""hi.;h allo'>l-s ""ater to llow be1ween A and 8 only under thc follo""mg c1rcurnstanccs: ""hcn c11hcr s1 nnd si are m thc on pos1t1on and '" is m che olT J)O$L\IOn; o r '>lhen s 1 and tJ are 1n lhc on posmon and 11 1s m the off po~111on

8 Lct .r1, .s:, and .s1 be on-off sw1tches m an clcctnl1n11 s""LILh1ng 11rrangcments :ind draw II d1agrnm sho""'"I how thc s""1tchn are oonnectcd (a) T he curren! llowi 1f and onl) 1í all threc ~w1khcs tire m the on pos111on (b) Thc currcnt flows 1í and onl} 1f J1 and s2 are m thc on pos1hon a nd JJ is in the off pos111on (e) The curren! flows IÍ and only if 1, and si are m thc on po~1110n ""ha1c,cr Lhc pos111on of o may be. 9 Lct /he a compound statcmcnt con~trudcd from Pi, .p, . Thc truth Yt of /iSJUSI the set of 1host choices oftruth \'3IUCS PL,. , P, for wh1l1n11 rornpound s111temen1s:

ror

(a) (h) (r) (d) (e) (í) (¡)

P1 /\ pz, p¡ V (P1 /\ p1). P1 -(pz •p,), (Pi V Pi) •pz, (p1 /\ pz)--•~p1. (p1 /\ p¡) • ~p,, (P1 V ~P1) • • ~ (~P1 /\ P;)

1 O A ~"' 1LL hin¡ networl. 10 rontrol thc lau!khmg oían I C B ,1 is 10 he dcs1gncd 'IO 1t can hc nuruged h)' thrtt s«onJ \icutenJnb In rnnfurmil) ,,_.1\h dc:mocr.itK 1rad1110n. the De¡xartment oí Dcfcn~ nul.C'l the follow• 1n¡ 8>)ump11on. lf an) t"'o of1hc thrcc \Ceond hcu1en1ant~ de,;1de 10 , tose 1hr1rc(lntrol s"'11,hes, 1hr m1~\,lc "'111 bt bu!khcd. 0n Lhc othcr hanJ, an) l'>IO of thc thrcc d,:cide not 10 do:.c 1he1r ,ontrol s,,_.11,hM, thc m1~1lc "''11 not he bu!khcd. hnJ an a¡,pr'>l1kh1n1 nct""orl.

,r

t11

1:1nd a ~tatcmcnt e.¡u1\alcnt IO the )lJltn"ll'nl p

m~Ol\6 the ,onlk""CII\C V and the ne¡.111on

~

• q ""h1,h only

t 12 l·1nd a iutrmtnt e.¡u..aknt to thc srntcmcnt p••q ""hk.h onl) 1mohc-s thc ronne..:l1\C V Jnd thc nea;iuon

~

13 Find a statemcnt eqmval1m1 10 the statement p /\ q whi,h only 1moh·ts thc eonnccl1\C • and thc negatlon

~.

14 Let p denote the statement, " I study re¡u\arly.. ; q thc statement, .., gel aood &l'lldcs in 51.hool.. ; and r thc statement... , he on thc beach." Wt1tc down thc íollowm¡ ~latcmen1s in s)'mbohc íorm usma p. q, and r (a) Thc C0n\CtSC OÍ " lf I Study rcguLuly. thcn I gel good araCt and xis.in itcm v, hich bdongs 10 thc set S. y,,: v,ntc

xeS

(1)

Formul-1 ( 1) 1s rcad. ··x is an dcment of S" or •·x is m S'' or "risa membcr of s· or ··x bclongs tos· or ··r i, a point m S • l f it i~ not thc case that ,\' is J mcmbcr of S. v,c "rite

T herc are l"'º somewhat diffcrcn1 ways to describe a se1 Thc fim is s1mply 10 list thc clcmcnts. For c11.ampk, thc set "hich con• s1s1s l)Í O, l. and thc Mjuarc root of 2 can be writtcn

:o. l., 2:

(2)

Thc curly brackct§ in (2) suggest that O. l. and, ·2 :irc containcd in thc set. A s,..--cond and gcncrally ffillrc u?>Cful notatmn for a set is 1n lcrms ofits "dcfini11g" propcrl) Thus, suppo!,C that Y. e are d1'>Cussing t.'t:rlain itcms x, :md for ca.:h .\' thcrc is a propos111on Jcpendmg on x ,md dcnoted by r( \'), whlCh is \erifü1bly e1thcr true or false. Then v,c (3)

(_\ r(.,) is true'. to dcs1gnatc thc set of .ill , for "hich thc pro¡x:mtion r(:r) 1s true. For namplc. thc ¡;el l•Í c,en intl·gco can be wrillen

(4)

[\ J , -

2y, }' ,san intcgcr:

In (4). wc h,l\c abbrc\Íiltcd \lightl). (4) ¡.., rc,1d, ··T he '-Ct of al\ x ,uch thJt , 1\ t\lt1cc y, whcrc ,- 1, an 1ntcgcr.' To pul thc prccedmg -..:ntcrn:c in thc forrn indicatnl m (J), v,c "1mld ha,e to !i.l). somc"hat pomkrnu...ly, ··n,c !>CS ··fl•r ncr), (in ,1 l>l!I S)p(\) 1s true." M ··Thcrc C\I\IS an , (in SI for "h11,:h p( ,11, true.·• Sud1 phraCnh S; 1t 1\ rart of the d1\I.: ri:¡,r,:,;cnung I Jnd hoth S .ind are m,1Jc thc '"uni,cr\.11" d1)C rcpr.:)Cnting U. lt 1\ not al...,.t)\ m:..::c,ur) to dr,1..., thc o;.e1 l..i. Although \'cnn d1agr.in1s are hclpful 1l1uMration\ nfn:l,1ti~111~1h~ hl:t...,ccn :,ch. thcy .irc lln\) illu)trntion~. 111.'\CT pu1of). Rc;1\izc 1h,1t thc

5 Let 1V bethesetofall10.omen For:cE ll ' letp1(t)bcthe\latcmenl, ··.\' IS 11. blonde;.. Pi(x) lhe SU:ltement, ··x has blue i:yc1n(.ru/1;

(e) xn

(d) .\UY

t

en

cv

rn"

r;

(h) nn nn'

xnu

(J)

ruu

(kl .rno

\;

u: O;

xn

rn.c';

.2 I Set.s

43

(1) XUO -

(m)(\"J'

X; X;

(n) ((X')')' • (o) O' U; (ri) U' p

.\';

1 l Lc1 A :rnd H be sub..ct\ of the um\ersc U l'rove that the follo"m¡¡: cond1tiOn\ are nll cqlU\;1lcn1 10 1ht mdu,1on A C 8: (11) Aun (e) B'CA'; (e) BUA'

B;

(h) (d)

sn

-1

, no•

1; U;

U.

12 Draw Vcnn dm¡¡nms \\hich 1llu~tr:1te the follo"1ng ~tatements conccrmng lhc subscb X. 1', and / of lhc unt\Cl"IC U: (a) xn Y; (b) .YU Y: (e) ,\'n X' - U; (d) ,\'UX' U; (e)(XU)')' X'nJ"; en cxn Y)' x·u r: Ca) XU(Yn/) (.\'U Y)n(XUL); (h) Xn(YU/) • (Xn )')U(.\n/). 1 J Lct U be thc set of all "hole numbers. Define subsets \ . Y, 7 of U ~follow~: x- :n ; n 81, ,eu:: 1' (n " 61, ,eu:, n • 48,, 1€(/),

/

l.s 1t true th:it Z

rn

\" n J'? Explain )Our answcr.

14 In a s.,mple oí 100 Studenb, the following infonniuion .....-as ob1aincd (1) 12 1nl..c: Engl1)h. math, •nd ph)'~ICS: (11) 2.? tal.CJU~t Cngl1~h and nulh ; (111) } lal..e JUSI Engll)h and ph))ICS; (i\) 7 tal.e JUSI math and ph)-~1cs; (~) ali HXI 1nl..c at lca~t onc oí l::.ngll~h, malh, nnd ph)Sto l low man) s1uden1s 1al.e prcci,dy onc oí the thrcc subJC("t~~

15 Let s 1• s~. S1. and S, be sets con,imna oí 1in1tel) man) elemcnH. Dcri\C a formula 11na1ogous IO (40) for •CS, U Si U S1 U .S-1) (ll mt : Apply (4ll) \\llh S 16

repb,;cd by S U S ,_)

Ll.tabh)h formulas (41) und (42)

17 Let S. denote thc ~t oí ull ¡,0~111,c írudK)n) betwecn U and I w-hKh ha,e dcoommator 11 \\hcn e~pr~'d m lo111e~t term, \\ hat 1'>

C™f!IM 7 ¡

fund;1m,

•

"

1.3 Functions

LC't X und Y be I\I.O sub..c1s of the uni\erse U. We are interested in con~idcring 'ii!IS of pairs ofelemcnts 1n which the first mcmber of the p:,ir come, frorn X and thc ~ond from Y. A pair of clements 1s c;11led an ordrred pair 1f it possesses the important propeny that the íirst and S(.'Cond mcmbcrs are d1st111gu1~able by 1he ordcr in which they appcar.

Delinition 3.1

O,J.,,, J pair lf .t and y are elements of U, then thc notation for thc orderrd puir with first ekment x and sccond ekmcnt )' is

(x.y).

(1)

lf (x 1• )' 1) and (.t 2 , n) are ordered pairs then (.t ,. )' 1) means that

•

(x:, y,)

(2)

The notion of a set of ordercd pairs is fundamental throughout mathcmatics.

Definition 3.2

Gtrtrsian prodi,ct, rtl11tion The Cu,1esia11 produc1 of the seis X and Y is the totality of ordered pairs (x. )'), where ;e E X and y e Y. That is, the C;1r1esian produc1, dcnotcd by X X Y. is thc 5el

xx

Y - {: 1:

-

(x.y).

te

X. >' e >'l.

Any subset R of X X Y is called a relmion on X 10 Y. Thc domain of a rdat1on R is the totahty of íirst elcmcnts oí pairs in R . Thc ra11ge or r:odoma111 of R i,; thc lotality of second ckments of pairs in R. Thc domain of R is deno1cd by dmn R and the range of R is dcnoted b} rng R.

Exampfe 3.1

Find thc rangc ofthc rclation Ron Xto )' for which dmn R • X {1.2,3,-',5, -5) and R .. :{t,y) t EX,)••x 2 :. The pairs in R are (1, 1), (2, -'), (3, ()), (4, 16), (5, 25), (-5, 25). Hencc

rngR .. {1,4,9, 16.25:.

:i /

func11o,is

45

O~n·c that wc do not count 25 twi parallel to J'. As i~ cu~tOnury in geometry, ..,.e eonsidcr C\ery ~1raight linc to be parJllel 10 it>elf and ther.:fon: R is rtne, i\e, lt is also dcar th,11 R 1s symmctrie .im.l tra.ns,ti\c a.mi hence ,s an equi\akncc rdation.

r 3 / Func1,ons

47

Example 3 .5

Lct X denote the sc:1 of ali men. and define R 10 be the totalily of pairs (.\,)') for which x and )' ha,c at least onc parcnl in common. Is R an cqu1\'alcnce rct,11,on w1th the agrecmcnt that cach pcrson is considered 10 ha,e at kasl onc parcnt m common w1th himself'I Thc rclation R is clcarly rene:.:i,c .:md symmctric, but 11 is certainly not 1ransiti,c. For, y could be the half-brothcr of .\'. and =could be thc half-brothcr of }', and ye! x and =nccd not ha,·c any p;ucnt 1n comm(ln.

Supposc X and Y are two sets. cach with a finitc number of elements, say, i,(X) • 11. 1'()') • 111,

whcre X - (x, ..... x~) and Y - {>• 1. . . . . )'..,). Lct R be any rclation on X to Y. Thete is a ,cry simple tabular or mu1rix fom1 in which wc can e-.:hib11 the rclation R: construcl a table wilh column hcadings ., 1• • • • .-·~ and row hcad ings y 1••• • ,y.., and at thc intcrsection of row I and columnj of thc matri, pul I if (.i,. }',) E R and pul O if tx,.y,) i: R.

Example 3.6

Lct X -

p, 2, 3, 4) and lct R - {(i,j) i E X /\ j E X /\ (i - J) is c,cn).

Wceasily ched that R consists ofthc following set ofpairs: {(1, 3). (3, 1), (2, 4), (4, 2). {l. 1). (2, 2). (3. 3), (4, 4):. Thcn thc matri;i¡ for the rdJtion R is 1

3 4

¡l~ i ~ !ll· 4

Example 3.7

O 1 0

A pair oí honcst dice is thrown so that 36 possiblc ordcrcd pairs oí numbcrs can .ippcar. Define a rcl.1tion Ron 1hc set ( l ..... tocon\i~t oí ,111 p.:iin (i.j) for which (1 + j) is 7 or 11 . Find thc matri\ for thc relation R Thc pai~ in R are

6:

(1.6); (6.1); (2.5); (5,!); (],4); (4.J); (6,5); (5,6)

Thc ma1rii for R i~

1 23456

o o 3 0

Delinitlon 3.4

oo o 1

o

o

O 1

0

1 0

0 1 0

O

1

:e~:

lncidence r,r4trix Lct X - (:c 1• • • • and Y - [y 1, .. y,.!. Lcl R be a rc!Jtion on X to Y. Thcn lhc mcidence 11w1ri:c fur R. dcnoted b)· A(R). 1s thc m X II arr.1y m which the entry m row, and column J is I or O accord ing a~ (x,. y,) e R or (.,,.,-,)e R. TCl>pt"Cli\dy.

Thus. 10 form thc incidena:: matri,; for thc rdation R. wntc .Y1, •••••Y~ as column hcadings and )'i, . . . • )',., as row hcadings. Thcn thc cntry oppositc y, and below x, i s I if (x,. y,) e R and is O ií (x,. y,) E R Obscnc that thc incidcncc matri, A(R) coniams a cornplc1c desc:ription of thc rclation R and ali propcrtics of R appcJr as propcrt ies of A(R). In thc prcceding c,amplcs and dcfinition. thc matrices conSi!,tcd only of thc numbcrs O and 1. Onc can CJl>lly con O an O, it is clear real numbers.

The question is \\hcther thc range consists of ali positi\e rcJI numbcrs. In other "-Ords, can .:. assume an) po~iti\C: \a]uc )' for an appropriatc :e > O? Of coursc, the ::inswcr is C:lS} because gi\CO a po~iti\e number y. we simply choose x to be

1

.

' Example 3.11

Lct X be thc set of ali non-negali\e real nurnbers :md lel Y be the imcnal of al\ real numbcrs .r sati~f)ing O 5 y< l. Consider thc function f. X-· } gi\ en by f(x)

."C~

1

••"CE

X.

Show th;II each number in )' oc-curs in pre-cisely one pair of / Sincc x + 1 cxcttds x. wc know that -·~ - is less than l. and hence x+I /(,;) e Y for un) x e X. Rccall th,ll thcclemcnts in/arcjust p;1irs ofthe form

(x. 1 ~ x) ·

(x, /(:e)) -

Supposc that

IWO

. pa1rs

(

X1,

,,

1+ SJmc Sttond members. that is.

)(

.,;,

•

X2,

,, ) in/h:'I\C the

1+

X2

.''• - J(x 1) ~ J(x2) • y 2 means that x 1 and x 1 havc dilfercnt social securily numbcrs and hcoce x 1 and x 2 must be d iffcrcnt pcoplc. Moreo,cr.r 1 is 01110, sincc to each social security numbcr thcrc corresponds a workcr. 1 ghcn Frcqucntly it is rcquired to find a ••formula" for g • a formula for f. In Examplc 3. 11 wc saw 1hal if X is thc set of ali nonnega1i,e real numb:rs and Y is the in1crvJI of rcJI numbcrs O 5 y< l. thcn/: X-. Ydcfincd by the formula

r

r

/(x) -

x-t

1

r'.

is 1 1 and onto Y. Wc can asl. for a similar formula for i.c., g1,cn J' E Y how do wc find r'ú-)'1 By dcfin1t1on r'(y) 1s thc clcmcnt x of X for which /(x) - y. llcncc, in thc cquation ' - /(x)

Ch.tp the pair in¡-· 1 ....-hose fim mc:mbcr isf(.t). Thc: rcar.kr ~hould pro\ide a similar argumcnt for (15)

Example 3.12

Lct X be thc ~t oí ull iw~ks triangks in thc planc wllh \CrlK.-es (0, O), (1, O), und (!, 1). us 1 \Mies O\Cr all J'I05iti\C real numbc:rs

1.3 /

Fvnct,Ofl$

59

Lctj: X-· P be thc function which associates v.i1h cach triangle A E X thc arca of A, 1ha1 is, J(6) - arca of 6

- ¡,. (Thc arca of a trianglc is ½1hc base times thc al1i1udc.) Thc set P is the set of positi\C real numbers. Obscne first that f is 1- 1 and onto P. For. lc1 A and A' be two trianglcs in X(A' docs not mean thc complemcnt of A hcrc 1 ) and assumc that f(A) - /(A'). Then ;, - ½t' :rnd hcncc 1 - 1'. Obscnc from the figure that A and A' 1hcn ha,e the samc \erticcs, so 1ha1 6 • A'. 11 is also ob\ious tha1/is 0010 P: for, if pis any positi\e numbcr, thcn thc 1rianglc A E X whosc third \Cr!Cl[ is(½, 2p) has arca p. Finn\ly,] 1 is the function v.hose ,atuc for any positi\C number I is the isoscclcs trianglc in X whosc ,crticcs are (0, 0), (1, 0). and (½. 2t), tha t is,

¡- 1 -

((1.A)i1>0AAEXA(Ah:mcrtia-s(0.0),(I.0).and{!-,2t));.

Letj:

x __,

Yand g: Y__, Z be 1wo functions.

lf /has domain X and rangc Y and g has domain Yand rangc Z 11 1s C\-idcnt that wc may form a compo$itc h: X__, Z which as.so• ciatcs with caeh clement x e X thc dcmcnt g(/{x)) e z. In othcr words, 1,(,) - g(/(x)) Thus h mars \ into thc rcsult of t\·aluating g at J(.,). i.c., first pcrform/and 1hcn pcrform g on this an~wcr. For cxamplc, suppo:,,e Jis gi\cn by the formula /{.,)., x2

and gis gi\cn by the formula g(x) • 1 ;. .~

(takc thc domain and rang,;: of both functions to be thc set of non• ncgati\C rc;1l numbcrs). Thcn the \aluc of thc compositc h of /

CN,:,rw 1 /

fundM!"ttlfl1.tls

50

;md gis found by comput1ng thc ,a1uc of g at /(x): h(x)

~ g(J(x)) ,. g(x:)

¡

x'

+ (.\'l ),

-~x'

In words: thc \í1\uc of h at x is

or cqui,akntly. /, consists of the ordcred pairs

or finally, in sel nolation

Definition 3.7 (16)

Composition o//1mctio,u Lc1 /: X - Y and g: Y - Z be two funclions. Thcn thc rclation /, C X X Z. h •

:(_\', :) 1 x E X /\ : • g(f(.,)):.

is called thc composit~ or prodiu:t of /and g. Wc writc (17)

" - gJ.

Thc nolation (17) does not me.in that thc funct1on \Jlues of g and / are to be multiplicd togethcr to obt.iin thc funl·tion ,.11uc of h, c~cn whcn such an opcration is possibk

Ex•mple 3.13

Lct g be the function which as~iJ.tcs with each posilÍ\C number t thc cirdc in thc planc with «nlcr at thc origin (0, O) and J.rca 21 Lct/~ thc fulll.:t1on in E·um¡,lc J l 2. Thcn thccomposition /, gJ of/and g 1) thc func1ion which associ.ttcs w1th cach trianglt' t::. with ,crtuX\ (O.O), (1, O), and (!, 1), the circk with ccnlt"r al thc origin whOl.e area i~ t. O~nc that h(t::.) i~ thc cirdc of arc.1 t ami thcrc• forc its radius mu¡,t

be\:·

, .3 ¡ Funcr,ons

11

1

,;.,,

V\ ¡(O OJ

(1.0)

1

Theo,em 3.3

Thr composi1io11 of fu11c1ions is 011 auociaticr ap,:ratian. Tha, is, if /: X - Y. g· Y - Z. und k: Z _...,. IV urr funoíonf. then

(18)

k(gf) • (kg)f.

Proof Let x E X. The '>aluc of the lef1 side of (18) al ."I'. is (k(g.f)')(:c) 1,1,hich. from thc definit ion of function compo~1tion. is .l((gf)(x)). F urthcrmore. g/("I'.) is b) defimtion g(/(\·)). Thus k((.ef)(x)) • '(g(/(x)).

Onc can check similarly that ((kg)/K /(x} .. 1

¾x.

g(\'.)

x1,

k(x) -

!.·

Thcn

g/(.,)

g(/(,))

•e~J x'

• c1+xp·

Thus (k(gfl'J/..,) • k(gf(x))

- k((I ~\p) 1

- ((i-~',)') kg(x) • k(g(x)) k(x2 ) 1

xi. Hcncc ((kg)/)(x)

kg(J(x))

••(, ·~ ,) 1

-(, ~,)' ,.. (I_± x)~. x' We sce then that

('(gJ)KfXx).

Show that although íunction composition is associati\e Thc:orem 3.3). it is not commutatice. This mc:ans that 1í/: X and g: X-.. X. it is nol necess.irily the case that fg gJ. c:xample, lcl X be thc: totnhly of pairs oí real numbers (.~1, Define

and Then

Js«-' i. .,in - f{x«-,a. _,,m -f(( , 1. x1))

-(, 1+ :?•.,. + J)

(~e •X For x1 ).

I 3

f

Fvnc1,,:xu

13

g/((.-.:., .-.:1)) - g(/((.t1 ..t 2)))

+ 2, X2 + J)) + ~. A¡ + 2),

- R((X 1 •

(X2

A simple but import,mt function dcfincd on a set X is thc identil)', l x. The ident1ty function is dcfincd as follows:

1,- .,.

:(x.x) 1 x

Ex:.

T hus l t(X) • x, x E X.

Theo,em3.4

l~1f

X

• Yb,a 1- 1 01110/u11ct1on. Thtn

ond

This result is \cry casy to pro\c and ...,¡l] be lef1 asan exercise (sce (14) and (15)).

1

As a mnemonic dcúcc, wc can think of sets as "H:rti«s'' and functions a§ "arrow.1," connecting thcsc ,crtices. Therc are sorne pcrtincnt propcrties that thesc ",ertices" and "arrow~" mu~t h,1,c. (i) tí /is an arrow from ,ene, X to \Crtc, Y and gis an arrow from \Crtex Y 10 ,crtc, Z, then a "composite" arrow. gf, from X to Z mus1 exis1.

•' In other words, function composition is defined. (ii) lf X, Y. Z. IV are ,ertices./is an nrrow from X to r. g an arrow frorn Y to Z. and k on arrow from Z to IV, then k(tef) - (kg)f.

A{gf)

This is the Msociati, e law for function composition. (iii) For ,ertices X :and Y there e,dst idenllly arrov.) /.ir: and /¡such that for any arrow Jfrom X to >',

• An Elementar)' Thcor)' o! 11,eCatqioryoíSds. Proettdinp oí Thc Nanonal AcadcmrofScim.::esoí1he U.S.A .• V, l2 (196,IJ, l'fl. l~UII

A collcction of ,enices X. Y, Z ....• togcthcr v,,ith a collcction of arrows f. X--. Y is calkd a cattgory of sets if the prcceding lhrcc uioms are s.itisficd. Ahhough wc do not do so hcre, thc usual a ,ioms about si:.ts can be replaccd by uioms conccrning catcgories. Thi) was done by F. W. Law\ere in an articlc pubfühcJ in 1964 • Some of the topics we now int roduce should be of particular intcre)l to thOSé rc,iders v.ho require that abi,tl'JCt mathcmatics h.i,e practiC'.11 and imponant applicauons. Ccrtain closses oí funct ions that wc now di'iCUSS are of great ut11ity in analyzing dice gamcs and bridge hands, in doing cryptanal)"sis. and in deal mg v.Hh many other manifest.111ons of human gulhb1hty. Thcsc func• tions wi\l be introduced now and considcrcd in detail in thc ncu scction. Assumc th,tt )' is .1 set :md kt V, be thc set of integc~

(1, 2.J.

N,

~

Then any function f. /1., Y is callcd an r•samp!t• of Y. For C.\lllllpk, if }' :-.,. -5, -7: 11nd r - 4, thcn thc func:tion f defincd by /( 1) - -7, /(2) -5, /(]) - -5, /(4) -)

, .3 /

Fvncr,oni

es is a 4-samplc of Y Thus an r-sampk can be 1hough1 oías a labclhng of :wmc of thc ekmcnn M Y with thc intcgcrs l . . . r. Tha1 is. \.\C labe\ thc ekmcnt }' E )' \.\Íth thc intcgcr J.. E ,'1;, if /(k) • y. Smcc no add1tional assumpt1ons are rnadc on / it ma) wel\ be thc case 1ha1 a parucular element y E Y 1~ labelled by more !han onc intcgcr (/is not nec-cssarl)' 1 1) and \Omc elements of J are not l.1belled at all (/ is not ne«sr.:mly onto). lf /isa 1 1 function, thcn/ i no intcger i

1 3 f Funcz,on,

&7

(e) In how m.1ny wayscan a da\\ of!0~tudcnts be di,u.kd in10 1hrcc c.liscmsion sections consisting of 7, 7. and 6 students, n:specti,ely., Any such arrangcmcnt can be ident1fied .,.,ith r-s.m1plc /: N2u - N, in which 1 •

t.!.

and

(d ) In how many WJ)'S can 6 bool.s be arrangcd on a shclP Any such arrangcmcnb can be idcntifü.-SOCl.all'S \\llh each clemcnL oí X the ordered pa1r (x, x), 1 c .• f(x) • (t, x), and K l.) 1hc funcuon "ho!.t! domam 1) \ X ,\- 11nd "hose value at any ordcred pa1r (x, ,) 1s 1 or O a,i.:ording u .t - y or x ¡o1 .i-. thcn the rom~11c fun.:uon sausties ,cf(x) 1 íorallx~ \'. :0, 1,2: andK :cO. l ),(l,2),(0,0):, 1heng1safunct1on 8 lf.\' onXto.Y 9 11,c numbcr oí 2-pcrmullllions oí 11 111,0 clcmcnt JC:I 1s 2

10 Thc numbcr of l-~mp~ of a t10oO dcment .-.et 1s 8

Exercises

1 ldenhí>· calh of the follo\\-m¡ 1oet~ a~ a fun.:·tion ora relahon 11,h1,.;h 1s not II funcuon, g•~ m¡ rca.l(lru,; (¡¡) !(0,1),(1,2):; (h) '.(O. 1), (O. k) :(O. I). (1,2), (2. 1), 0,2):;

-1):.

{d) :(.t,1)

.t~;

_1· -

1 A -1

/\ 1 .,:. lb)\ :n,1.-1,2, -2,J.-J:, > - :0,1,2,J:.

14

:i:.

R

:e,. 1)

~ i=

\

/ \ .1 E

) /\ _1 •

-.,

;

1:J / Funct,ons

71

(e) X ,s a íacmly consistmg of two parrnts. 1wo malc ch11dren, R •

:(x,y) !x e X/\>' E,\ A

'""º íem.1lc thildrcn, and

ü· 1s thr brothcr ofx)'.

1 5 Lrt ,\ be

a thre-e elcme:nt set Statc 'whi,,;h oí thc íol~m¡ 3 X 3 matrices are 1ncldcncc m:itncrs for íuncuons/: .\ , r f'urthcr, ""hich are mrnkncc matnccs for 1-1 func11ons~ For íunc11oru or110 \ "

[i ~]' (e)[: ~](a)

!] ' (al[! ¡] (e) [~

(b) [~

(d)

({)

rn

o o'] '

o

il ·

[: "] 1

•

1

• 15 Ut.\'bcafimtcset,X • :x1, .. ,..-.!. (a) Show that an " X II m:itrni:, A(fl, 1s an 111("1dencc m:itrl'( for a íunc-t10n /: X ..... X 1f and only ,r eal·h column of A(fl has prtc1'>Cly onc I and 11 - 1 lc:ros (b) Show 1ha1 A(fl is thr: 1nudenct matri( for a 1 1 fu!khon / 1f and only 1f rach row and rach colun1n of A(/) contams prtt1?>tly onc I a nd " - 1 icros

17 lct R be a relation on a fimte set \' 10

X. Wha1 propcrcy must the mc!dcncc matm: A(R) ha11c 1n ordtr that R be rcfle~1\"C., (Scc Dcfimtmn 3.3 (1).) AnsYl-'Cr the saffil' queshon for thc S)ffillllClri,.· propert)'.

18 A physic1an hnds th:u among thrcc pa11cnu, PI, P!, and p.,, any l. of thc paticnts cxh1b1t at lcai.t k of 1he i,ymptoms of d1sea~ d1, di, d .., andd1forl. • l.2,or3 (a) Define a rela11on R as follows: (p,, d,) e R 1f and only 1f p, c:xh1b1ts the symptonu of thc discasc d, In thc 1nc1dcncc num,. l(R), hov, does 1hc fact 1h;1.1 an)' l. o( thc patitnts exh1b1t al lcast l. oí 1hc S)mp1oim mamfest use1r~ (b) Wh,111~ 1hc 1ntcrprc1a11on ofa 'iCI ofthrcc 1•~. no tv,ooí .... h1amc row or thc S..llllC colu11m l 19 In a um~er~1l)' 1i.. mg ¡roup, four swdcnts carne dov,n wnh 1hc meas.~. thrcc othcr siudcnts ~ere qucsooncd a\ 10 v,hctber thc) h:id h.id contacbv,uh lhc fourv,ho,,.,c:rclll, thcn füc morc~tUhtch lhc entry m column j and row f 1s !he total number of expo!>ures of Muden! q, 11,h1ch C"'J.n be traccd 10 student m., 1 - 1, . ,4, j - 1, .. ,5. 20 With rcíerencc: lO Examplc 3.9, flnd anothcr !M:lcction oí four nm1ures m \l>hich each spcc-ies 11,JII lind at le:ist one acr-eptable m1tture. t 21

Pro\e Thoorem 3.4 ( H int: Refer to the «¡W1l1\lt!S (14) and (15).)

Let X ~ thc >C!I o( real numben.. Fmd thc compos1tc g/ of lhc íollo11,·1ng p:11rs oí íunc11on~. ['(press )OOT answer 1n terms oí a formula forRJ(x), xEX (a)/(.·() :e, g(:c) -x; (h) /(.t ) 1, g(\') • 2; (c)/M :c1 ,K(:c) x I; 22

(d) /(.,;) • (e) /(d

~

(í) j(:C) (¡) /(.f) (h) /(.t) (1) /(:e) (J) /(.l) • (l) /(x)

1 ; xl' g(:c) • /(:,);

x - 1, g(.,;) X 1 .\' 1 •

(x :ci

~

1, g{.l) •

ll'M l )(x

+ 2.r

x

+

1;

{(X);

-/C,

t

.t1 , k(l) •

1). g(x) .,'(r 1, g(.r) 1;

+ I);

.\"J;

x', K(X) • x3.

:1, 2, J; ond 1dcnufy cach ofthe íollo-.1;mg func11ons asan r•!,amplc, r•pernlulllhon. r-~lcc11on. or r 1 + 2a.

Again, if wc takc n - 3,

(1

+ a)3

..

1+

> 1+

.la+ Ja2 + a3 3a,

and wc sce that p(J) is true. We would like 10 pro,e that (5) ho\ds for e,cry natural number n ucceding l. Out the principie of mathcmat• ical induction is not quite set up that way. 11 s«ms to require that

p(I) be true No doubt lhe rcadcr has gucsscd by no11i 11ihat thc appropria1c modification of thc principie should be. let n 0 be o natural munber. lct q(n) f>e u proposilio11 ·,,:hü:h ,r eilher m1c o, false fo, euch nuturul 11111nber n, 11 ~ 11 0 • // (a) q(no) ,s mie ond1f (b) 1he imphcatio11 q(k) -

q(k

+ \)

is ,,11efor coch 110111,0/ numbe, k. k ~ n 0 , 1he11 q(n) is 1,uefor ali natural numbers 11, 11

~ 11(1.

Thc ract 1hat this modificd statcmcnt oí the induclion principie holds is a d1rect con~qucncc of1he original statement. For, simply define p(n) to be thc proposilion q(110+11- 1),

i.e., p(I) • q(,,o), p(2) - q(no + 1), p(J) ~ q(no + 2), .... Thcn to prO\C by mathematical induction 1ha1 p(") is true for all natural numbers n is pre 1 + ka.

This inequality is prescncd if wc multiply both sides by any positi\e number, in particular by the posill\C number 1 + u;

+ aX I ( 1 + a)"·

(1

+ a)t > ( 1 + uXI + l.o), 1 > 1 + ka + a+ ka 1 • 1 + (k + l)a + ku~.

Now, ka~ is a l)OSIIÍ\C number and hcncc (1

+ o)t

1

>

1

+ (k +

l)o,

and p(k + 1) is true. \\ e h1nc pro\ed: if p(,k.) is lrue, 1hcn p(k + 1) i~ true for any natural numbl:r k, k ~ 2. Hcncc thc propos1tion p(n) is true for :iny muural num~r n. 11 ~ :?.

14 /

lnduc11on , nd Combtn,ux ,

19

Thc principlc of mathcmatical induction is particularly uscful in analyzing combinatoria\ problcms, but bcforc going into this inleres1ing subjcct, wc will introduce sorne classieal languagc and notat1on. Definition 4.1

Farrori'lll, binomial co,fficitnt U:t n be a nonnega1hc intcgcr. T henfimorial n or n-factoria/, wriucn n!, is the producl oíthe first n positi\C intcgcrs, ií n ~ l. That is, n!

1 · 2 · J · 4 · · (n - 1) · n.

Thc number O.factorial is ddlncd to be l. i.c.,

O! - l. lf r and n are integcrs, and O $ r :S n, then the number

"'

(n ...;,.· ;:-)!r! is calkd thc binomial t:04'.fficient n oc:er r and is writtcn

The main task in this scction is to pro\·e Thcorem 4.2, which will show us how to count thc numbcr oí samplcs, pcrmutations, com• binahons, sclcctions. and su~ts of a gi\cn set. Wc bricny re• eapitulate the defin itions oí thcse items ghen at the cnd of Scc· tion U, and add sorne rcmarks that should hclp to clarify thcir meaning U:t X be an n•set. An r-sampll' of X is a funct ion /: N, • X, N, { 1, ... , r}. T hus an r-sample can be thought oí as a labelling of thc it..-ms in X with thc integcrs 1, ... , r (i.c.• if /(k) - x, thcn x is labclled with k), in which any itcm ma} ha\C more than one labcl but it is not ncccssarily the case that C\cry x E X ha\·e a labcl. In other words, an ,-samplc / is not ncccssarily 1- 1 nor is it ncc• cssarily onto X. An r-~nnutation oí Xisju~t an ,-samplc.f: N, - X. that is 1- 1. Thus wc insisl for an r-pcrmut.ltion 1ha1 no itcm be labcllcd more than once Supposc thc ckmcnts of X are dcsignnted in sorne ordcr, say X - {x 1 , . . . • x.,:, and kt /: N, - X be an ,-sample: /(1)

"

Xt¡,

/(2) -

Xl 1 , . . .

,/(r) -

Xl,•

e,,.,,,« 1

Fund,1m ,,,,.,s

10

then/iscalled an r-selwion of X lf/is also 1- 1, w th.i1

then/is eulkd an r-f'ombinmion or r-fübSel of X Therc is a very hc:lpful and informa1i,e way of designa1ing ,-sampks. lf X {.t 1, .••• x,.J and /: N, - X is an ,-sampk we can wnte out the rnnge of/lU a stquence oflcngth, (/(1). /(2). /(3), .. , /(,))

Bu1, after ali, /(1) is sorne ekment of X. s.ay /(1) is the element numberc..-d k 1 of X:/(2) is the dernent number~-d k 7 of X: etc. The preceding sequcnce can 1hus be wrinen {.tt1,

.t1,, Xt1, • •• , .tk,)

wherc A i. • . • • k, are jusi intcgers chosen from 1, ... , n, al\owing repctllions. E3ch ,-sample / de1ermines exactly one such sequencc and each such sequence dctcrmin~ cxaclly one r-sample. lf / 1) an ,-permutation then the intt'gt'rs k 1, . , •• k, must be dist1nct i.c., / is 1- 1. lf / is an ,-sclcction then k 1 5 · · · .S k,. lf / is an ,-combin:'llion thcn k 1 < · · · < k,. We can further simplify and organize the preceding remarl,s After ali, giH·n A 1 , • •• , k, from the set {l. . .. , n), thc scquence (xt,, -'t,, X1,) is uniqucly specified and hence the r-sample of X is uniquely specified. Thus 10 counl the number of r-samples of X we nced only counl the number of sequences

x.,.... ,

(6)

wht'rc k, E (l . . . ,n;, i • 1, .. . ,,. lf, for e,ample, wc .... ant to count the number of ,-selec1ions oí an n-sct thcn ...,e can simpl)' count 1he numbcr of sequences of mtegers (6) for .... tlich

\\e summarizc tht.'SC rcmarls in thc follO\\-ing 1hcorcm Theorem 4 1

l et X be 1he 11-set

Correrpo11d1ng to u11y r-samplt f: N, o/1111egerJ (7)

• X 1hert if a un1q11e srquenre

1 4 ¡ fndurt,on •fld Combtn•r,-,,,c:r

11

1 $ k, $ n, definrd bJ j(i).,. Xt,,i"" 1,,

(8)

C011~·ersely,for any such scquencr (7), 1/l(•re it predsefJ· Qlll' r-r;ompfe f. defi11ed by (8). Mor,·ot:er. (a) / is 011 r-perm11totio11 if ond onfy if k 1 , • , , k, ore dis1inc1; (b)fisonr-cvmf>i,uuionifa11donfyifk 1 < k 2 < · < k,: (c) fisa11r-sefec1io11ifm1do11lyifk1 5 k2 5 · · 5 k ,.

\\ e can now go on to thc m::iin rcsult of this section v,hich tells us how to count these functions. In thc proof of thc follov,ing theorem and su~uently. v,e use a ,ery ekmentary combinatoria\ principie for countmg the total numbcr of outcom~ of tv,o independent e,ents. l f the fi~l e,ent can happen in m ways and thc sccond in II Wll)'S, then thc two CH'nts can h:.1.ppcn in 11111 W:l)S- This statcmcnt is callcd the m11/tiplicotio11 pri11ciple. For c,ample, from fi\e differently colored skirts ::ind SCH:n dilferently colorcd blouscs, a total of 5 · 1 - 35 diffcrem outfits can be sdccted. Theorem 4 2

Let X be on n-set. Then: (a) the n11mber of r-samples o/ X is n': (b) if 1 5 r 5 n, then tfte 11w,1m'r o/ r-permu101io11s n!

o/ X

ir

(n - r)!:

(c)

if 1 5 of

r

5

n. rhe11 the munber of r-combi11atio11r or r-tubstlf

x;,(;):

(d) the munber of r-selecrionr of X

is("+~ -

1 ) :

(e) the n11mber o/ subsets o/ X is 2".

Proof

(a) Lct X • {xi. . . , x,.} l!lnd kt s(r, n) denote the number of ,-samples of X. Actording to Theorem 4.1, s(r, n) is the number of scquences (9)

thJt cJn be formed in which 1 5 k, 5 n, i • 1, . . , ,. Now ií r - l. then (9) simply becomes a scquencc oí length 1,

andk 1 may bechosen in II v,ays from {l.

. n1 Thus s( l,11) • n

In othcr word'>, s(r, 1t) - 1t' 1n case, 1 and \l>C ha,e thc begmning ofan induct1on argumcnt on ,. SuppOSe thcn that J(r 1, n) n•- 1• With any sequence (10)

wc can form a scqucncc of leng1h , by adjoining any one of the n intcgcrs 1, ... , n to (10). Thus eaeh of 1he s(, - 1, n) - n•- 1 sequenccs (10) produces n sequcnces of the form (9} and it follows that s(r,n) -

s(, - l,n)n

- n'. \Ve ha\C pro\ed that therc aren sequcnccs (9} of length \, and on the assumption th.it there are n•- • sequences (10} of length , - 1 it followi that there aren' sequenees (9} of length ,. This prO\C5 (a). (b) Lct p(,r, n) denote thc number of r•pcrmutations of an n-set. lf, -- \ , then p(I, n) is JUSI the number of sequences (9} of length 1 and, as abo\e, this is obviously 11, that 1s, p(l,n)

11

_ ,,.(rr (n -

1)1

1)'

n' (11 -

Thus, assume that 2

5

r

5

1)).

n and

(11)

Aecording to Thcorem 4.l(a) 11>e v.ant to count the number of sequcnces (9) in which k 1 , • • , k, are dislincl. We ha,·e ,erified that for r - 1 there are p(l,,r) - 11 of them, and v.e are usummg n' that there are p{r - 1. n) ~ ) ) ! of them of Jength , - l. Now, gi,en a sequcncc (10) in which k 1, . , k, 1 are distinct, \\e can forma sequencc for 11>h1ch k,. ... , k, __ 1 , k, are d1s1inu by simply choo-.ing k, to be any one of thc n - (, - 1) ... 11 - , + 1 intcger; in {1.2, .... n} othcr than k. 1 • ,k, ..1, and adjoining it to (10). llcnce v.ith each of 1he p(,r - 1, 11) dilfercnt sequcnces of the form (10) of di~tinct intcgcrs, wc can con5tru,t n r+ 1 d,mm::nt ~ue™.:c~ of the form (9) of di!.tinct intcger.,. r rom the

T4

J

lndvcr: m and Combina1oncs

13

induction assumption (11) we ha\e p(r,n) - (n - r

+ l)p(, -

- (n - , + ■

l,n)

l)(n - (:1 _ I))!

,,

(n -

' + I) 1 · 2 · 3· · · (ñ - ,xn - , + 1)

(11 -

r)! ·

Wc ha\c pro\cd that if the formula in the sta1ement of 1he 1hco rem worl s for, - l . thcn 11 works for ,. llowc\er. 1his docs nol fully mcc1 thc rcquircntcnts. After ali. we ha\C only the n propositions

,,

,

p(,, n) - (11 - r)!

1, 2, 3, .. , 11,

but thc induction principie requircs that wc h:nc a set of proposit1ons, one eorresponding to each posilivc intcger. This difficulty. howe\cr. is f'asily r~ohf'd. for, we can simply define a proposition for n + l. n + 2, ... to be any true statcment , c.g.. 1 l. Wc will thcn ha\c a hst of propositions. onc for cach positi\C integer n, say q(I), q(2), .... q{n), q(n + 1). q(n + 2),. . Thc argument in the proof has the following form : q( I) is true; 1) .... q(r) is true, q(k) i~ true,

q(r -

, - 2,, .• , n; k n + 1, 11 + 2, •..

lt is ob\ious that for this set of propos11ions, q(r -

1)--- q(r)

is true for ali , ~ 2, Thu-s wc are able to apply 1hc induction principie. This lcchnical dc\icc of Simply adjoining a set of true propositions to a fimtc set of propos1tions allow:. us to use the principie of induction to cslablish a finitc sel of statemcnts. This modification of thc principie of 1nduction is calledfi11ire i11d11ction. (e) By Thcorcm 4. l(b) our task is to count thc numbcr of scquences (9) for which k 1 < · · · < k,. But this is easy now that we ha\C pro\00 part (b) of thc prescnt thcorem. For, gi\cn any scqucncc (9) in which the in1egers are incrcasing. wc can form ,! sequeoees in which thi! integcrs are di_..tinct. Thc reason for this il, that {k 1 , • . • k,) is an r-ckmcnt set, and according to Theorem 4.2(b) thcre are r!

Cfl;tprrr 1 I F/Jlld1rmn1a

84

(lakc II r) diffcrent scquenc~ of lcngth r of di~tincl integers thal can h.! formed from it. Also, CS in this w.iy. Thus. if c(r, n) denotes the numbcr of r-co mbinations of X. v.e conclude that c(r, n)r' is thc numbcr of r•pcrmut.itions. of X ;

"'

c(r, n)r!

c(r, n)

(n - r)'

"'

(11 - r)!r?

- Gl (d) A trick} dcúcc is required to prmc (h,h eath ofthe followmg 1den111ies for pos1t1\C: 1n1egco usm1 thc prmciplc of m.uhematKal mducuon·

1 (a)1'2+2 3+3'4+ (b) 1 2 (1ble b) 8.)

/(11

+ ]) -

9 1\ oommmtt of p pcople 1s to be chos.rn from ll g1rb and b bo}'s In how many v.a)~ th,~ be done" B) counung the numbcr of com• m111~ m \\fa,.-h thcre a re ¡)re...·i:.cl); ¡ir is and p - ; boys., O, • p, show that

,.:m

lt IS cnurcb ¡xM1ble that p 1) grcaler lhan g or b. Undcr the-.e cirrnm• stanccs, ho"' should

(!)

or (:), ctl· .. be delined m order 10 mal.e thc

precedmg equahl> corree\?

• 10

In a da\:> oí J bo)i. hnd 3 g1rl~. cach girl h:i~ httn mtroduccd lo ¡wec,sel)" 2 boys ond cach bo) has been mtrodoccd to precise\} 2 111rb. I n how m;in)' Wl)S can !he) pa1r o ff mio d:ince partners v.ho ha"e been pre• \ I0U~I) mlroduced 1

11

nie

1here

ore(~))

Om1nom.1 C,g;ircue Compan) h1h de,·1:.ed 1he íollov.ms ~uahl)' con1rol procedure. From a baKh of20 uaarette5. 6 ofv.hi.;h are deíccti,c, a sample of, ug.1ret1C\ 1\ selc,,;ted. Tñe \\hole ba1~h w1ll bc d1:;,.:arded 1f 2 or mo~ c1garettn m thc sample are defect1,c. In hQv. m:ln) "'ª)> can a s.ample be ~\el.:100 at random >0 lhat 1hc enltrc batch I) r(IC(:ted " (llmt · ~ mplo alto~thtr Coum 1he numbcr o í j-elemenl subseu

oom..1mmg O ofthc dc:fa:11,-c c1garc:IIC$ 1md lhc numhc:r conrnmmg I oí 1he defec:li\C cagarettb. Sub1ra,1.)

12

In how man) "'ª)) t·an 8 rcop1e t,c ..eatctl 111 a roum.1 tabk 1( onl)' 1he1r pc,-,.111on~ relat1,·c to onc llllQther nial ter?

• 13 In how man} v..i)"J can 4 g1rls and 4 b,;,)) be i,cated ata round rnble 1r bo)~ and girb mu~t ahernalc: and o nl) their ~111om rel111,e tQ one another matter ? 1 4 ~how that af "• ,, and , are nonncgame mtetc~. , sa11~í)mg

< r :5 n,

G) • C)· then r -4-- s

■

n

15 In ho"' m.11'1) V.il)l can an 11-man foo1NII ttam be cho..cn from 15 rla>er.' 1 6 In ho\\ lll311)' "'ª>) can n ,omm1tte.: of S pcople he lh~n from a aroup or 8 pcople 1f (11) 1hen: are ~ pco¡1lc "' ho di.hJ..c one noother and rcfu,;c to "Cf\C v,,,uh onc anotlk:r;

90

Chapter 1 I f¡,m1~r,

fN 1herc are .1 pc-oplc \\hO 1,kc cine an,;i1her mld '"""ton wnma IO~>t:lhn"

17

Lc1

re.O he thc rropo;,,uon, "'lf l..

1:. u nonne¡¡.at1\C mtq,.'t'.r. thcn

l' -

:z • ♦ I"

(al \how th.11 thc m1pl1ca11on r(J.) - r{l.. + I) (b) For \\h11,;h nonncgame mtc~r.." 1~ p(1t) 1ruc !

• 18 Con,1dcr thc 1ncqualil)'

\\here J.. 1:. 11 nonneg.i.rnc m1eger ror \\hat ,Jllb oí J... doe-. 1he mequaht)' arrear to be true? Formut.11c and rrovc hy m.;uhcn1.1til:al ,ndu.;tion !he apr,ro¡matc tho:orcm cOnl'ernmg the rdauon!>h,p hct1H-to 2 and l.. l_ 19 Prme the íolltl11o 1ng mcquahty for ali mtcgcr~ n m,11hemat1 in (11) be u.sed for thc choice of 1hc ,\" 1 tcrm, onc of thc foctors for thc choice of thc x 2 term, and onc of thc fac1ors for the choice of 1hc " 3 1erm? This, ho11,e\er, is equal to thc number of ordered panition~

n,

Ctiaptcr 1 / Fr.md~nt Is

S4

of4itemsin10Jsubsets(A 1,A 2 ,A,)inwhichv(A 1 ) 2,11(A 2 ) • 1, and 11(A 3) l. \-!ore e~phcitly, v.e can think of the sd containing the íour factors in (11) as partitioned into three subsets: A I is a subset consisting oí two fac1ors from v.hich v.e choose x 1 : A 1 is a subset consisting oí onc factor from whith v.e choose x 2 ; and A.i is thc ,;ubse1 consistmg of one factor from v.hich v.e choose x 3• According to Thcor.:m 5. 1 thc numbcr oí such ordercd partitions is preciscly

2'1!1' 12.

Exampla 5.3

In how many ways can the white chess pieccs be arrangcd in tv.o lines on a chess board? A linc isju5t a row of eigh1 adjacent squares, and there are 8 pav.ns, 2 knights, 2 bishops, 2 rool.:s, 1 queen. and 1 ling to be arranged 011 the tv.o lines. Imagine that the squares in the tv.o 1i11es are numbcrcd \, 2, ... , 16. \\e define an ordered pariitíon of the set of intcgcrs {I, 2, ... , 16) into si., subs..:ts,

where A I is the set of 8 integers represtnting lhe squares on v. hich the pawn:. are to go, A 2 is the \CI of 2 integer. reprC)Cnting the ,;quarehich k 1 + k 2 + k 3 i • 1. 2. 3. The rcquircd expression is

L

(19)

.,.. •1+•1• 16

xt• x~' xi•.

(1) Occasionally thc sigm.i and pi nota1ions are used 1ogether. Thus (19) can be Y>ritten (20)

.

E..1ch summand in (20) is a product, namely .

x1• .,~•.,!• -

g ./,•.

(j) Using thc sigma and pi notalions. find a formula for (21)

(.t,

+ X2 + ..\' + X4)\

To sohc 1his probkm (which will suggesl a general 1hcorem of this 1ypc), imagine the fi\C foctors in 1he expression (21) wnnen out. (22)

··X

·K··X

·)(·· -J.

4'~

1 To construcl a term of the form ~• x!;• in thc cxpansion of (21). Y>C ehoos.e x 1 from k 1 of the fnctors in (12), -'l from k 2 of the factors, x 3 from k 3 of the factor., and x 4 from k 4 of thc factors. The problcm, thcn. is to decide how many wa)s this can be done. Think of the fhe factors in (22) as being part1tioned into four sub!.ch, (A 1. A1,Al,A 4 ): A 1 is thc subset consi!it1ng of the k 1 factors from which ., 1 is choscn; A 1 is thc subsct consisting of thc k: factor~ from which .\· 2 is chmen; A 3 is thc subset consisllng ofthc k 3 factors from Y>hich x 3 is choscn; and A 4 is thc subsct consis1ing ofthc k 4 factors frorn which ."C4 ischoscn. Sincc thue are fh e factors altogcthcr, k 1 + k 2 + k 3 + k 1 • S. Thus the number of times -~•.'11/i•.t{• appears is just thc num!x-r oí ordered partitions of a 5~kment set into four sub!.cb, (A 1 • A 2. A 3, A 4 ). such that v(A 1) • k 1 , 11(A2) = k2, 11(A3) = ka, and 11(A 4 ) • k. 4 • According to Theorem 5.1. this number is preci~ly thc multinomial coefficient

Thu'i. (11) Í'i equal 10 (23)

( k) Find the eoeflkient oí the mm .\ 1A1x} in (23). For this 1erm k 1 • l. k 1 - 2, k1 2, anJ k,. • O. Then

"

1~2!2!0!

"4 • 30 (1) Find the \ aluc oí

This c~pr~sion is 1he sum oí al\ thc multinomi.il coc:flicients v,, hich appear in (2J). \\ e cou\d sohe th1s problcm by \\f1lmg out ali thc cocfficients and then 11ddmg. This pro,J«t v.ould be 1cdaous but íortun:itcly ano1her technique is a\,ul:blc. lí \.\C set x 1

x~ - .,, - 1 in (23). thcn C\crytcrm

I). :,~- is l . ,rnd (:?J) becomcs 1

(24) On thc othcr hand. v.c knov,, th.it (2J) is equal

(25)

(.\1

10

+ X2 + .\'¡ + .,.)\

so 1í we ~t , 1 - -'l ,3 ·'• 1 in (25), v,, e obtain thc \J]ue 4' ll follov,,~ 1h:i1 C:4) must be cqual 10 4) 1,014, :ind thc ~olution is complete. (m) Compute 1he toial number oí ordcreJ p.irtitions of a ~ ckmcnt set Sinto fouror ícwcrsubscb. Lct I,; 1 • J.. 2 • k.lí m a IJ.bor:itor) hJ.S ,pace for a.1 mtht IO bottle. Thete are , bo111e, oí lhem1ca\ 1, 6 lxmles of lht-mKal 8, and J bolllb of lhellUCal C IL i, requirtd 1h:11 111 le:m 2 bo11k1 of each of 1he l d1ffcrt'nt chem1,:ah hc: 1mt on thc )hclf In ho\O, many"'">'~ can 1h1, he: done 1 (Order docs not matter.)

6 In a \t11ndard cki.k of ,2 canJs, how nun) d11krcn1 2~ard hand, are iherc that ~on,1,1 oí· (a) 2 fan'Glfd);

(h) noíac.:ccards;

fd ,·ard, \\hO'>C numcrical ,alun 101111 b, than ~: (di at lea,t one 11c.:e; (e) n,o klOlf\ .•

7 i\ hOU'-C\IIIÍC IO.l\hC-. to '-CI ~r 4 l.'oluc. ~ red, ,rnJ blk.lrdrmlmg gl.mb 11hool u rouml tabk. In ht>IO. man~ v,J\\(Jn th1\ l,e done: ifonl\ thc relame f'O\lh(m, uf cho.- ¡¡l,i\,n maHtr' (A,)umc 1h.11 al.1,'>t$ oí 1hc \Jme c.:olor are 1ndi-cm11:m~hal.'olc)

1.5

f

Pa,1,1,onf

107

8

Abo"< l'Ontaio~ 5 )CIIOY.. 4 ¡rccn, 1md 6 h\uc c:andb. In ho11, m.1ny ,an 5 ,andle\ be M'.lt.•.:ted su.:h Lhat 2 are )tllow, 2 are gr...-i:n. ami I t ) bluc'! Y.a)-S

A da\\ of 16 ~tudcnh 1~ 10 be d1rnJed 1nto d1:;,;u,)ion ..,.3)) can 1h1s be done·•

9

111.

1 O faaluate the íollowmg: ' 1 (a)~ ; :¡::- 1;

(b)

! '

11

,i·+1

1:

"(c)~(g,)'c J'¡ (t •) ; ''' ".~{ ql). ·m .Ú, (,(\ (t ,)) ' 0

(d)

{g)

(h)

E(-1)';

ú.

(-1)';

"(,) 1\(ú,), ·e,) 1 \(,i'r,e)(:)), (k )

t (') + E (, ~) ; 1

.1

c1> _l_l, x': (m)

(n)

(ol

1'\ :e;; ,lj -l;

ú

r-';

,-1

)

(p) 0

(q)

Ú·"•

1\

t'.;

'(r) ,(~ f,;

' b) (1)

E x'';

txi;

(u) ~

,T;;

' (\·) ~x: (w)

t '"-•;

"(x)t-'":; 0

M

t.

T~:

11

faalua1ctachofthcfo\\010.mgm""htchli.1.li.i. • ,J...r:mgco, cr 111 nonn,c¡;Jli\C mlC¡tl'5 ~ll~Í)lnll thc 1nd1"ucd C\¡ualitin:

ca> ,,.,L,-.1 l•J'•;

L

(b)

2'·i'411;

11• •1••1-l

(d) _., -~-:.

(e)

.,

(,1\);

11 ,:• .\~';

'

1.5 /

PllrlJr,, 1$

109

Ch)

ti 2

C.

L

¡\ ,..) ~ l·.

1

!>ho" that Theortm 5.2 bc.;on11:1, thc d,1~~i.,;3I bm,>miul tMOrrm,

+ r~f -

(q

~ ) (;) x 1 '.rJ•-l,

b) t:ilmg r • 2 m (28)

13 Show th:n

t,, (")' ,· b) c,ommmg the nprc

20 T.,.o di..--e are t{)';'),C(j thrcc llmti m Slk:Cc namcd Tom. Di.c-l, and Harry. In how many "'ªY~ can th1'> be done 1í or thc bo)s get three marble-s cac:h? In how many ....-a)$ can th1~ bc done 1í lom ge1s four marbles and D1ek and Harry 11e1 thrcc cach?

'""º

25 A dcd, oí 52 cards 1s dcalt 10 four p1a)cTS. Ho"" 100n) 1111ual s,1ua• 1ions are ~

iblc íor a bridge ¡ame~

•26 Show that 1í r

(Hmt ; Set m (28).)

X¡

1~

(

11ml: Thc an~11,cr 1~

an C\.-Cn ~1ll\'C mtcger, r -

"' )

c"ii!j~

2p, thcn

.\1

27 Show that

28 Thc ro110.....1n¡ ¡ame l\ pla)cd. Thrcc Y,hllt halb aOO four hlacl: bal\s are pl.:m:d m 1111 urn A playcr roll, two dice and 1hcn dru11,, a hall from the urn. lle 11,m~ ,r and only 1f he lhrows a !lt\en or an ele\en and dra""s a bla.:l ball How m.1ny oull'Omcs are po'>.'i1blc m 1h1~ ¡¡ame~ llow man) oí lhCl>C are ""mmnK outcomci,'1 lf l~re are'""º f!COplc pla)'mg, how mu..:h monc) ~houlJ u pl.a)er denumJ to be pul up a¡amlt h1l n len dollJn?

º""

29 íhrtt dKe are rolled ,,..w:c. In how man) ""ª>~cana .tanda 5 come: 011 m .orne orJer?

16 /

lntroduct,on to Probab,/,fy

111

1.6

lntroduction to Supposc that in nipping a coin n timo ,..,e obscne that it comes up Probability heads / times and tails n - f times. Thc relorice frequenC'y of thc occurrence of heads is defined 10 be ~ 1f n is '"largc enough.., we belie,e 1ha1

!

is a reliablc es1imate upon whieh 10 base a prcdiction

of what will happen tH any gi\ien toss. For e~ampk. if il is obscned that this ratio tends to be near ½for a largc number of tri31s, then it would be sensible 10 bct c,en money, c.g., a dollar againsl a dollar. 1ha1 it will come up heads on any gi,en tria\. Ofcourse. before v.e are prepare ~I to pickup your rnonc) and go home, anyw.1y). Probability thcory i'> concemcd wnh thc prnbkm of making rntional dccisions about c,cnt'> in wh1ch thc outcomc cannot be pr001c1eJ in ;1d,ance \\lth any certainty. As ""llh an) ma1hcmJt1cal theory \\hich purpom to ha,c applk-ationl in thc rc,11 ,..,l1rld, thc

Chapter 1 /

FiS a coin. Lct U - {//, Tl be the set consisting of thc two lctten //(hcads) and T(tails). Lct ~ be thc sigma ficld of all subscts of U:

~.,.{O,(//). {T} . {//, T}J. Define a function p: ~ - R+ by p(O) • O.p( {11)) • ½,JJ(;(O, e), (l. ,):)•¡>( :(O,,):)+ ¡>(:(I, e):)

- !+! • l. In other words, the probJbil1ty th::it Junior will get a chocolate (or \'anilla) cook ic is j. O~ne that the probabihty measure p does not h:ne the samc \ia lue for C\Cí)" one of the clemcntary eu·nts {(O.e);. {(l.c-)). {(1,c):. {(2,c): . l-lo VrC\icr the \alues of pare pcrfec1ly rcasonable in terms of the possible outcomes of thc expcriment. For, oncejar O has bccn choscn there are in fact two chocolate cookies á\ai lablc for Junior's grab, ~1y e I and c-1. Both of th~ are lumpcd 1ogether in 1hc single symbol (0. e-). We can furthcr identify the separate cool1es by: ( l. c-3). (1,v,), (2, e,), (2, t:3). Thus

is a son1ewhat more relined samplc ~pace than U. Now 1f the jars and cookies are choscn at random. thcn any onc of thesc 6 pairs is as li kely to occur 11s any other, so that the assignment oí a measure oí to each elementary C\Cnt consisting of one of thesc pairs is rcasonablc. But in our original samplc space thc pair (0, e) accounts for both (O. c 1) and {O. c-1 ) a nd thus the \i:tluc ½ • + is assigned to {(O, C')}. Simil:ir rcmarl;s can be made for each of thc othcr dcmentary C\Cnts in U. \\ e d1d not srnrt with U 1 becausc thc ques1ion askcd only for Junior's chances of gctting a chocolate cool.ic and no t his chances of gctting any particul.:ir cookic.

t

t

Defínition 6.3

t

Eq~i-probablt spau Lel (U, p) be a finitc probability sp:icc. Then (U, p) is an l't/UÍ•probable spau if the probabihty of en:ry dcmentary e,cn1 is the same, In other words, if x and J' are any two elemcnts of Uthcnp({x)) • p((>•:).

As we remarlcd bcfore, thc dclinition ofthe probability me3liure p on thc spacc U can be arbnrary as long as the eonditions of Dclinition 6.2. (i) and (ii), are satisficd. lt should be emphasizcd tha t a spaec is cqui-probablc only by assumption. For e,i:amplc, in flipping a coin to decide who pays for thc drinl.s, most friendly people \\Ould hcsitalc bcforc dcmanding a long series of trials 10 es11matc thc rclali\C frcqucncics of heads and t.:iils; instead, they would assumc the sample spaec to be equi-probable

Cl'l1pt('r

I

f

F,md1m-,ifi1/J

1 111

Thc main result in this soction is thc following sequcncc of ele• mcntary fac1s about finitc probability spaces.

Theorem 6.1

le1 (U. p) ~ afinite probability space. Th.•11: (o) p(O) O; (b) if X is an evenr, then

p(X') • 1 - p(X); (e)

if X and Y are ece111s and X e Y.

then

p(X) :5 p(Y); (d) if X 1 , ••• , X, are pair1lise difjoi111 t'ce11u (i.c., X, n X, - O for i ',ii ;). then p

(.Q, x,) • t p(X,);

(e)

if X 1•

(í)

if X and Y are ecents. then

• • ,

X , are arbi1rary eL'tnts, then

p(X n Y')

(g) if X -

p(X) - p(X n Y);

{xi. ... , x,} is an t'~111.1hen p(X) •

,t.

p((x,) );

(h) if(U.p) is an equi•probable ipace and X is an e1:e11t then

Proof

(a) Sincc p(U) - 1, U

uO•

U, and Un O - O. ~e ha,c

1 • p(U) p(U U O) • p(U) + p(O)

• 1 +p(O) lfcncc p(O) - O. (b) \\ e kno~ that X anct X' are d1sjoin1 and 1hat X u X ' • U.

1.6 /

lntrodur:r,on 10 ProN1J,l,1t1

119

llencc 1 - p(U) -p(X u X') - p(X) p(X'),

+

Thus

1 - p(X).

p(X1 -

(e) Sincc X

e

Y, "we can writc

r-

XU(X'

n

Y).

MoreO\er, X

n

(X'

n

Y) - (X n X ')

n y

- o. Hence X and X' n Y are disjoint, and we may apply Ddinition 6.2 (ii) 10 obtain p( Y) - p(X U (X' n Y)) - p(X) p(X' n )').

+

(2)

Now. p{_X'

n

Y) ~ O

and hencc from (2) we conclude thal p(X)

:5

p( Y).

(d) We know from Definition 6.2 thal in cate , - 2. the required

equalny holds. The proof of the theorem for arbitrary , is a ,er) easy induc1ion. For, wmc

lhe facl thal the X, are pairwise dis101nt ímphcs thJt

x. n

•º2

X, •

.~2

(X, n X,)

- ,Q, o llence (3)

- o.

,(,Q, x,) - ,( x, u Ú, x,) + (,Q, x,)· - p(X,)

p

Now. 1hc imJuct ion as-,umption )tate-, th.it for , dhJ('lnl C\Cnb,

r( Ú x,) - t ,_,

.J

1 p;iirv.i\.C

p(X,>

Combining thi., with (J) C'>l,1blbhc~ 1hc rcquircJ cquality for rC\CnlS

((') Let X 1 anJ X 2 be arbitrar) c,enh. T hen X1 U X. • (X, 0

x;) U

xl

:1nd

Hcn1.~. by Ddimtion 6.2 (ii) and p.irt (e) of thi> thcorem. (4)

p(X1 U X2)

p(,\'1 0

x;) + p(X2)

$ p( X 1)+p{X1),

n x;

sincc ,\ 1 c X 1. T hisprO\CS(c)forr - :!. T hcgencral ca~c is pro,cd b) induction in t he samc wa)· as pa n (d) 11nd is kf1 asan (í) We "rite

X • Xn U

- x n" ') - (Xn Y)U( X n Y')

\-lorcoH:r. X

n

Y ¡¡nd X n Y' ,m: d i.,Joint. 1kncc p(X)

p( \ 'O)')+p{ X n l'' ).

the required equa lity. htl Ob:. appl)i11g {g) 10 U, wc ha\C (5)

1

p(U)

.t,p({,,j). ll) thc definition l)Í iln c-qui-pfl)babk ,paa. ¡>(,' ,,: ) i~ thc umc for cao.:h i. 1 • l. • , 11. ;md heno.:c fwm (5).

16 /

lntrodi.Jr:tJon to Pro~bi ,1y

1 21

Similarly. b) again applying (g), v.c haH; p(X) - ,

rn,

- ,_,t

Example 6.4

(x,))

p((.,,))

In a group of 5 pcopk. v.h.n is the probabihty that at least two of thcm v.ere born o n the same day of the )'Car~ (For simphcity, we as~umc C\Cry ycar has 365 days.) T o ana lyzc this probkm, 1,1,e kt thc sample sracc U be thc (J65)$ 5-tuplcs of dates, (d1 , d 1 , di, d~,d~). and \I.C assumc that the sp,icc is cqui-probablc. i.e.,

~hJ tc\Cr the choice of d,, i • \ ,. • 5. In othcr words, 1,1,·c assumc that fhc pcop\c are cqually likcly to be born on any choice o f 5 days. Thc C\Cnt X which intcrests us is the totah ty of 5-tuplcs in 1,1,hich d, - d, for at lcast onc pair i,. J. i.c.,

X• {(d¡, ... , d$)

3 (i ~ j

Ad, - d,)}.

•. /

Ra ther than compute r(X), we compute p(X') and thcn use Theorcm 6.1 (b). The set A" is the 101aht)· of 5-rnrks (d 1• • • ds) in 1,1,hich thc d,, i 1, . , 5, are all diffcrcnt . Counting thc toul numbcr of such 5-tuplcs is cqui\alcnt to counting thc numbcr of 5-pcrmutauon) of a 365-clemcnt sel According 10 Thcorcm •L! (b), this numbcr is

365 364 36:l 36:.'! Jól. Thus by Thcorcm 6.1 (h). p(X')

365 364 363 · 362 361 (365P

and appl)mg Theorcm 6. 1 (b). -.e Know 1hilt p(X) - 1 -

p(X')

_ , _ 365_. '.'64

ó~6s\.\ 362 . 361

- 1 - .973 • .027 (approximatcly).

Thc probabihty is rather small that at kast tv.o peopk ha\C the s.ame birth d,llc in u group l'ffhc. Thc samc argumcnt w11\ show. hov.c\cr. that in a group of twcnty•thrcc or more peopk. thc probab1ht) of ot kast two ha\ing thc samc birthday c,cccds ½,

lt is frcqucntly the ca~ thm in considcring two C\Cnti X and Y, thc lnov.lcdgc of Y atT~-cts 1he probabahty of X.

Example 6.5

A pair of fair dice is rollcd. As wc h:l\c compu1cd many time,-. 1hc sampk sp.1cc U can be choscn to cons1,t of 1hc ~ ordcrcd pa1rs (i,j), 1 :5 1 :5 6, 1 :5 J :5 6. Thc as,umpt1on that 1hc dil"C are fair is cqui1alcn1 m $.:l)mg that wc ha\c an cqui-prohablc spacc (U, p), whcn.:

foral! i andj. Wc also l now that 7 can come up in si, v. ays and 11 in tv.o ways. Thu,;, if X is thc C\Cnt consisting of ni\ pairs (i.J) for v.-hich i + j is 7 or 11. -.e e.in compute that p( X) • ~~:')

-1,\

So for wc ha\c \:ud nmhing ncv. Out ~uppoM! v.c pLt) a gamc m whKh ll is rcquircd that wc compute thc: probab1l11y 1h,11 a f.11Cn r••ir of d1..:c comes up 7 or 11, h,nrng bccn ll'ld th;lt thc sum ap¡,car• mg on thc dice 1s odd Then v.c can no k>ngcr )J) 1h,11 p(.n - 1, In othcr wmd,. if ) i~ 1hc ,.et of pair) (/,;) for v.h1ch i + j ' " odd, thc lnov.kdgc llÍ )' JIT~"..:l'> thc prob,1b1IH) oí \ \forc prcr.:i~d), thc ¡,cr1111cnt ckmcnt.tr)· Clenh (111 1hi~ c;1!>C, pairs 1mahng 7 or 11) are no ll1ngcr chtl~cn from illl ¡,,1..'sS1blc pairs m U, bu1 rJ1hcr frnm tht1~c v.-h1..:h are lnov.-n l\l ha,c an o,JJ ~um. Thcrc are líl sud1 p,m\, 1.c., 18. Thu, it i, rCJ'(XJfor 11ny X 1..... Li, then ((J,q) 1s a timte probab1ht) space.

Exercises

1

In ca,,:h of 1he follo"'mg c,r,trimcnh, dccuJc upon un arrropnatc probabthl) l&ned ah1lit} 1hat no ,tudcnt tale~ ht\ a\\lgncll !>Cill (e) í"ind the ¡,robab,lil) thJt at lca,1 onc Mudent tal.es h1s u,1gned

"''

(d) 1 1nd the prolxtbility that c:11:aol)· l\\O ~tudcnt~ tal.e thcir B\\i¡¡ncd

t 1D Let((,.p)beafinitcprobab1bt)'f'ªttandlet \and }bct\\OC\CnlS. ~ho..., that rnun r(.r) l'{n-rnnn 11 A ¡,a1r of foir d1,e 1~ rol!cd anJ \\C are told that thc total apricanna o n 1he dK't l l eH~n. Know-,n¡ 1h1\• .,..h,11 ,~ the prohab11it) 1h:u the total C\tttdi. S"! 1 2 A die 1~ \\Cightc-d !o() th.at the C\CO faces appcar I\\KC Bli ofien aj¡ theodd fa,c,,. lhe d1c 1~ ro11cd I\\ICC and ll 1\ ln0\\0 that the ~um of thc numbo.>n nppeanng 1n the 2 throw~ 1s JI lea\\ 7. What 1s the prolmbilit)

13 Thrce fa1r d 1~-e nrc thrown and thc samc numhn ap¡,caf'i on cach dic. ,\ ha1 1, the probab1hty that the ,um 1, not 6 .' (l lmt: s« Examplc tí 7)

14 A ho\ cont;un\ 12 hglu 1;,ulb\ OÍ\\hKh 2 are lno\\n lo he dcfcxt1,c r our hW)t bulb\ ,ne 1,cle~tcd m su,,e-.,1on from the ho,: W h,1t h thc rrobab1ht) that nonc oí the .i are dcfcdt\c·• (111111 1h1, 1, an apr,hcauon of lhcorcm6.2.)

Ch1p1er T

f

132

Fund,1,r, nt:,ls

15 ll is known 1ha1 ~é of thc men and 90Sé of 1hc 11oomen are O\U• .... e111,h1 m a cenam c:thmc airoup. Moreon:r. ssi·; oí the populat1on oí 1h1s &JOUP are 11,omcn. lfa pcNt(ln is scl«tcd at ranJom from the &JOup and is found to be O\·erv.e1¡ih1, .... hat 1s thc probab1ht) that lh1s pcrwn is a .... ornan'! (l·l1nt: Th1s 1s an examr,lc oflbeorem 61. 1n par11,ul.ar, formula (11})

16 In a semmar coll)l~tmg oí S studenu, s1, .. , s.i.. 11 u, lno...,.n that m comp1hna a bibhograph)" for a JOtnt rcscar,h rro,«t, s I docs 3(')';1 of thc ...,ork. si and SJ each do 2S% oí thc ...,ork. and .rt ond 15 cath do 10% oí thc 1a,orl ll i~ al:.o l ~ n that t1 makcs nu..,talcs 20% of thc 11me, s2 and 13 makc m1stalcs IS~. oí thc lmx, and St and s~ cach makc

mis1alcs So/c. oí thc time. lían entry from thc btbliograph> is chosen at rundom, what is the probab1ht) that II is wron¡'.' (H1nt: Apply formula (10), Thcortm 6.J.) 17 Thrce card~ are sclcctcd m succcs.s1on from a standard dcd, oí S2 carlh. Whot 1s the probab1ht)' 1hat all thrce are facc cnrd) '.' (H1nt : Use Thcorem 6.2.) Let X and XI be events m a fin1tc probab1ht)· spacc ( U, p). As~ume thal p(X1)p(X I X1) + p(X;)p(X x:) > 0. Show that

t 18

p(X1IX)p(.Y1}r(X Xi)+ p( Xí)p(X/Xí) (Hmt; Takc 11 - 2 1md U

X1 U

x;

1n Th«>ttm 63, Formula (11).)

19 A r~ntly dcH>ed tht for thc detection of canccr has bttn found to ha\·c thc follo11om,g rehab1lity. Thc ICSI dc1cc1s 7S';f of thosc pcoplc 11oho ha\,: cancer and d!X$ not dctect t hc discasc in 2Stj. of 1h11 &roup. Amona th~ pcoplc 11oho do oot ha\'C canctr, lhc ICSI de!C.:b SS~. as nOl ha\mg cancer, bu1 erroncously detccls IS% of 1h1s group as ha\'lnJ 1he d1scasc. lt is kll(W,·n that m a largc s.ample of 1hc populauon, ~ ha\-c canccr. Supp,()l,C' 1hat a rundom individual is gwen thc te)t íor canccr anJ rcghlcrt as ha1•m¡ thc disca.e. What 1,; thc probab1hty 1ha1 he ac1Ually has can..--er? (H mt: Le! XI denote the e1cnt consis1m¡ oí !hose peoplc ha\flng canccr. and lc1 X denote thc e1-cnt ooru.ist1ng of thosc pcoplc 11ohom 1he test ~O~)a~ ha\1ng thc d1.1,easc. We are ¡1ven that p(X1) - .02, p(X[J - .98, p(,\ X1) - .7S, p(.\'IX{) - .IS. Ar,r,Jy !he rcsult of !he r,rcccd1ng cxen:1i.e.)

20 An au1omob1lc re:ntal c:ompany ordcr) thre:c cars fron1 a lot oí 11\'c cao oí thc follo11om¡ colors; red, g.rttn , blut, )Cllow, bla,i... What is the probab1hty that !he con1pany 11o1II recen,: red aOO grcc:n cao; bluc a nd }r'llow cars: or bluC', )dlow, and black car,,? (1 hnt: LC't X I be thc M:t oí J-clement sulbth oí the set of fhe can 11ohi..h c:ontain a red anda ¡rttn car: let X2 be 1hc !>r'I of J-clement subscb oí thc ..c-t of fo·e cars 11oh1ch cont:un a t>lue anda )Cllow car; \et ,\ l be thc set oí 3-clement sub!>ets o í thc i.e1 oí fh·e cao 11oh1,h contJm a bluc, a )tllow, aOO a bla,k car. Of c:ourst, ,\" 1) a 1-clcment subse1. The oond111on) oí thc p roblem 1mply that

16 /

133

lntroduct,Ol'I to Proti•b ry

\ . n ( .\'i U .\ 1) O t.:~ thc equi-prohablc mt'll,urc and thc re.ult of E,er~hC IO to rnmpute p(\"1 U .\~U.\.,)

PC\1 Uf.\,U .\,)) r< \Ju \ ) rcx~i-'- p(.L) 1 . .

- p{.\'1)

- rt.11ncd her ho, oí tw from a u ate l"Onlammg t .... o bc>,Ccqucncc of C\[)Crimcnts i~ pcrformc seleucd from n ;11 random ,111d pl,1cC4.I in thc othcr jar. Finilll) a m;,¡rbk Í!> !>clcctcd at random from thii, l,lllcr Jar. What i~ thc probab11ity lhat 1hc two nwrbles t,1ken from thi: pr-, are oí op~ite color!>? Wc com,truct ;1 trce lO mJacate the initi;1l p,1rt of 1h1s c,pc:rimcnt

r

Thc n11mbcr ½ that appcar. on c:ich br.inch il> J. translJtron oí thc 1>lJtcmcnt thal a pr 1s choscn ";it r.iodom". 1 e. lhc numbcr ! rcprc:\cnh the prob.1b1hty of piáing Jar I or J.lf 11 To c.m,truct thc trcc rcprcl>CntJt1on of thc st-cond i,tJgc oí thc: c,pcrimcnt, ...,e mu!>t ,1ccllUllt for ,111 ¡'Kh\1b1htie-.. Sur~c thcn th.ttJar 11 h.1, bl:c:n 1 1 J N ;~~:c~·t!~"r~'.::l: :~ •;~:~!c.1~:; : 1~,:~ : ~-J,.r~t:~~ • b J,lr 11 and hcn,;c thc prnbab1l1t) oí !>ekdmg c1thcr " or h 1;. !S1m1l.1rl), thac arc 7 m,1rbk, m ¡.ir l. '>ll thc prllbab1ht) \1f,ckctmg • ...,lutc nurbk 1\ l :111J thc prt1b,1b1hty l,t -.cloctmg a hl.11.:l m,1rhk 1, J. Thc trcc gro...,1> four ne..., br,mcho. , !.-----"" Thc ~k.:lllln ;JI 1hii, ,tagc ...,¡11 aff1......:1 thc ~,ut.:omc of thc n..::,t !>lJg,: ' of thc c,pcn11K·nt. n,1mcl), thc prnbab1lit)' (1f llbtaming .i ma,bk of

~-~;t;

11

--b

1~~bi:'.~~~

a gÍ\Cn color from 1hc othcr jar. For C:(amplc, supfk.»C y,c are on lhc lop brall(h oí 1he preccdmg trec:, i.c., a whitc marble has bccn choscn from jar L According to 1hc dcscription of the C\pcrimcnt. this marblc is now placed m J3í 11 and 1hcn a marble is randomly sc:lec1cd from jar II Puttmg w in jar II rcsults m jar 11 h;l\ing 3w and 2b. Thus thc probab1hty of obrnining ta w Í\ J and thc probability of obtaining a b is f. Wc can add two bmnche, to thc top branch of the prcccding tn.·c.

We can complete cach oí thc rcmaining threc: branchcs by prcciM:ly thc same kind of analysis. For c: which are as.signed to these nodes yields the result; P1P11P111+P1P11P112 + P1P12P121 + P1P12P1n + P1PuP1 s 1 + P1PuP1u • P1P11(P111 + P112) + P1P12(P121 P1u) + P1P1s(Pu1 • P1P11 + P1P12 + P1Pu - P1(P11 +Pu+ Pu) • PI·

+

+ Pu2)

S1rnilarly, forming the sums of thc probabilities that are DS'>igncd to nodcs that are connected to !he second and third nodes al the first outcome yiclds p 2 and p 3 , res~ti,c\y. Adding t ~ sums we get P1 +Pi+ Pa -

l.

Putting thi:. in a diffcrcnt manncr. thc a~ignmcnt of probabihtics in thc trec (4) does in foct, r~ult in il probabihty mcasurc. Thc probabihty ~p.icc in this case: ,..,¡1] again be a set of J-tuples:

ro. 1. 1>. c1. 1. 2). (1. 2. 1>. . c2. 1. 1>. (2. 1, 2). (2. \, 3). (2, 2, 1). (2, 2. 2). (J. l. 1), (.1. 1, 2). (3. 2. 1). (J. 2. 2). (3. 2, 3). (3. 3, 1), (J. J. 2l: Thilt is, ((i,j, A:); is thc dcmcntary C\Cnt m which , i!> thc outcomc nf thc first tria!:} is thc outcomc of thc second tria l. gi\cn that i has occurred at thc first stagc; and J.. is thc outcomc of the th1rd tria!, gi,en 1h,1t i and j li.t\l! occurreJ .it thc fir~t anJ '>l.'Cond i.tag1;,':s, re• specti\ely. lt is frcqucntly the cai.e that thc outcome oí a gi\cn tria! of an expcrimcnt is "indcpcndcnl'' of wh.it has occurrcd al prc\io us stages. In Examplc 7.1 this ,..,Js dcfimtel)' not thc situatio n : thc ,arious outcom~ ~i.ibk íor chool,ing the fif'l,t marblc affcct thc ~sible outcomcs for choosing thc scco nd marbk. But it i\ c:h)' to concci, e of a series of 1ri.1Js which ,..,e ,..,ould call imkpc11 - p(X1), and similarly, p(X21 X1) •

p(~(~1( 2) p(X,)p(X2)

•

P

J(.l6)1.

As a final c~ampk in th1s scction. \\C con-.idcr a problem in ...,hich thc probjb1hty sp;ltt can no lllngcr be a,,umcd fimtc. Although thc techniqucs ...,e ha,c dc\elopcd are not adcq uatc 10 dcal \\1th such a probkm, thcy do. nc,crthclcss. suggcst ,1 mcthlxl oí a11ad.

Ex11mple 7.7

A match~ti.:k "brolcn 111 two places. V. hat 1s thc probab1l1t)' th,H thc thrL-c píeccs so obtainc the !,malkst sigma ficld of su~b oí thc unit i.quarc ltU e~amplc. Thcsc h,l\c to do v,ith the definition of the ~igma fielJ ~ . For 11 is not clcar precisely v,hJt one means b}' area and by sets wh1ch h.i,c an area. lt is th1!, problem which JHC\Cnls us from systcmatically in\e51ig.'.l1111g probJ.b1lity mcasurcs on spaccs "'hich are not finitc

Ouiz

An,,wer true or íabe

1 A com is tosscd 3 limo. The nurnber oí bran..:hcs 1n !he trec: diagram for this iexpenment 1s 6. 2 In a trcc d1agram, the numbcr of branchcs e:dendm¡¡ from a ll\"Cn nodc 1s thc same for e,,ery node at 11 ¡i,"t'n 1rial

3 lf X1 and X1 are C\"t'lllS in a finite probabihty space (U.p), then X, and X 2 are 1ndcpenden1 1f and only ,r p(XI n X~) - p(X ,)p(Xi) 4 lf X1, X,. and \', are evcnts m (U,p), then X,. X2, and ,\'3 are rndepemknt 1f and only ,f p(X1 n X2 n \"1) - p(X1)p(X2)P(X1)

5 lf \'1, Xi, and Xi are 1ndependent C\"t'nts rn (U,p), then X1 U Xi and \'3 are indept'ndent event:. 6

lí X1 and X2 are mdependcnt C\'CII\$ m (U.p), 1hen

x; and

,\"~ are

mdependent events.

7 lfX1andX2aremdependenteventsm(U,p),1henX1nX2 •P 8 11 ,\'¡and Xiaree~-ents1n(U,p)and \'1 n X2 • O,then ,\¡and X2 are mJcpendcnt 9

The probab1hty or ot>t11imn¡ e~actl)' 2 heads 1n 3 tossn ora fair co,n is

G)(;)' 1 O The probablluy oí obuumn¡ no heads m 3 tosses of a fa1r co1n is ¡..

1.7

F, rtSw, ''"J' Proct1

Exercises

!

1 49

1 1-'or each or the íoll011o mg lim1e Mochas11c proceMa draw an 11¡,rrornatc 1ree dtagrnm. labc:\ each branch ,uth thc arrropnate rrobabih1y, and define II saml'le ~pace am.l probab1lit) mea.)urc (a) A ítur coin 1~ 10~,;cd 3 tunes (b) A b1ased co1n íor 11,h1~h head~ appean h11~c u írequcntly a, ca,ls 1,1~~4tnnes (e) ThrL-c marblC) are ~ucCC\\1,·cl:, drn" n at random from an urn con1111mng 3 11oh11e and .i hlad; marble-1011 >-2"21 units of i1cm 1, ). 1a 11 + ).~.ui 2 units of 11cm 2. and ). 1a 11 ).::"21 units of itcm 3. Thu~ thc \CCtOr dcscnbmg the combmetl productiHt)· of both upanded factorics 1:. (2)

().1011

+ ).::021, ~,u12 + >-::au, ).1013 + >-:iau)

Ch#Ptllf'2 /

lmHfAl~b.,e, by

and if thc add1t,on sig.n bcl\Oottn 1..., 0 \tttors desigmucs componcnt• wise:add11ion (c.g.. (u 1 , 02, a , )+ (b 1• b2, bs) (a 1 b,, a,+ b 2, a3 b 3)), thcn thc \cctOr (2) can be more compactly ...,nttcn as

+

+

Definition 1.2

Vutor op,ratio,u Lct u• (a1, .. . , a ,.) be an 1Meclor and lct >. be a numbcr. Then lhc ,cctor >.u, callcd thc scalar product of thc \CCtor u ami 1hc sea lar (ic., numbcr) >., is dcfincd by

Wc somctimcs writc v>. for >.u. Jf u is anolhcr '1•\CCtor, (b ., b2 , ••• , b. ). thcn the siun of thc \CClONi u and v, dcnotcd u + v, is dcfined by

11 -

u+ o

(a 1 + b 1 ,a, + b, . .

··ª• +

b,.).

Thc two \CCtor opcrations in Dcfinition 1.2 ha\c intcresting geomctric intcrprctat ions. lf 11 - 1, i.c .• if wc are talk..ing about 1-tuplcs, thcn ...,e can idcntify any such l-\cctor \Oo1th a poinl on thc lmc.

l f >. is a real numbcr, thcn thc ,..arious possibilitics for thc pomt >.u I look as follows.

Thoc diagrams pcrtain to the case m ""h1ch o 1 15 positi\c . For o 1 ncgat1\C or O, the moJ1fic,111ons w1ll he ::.ur,r,lied by thc ~tudcnt,

2.1 / V«1Qn•ndM• 1r1UJ

157

Add111on or 1-\cctors a 1 and b 1 isjust ordinary addition

A 2•\ector (ai. a 1 ) can be represcntcd gcometrically as a dirccted line segment from 1he origin to the point (a 11 a 2 ).

(O. O)

10, is a number, then X(a 11 a 2 ) is rcprcscntcd by thc following thrce diagrams for \arious choiccs oí X.

(it, . • ,)

(it,.it,) ~

(it,, it,-)

C~p/H 2 /

Lmur Algtbra

1 sa

Thc addition of 2-\cclors has a somewha1 more intcrcsl ing geome tric mterpreta11on. Supposc a • (a 1 , ad and b • (h 1 , h1) are two 2-\cclors. By Definition 1.2,

\\ e sec from thc preccding dingrnm that the sum ofthc two \CCtors is construcled in thc following way. Transl:uc thc hncscgmcnt dcno1ing b parullel to i1sclf unlil thc imtial point (i.c., 1hc poinl prc"iously at 1hc origin) coincides v.ith the point (a 1, a,). Thcn thc diagonal of thc parallclogram so formed is thc \ cclor a+ h.

e,-

Similarly, in lhrl.-e dimcnsi(lnS a scalar multiplc of a \CCIOr pand5, conlraca, or change~ 3. then no suitable gcometric intcrprctation is a,ailablc. 'lc,crthclcss. 1Hectors occur in ,cry praclieal silllatiom. e.g.. thc e,ampk of m factorics producing " d iffercnt Hem~. But ,cctor opc-rations are defined for 11-,ectors and írom t heir ddimt1ons we c,111 easily pro,e a number of purely algcbraic results which h:nc significant applica11on\ \\e fiht define what is mcant by thc ..,cctor spacc.. of 11•tupks

+

D efinition 1.3

Vretor spuu "f ,Mup/u Lc1 n be a fi,cd f)O'i1t1,c mtcgcr and lct R" denote thc tot.i.hty of 11-, cctors with real componenls. Then R~ togc1hcr with thc operations oí ,cc1or add1tion and ,;calar muhiplication is eallcd the L·cc1or I¡Kll'C o/ real 11-tuplcs. The ,ector whosc components are ali zcro i~ eallcd 1hc :ero i·ec1or and is dcnotcd by O.. (or ~1mply by O 1f n 1~ under.u • (>.111, • ..\11,.) 11>h1ch 1, obH01hl) an 1Htt!M. S1m1larly.1ft· • (t'i, .t·. ). thcn {a) le\ u

cn,p1er2 1 Lm1arA.'gt1br1

1&0

u+ v

(11 1,,

, ,11 11 )

+ (c 1,, •• ,L•,.)

(11 1

+ Ci,.

, , 11,.+

e,.),

which is also an ,Hcctor.

(b) Lt:111 ... (11 1, ... ,11,.),v • (v 1, ... ,v,.).andw • (M· 1, ... ,M,.). Thcnv + 11• - (v 1 + w 1 , • • • ,L',. + M',.)and hcnce11 +(o+ i.') (111.... ,11,.) + (v 1 + w 1,. ,t.·,.+ w,.) - (11 1 + t: 1 + i. 1.. u,. + L',. + w,.). A similar calcu\Jtion ~hows that (u + e) + w (u1 v 1 11• 1, ... , 11,. v,. + 11·,.). I n othcr ""ords, thc as:.ociati,ity of ordinar)' addition oí numbcrs implics thc associall\iity of ,cctor addition.

+ +

+

(1) Obscnc that ¡.w • (1111 1,. ¡.cu,.) and hcnce >--(1111) • (A(¡.c11 1), .• >-.(1111.)) • ((>--11)11 1, . ..• (Aµ)u,.) - (>-11)u. l-lere 1hc assoc1ath·it)' oí ordinary real numbcr multiplication implies (i). 1 A word of caution in the norntion of the follo.,..ing definition is necessary: u, is not the 1th component of a \CCI0r u. lt 1s the 1th ,cctor in a certnin SCl oí m n•H:ctors.

Delinition 1.4

lineilr comhina1ion Let u1 e R". t l ....• m, and let Ac be a rea l number for Ctlch t 1, .. , m. Thcn thc n-,cctor

is called a linear rombinmion o f thc ,ccton 111. t • l. ... , m, and the scala rs A1 are called the rotffirients. 11 i~ often con,enient to use the sigma notation 10 denote thc linear com bination (4):

Example 1.2

lt is obscncd in a nurscry 1ha1 60"; of the ~ds from )cllow poppicsyield yc-llow poppics in the ne\l gcncration ,and thcrcmaining 40C( )ldd ""hite poppil-s. \ 1oreo\Cf, 30'"'( oí thc ~~ from ,,.,hile popp1e,r; yield ycllow poppics in thc nc-.:t gcncrJlion, and the rema1ning 70C'"( } ield ,,., hite poppies. Suppo~ that J ~Jmplc oí 1'. 1 sced~ from }dlow poppies and A2 sceds from ,,., hite poppi~ i~ sclectc

I. A,1r, • l .,.

(·-·ª") ~u

m• . m•. +

+ ··+

rn·-·

In other words, the k1h column of l ,,.A is A'•', so that

Corresponding to scalar multiplicat1on and \CCtor addition, \\e can define similar operations for matrices.

Delinirion 1.7

Addirio" ª"d ual"r 1"11ltiplica1io" ofmalricn (•) Let A be an m X n matrh: and lct e be a number. Thcn the sea/ar produc,

is the matrix obtai ned from A by multipl)'ing e\ ery entry of A by thc number c. In other words, the kth column of e.A is thc Kalar multiplc of the m-tuple A ·1r,, (cA)'Jr,

cA ••.

(b) lf A and B are m X " matrie~, thcn thc sum of A and 8 Í!, the m X n matrit whosc kth eolumn i~ thc sum A u+ B *', k 1, . , 11 . Thc sum is dcnotcd by A + B.

Thus the entry in row i eolumn k of the matri'I A u,1,:

+8

is the ~um

+ b,l,

where a,1 and b, 4 are the entri~ in row i, column k of A and B. rtspect1\ely,, .. 1, •.. ,m :ind J.. 1,

Theorem 1.2

(a) l t l A M un m X n mulrix und lt1 B w,d C ben X p mquarc n1;1tn,; v.11h nonncgat1\t entnn. th,:n I" h,.h nonni:g..itnc cntnc,. íor an) ro-,.1t1,c 1n1cger p.

21

J Vectors and M,11,ctJs

175

(Hmt: For p - 1, A 1 • A has nonnepti\·c tnlries. Thu, surposc thal A•· 1 has nonnegati\·e t'nlries. lñcn A•"' AA•-•. so tha1 an) column oí A • 1s a lmear rombmauon oí columns of A u~m¡ t'ntnn írom A • - 1 as s.c1lc1r codlicicnts.) 8

lf A i~ an m X n m:Un1f, show that

9

Fmd thc transJ)O§CS of thc íollowmg matrires: (a)

rn'

(b) [.. Show that

( llmt: Thc (s. r) cntf} of (AD)'" is by dcfin1tion 1hc (t,,) entry oí AB. Arcordin¡ to thc pre«dmg txtrúse, th1s is Jlbt lhc ,ingle entry in the On thc other hand, thc {,, r) cntry of Br,.¡r i.s thc 1 X I matn" A.· single cntry 1n thc: 1 X I n\atri" (B1l,,(A1)' Out thc sth row oí B' is b)· dcfimuon thc sth column of B and thc 11h column of Ar 1s the rth row oí A, i.e., ( B'), • B' 11nd (.17)' - A . llcoce {B'). (.Ir)· • A, 8'1• Out we saw that A 8'' 1s thc IX I matri" Y.hose smglc cntry is the (•, r) entf} oí (A8)1 . Wc ha\'c ¡,rO\cd that thc (1, 1) rnlr) of(AB)T is thc &1rnc as 1hr (s. 1) cntf} oí srA1'.)

e•·.

Chapro, 2

J

Lin.M A!g,.-,,~

171

12 Let

"'" B •

r::: :::] bi1

bJ2

bu b,2 Wnte out the follo\l O, plJ}·er. Once again, to !.!IY th.tt d, 1 > O. d, 2 > O, d ..,~ 1 > O. . . d,. > O me.in~ that thc (1,j) cntry of A i~ posit,,c or thc (1,1) entry of A 1 is positi\c, pcrha~ both, J-,. s. J l. 2. . ,i_ That 1), (._-,•.,,)e R or there n1,1s a pl,i)cr .\",1; ~uch that (.x4• x,) R .ind (,:,. x.) e R. llem:c, c11her ,, defc.11s -"• or x, dcfrah x,1; ...,_ho in turn defcat'i,)'t,J'J,Y•l, and Z • (:: 1,: 2!. Lctf. X -- Ybc dcfincd by/(x1) - Y1,/(x2) ,.. y,. and/(xs) • Y1, andlctg: Y~Zbcdcfincdbyg(y,) ~ Z¡,g(YJ) Z1.g(y3) • l¡, and g(y,),.. z,. Thcn (g/Xx 1) · g(/(x 1)) • g(y,) • : 1 and similarly (gf)(x 2) - : 2, (&f)(x1) • :,. Wc cons1ruc1 thc incidcncc matrices A(g), A(/) and A{g/): Y1 Yi YI A(g) • :: [

~ 6 6

n:

Y◄

,·r ~l· x, x, x,

,, o

A(/)

,,

1

,. o

A(gf)

x,

,,

,, [ 1

o

z, o

x,

n

Compuling 1hc matrix product A(g)A(/), we haH·

A(g)A(/)

e

[~

~]

[H i] O

A(gf).

Th1~ cx11mplc suggests the following rcsult.

1 0

Theorem3.2 (8)

L:1 \ • ll'I f: .\

., 1 , • • • , ) 1111 taltn h)- X. A. h>· >. and .\ b)' Z. The rok of thc rd,111on R 1 1) pla)cJ h) g anJ the rok of thc rtlJtic-.n Ri IS ¡,la)'cd b) / In thc pr.-:'>l!nl theor.-:m, A 1 .. 1(,:) and .A z • .A(f) Thcorcm 2.2 st.ucs that 1lw (1,j) cntr)' of thc produi.:t -1 1.A J

(9)

A(Jr).A(/)

is cqu.tl to thc numbcr c•f 2-stcp conne..:tions from 1hc11h ekment of X 1 - X to thl' 1th ckmcnt of .\' L In mhcr "orJs. the (i.J) cknu:nt of A(g).4(/) is thc numbcr of dcmt"nb J'• ()f ) .\ 2 for "h1ch fr1,:,)eg

(10)

Therc is "iletl) onc orJ.::rcd pair in/" hc•sc firM mcmbcr is .,,. Thus ( IO) is !>Jli)ficJ b)' prc', \ ) - g(/(.,¡)) - g(_r¡)

Summ.irizing. "e e.in $a)' th;u for a fr,;cd i amJ J, cilher thcrc is n andg(n) • :.,or prn-1)(:I) une clcmcnty4 f" )' )uch that/(:,. 1) no such elcment e\1~1~ 111 the fir,t im1,m..:c. the (1,J) demcnt of l(g) ·l(f) 1s 1, Jnd b) ( 11) thc (i,j) ckment of 1(1{/) 1~ ah.o 1. I n the second altc:rnarne. thc:n.· 1s noelemcnt n l for whKh g(n) = : , ;md hem.-.: (Kf)(\,) • ,:(/(\,)) e.inn,,t be :,. for otherw1.) t, thc I in column; must be in a rov. wh1ch is not abO\c: thc: row v.11h thc I in column 1. Morc:O\Cr. sincc /is n funclion. C\Cry column contains ¡,rcciscly onc I Con\crSCI}, an) matri:c. "'hio.:h contains ¡,reci:,OSses:.cs a set oí 1's in pos1tions {(i 1, 1). (1 2 • 2), (i3 • J). (i 1. 4)!. Thcn II is clcar from (12) that lhcrc ,He threc d1ffcrcnt ro""s of A. namtl) rov.:. i 1, / 2 , 13 , )Uch th,n in column J of A lherc 1) a I in row i,. j - 1, 2, 3. T hus the que~t1on becomc): Doe!. S pos~~ J d1.1gonal con~1~11ng en• urdy oí \\e shall ~how , n thc nc,1 M!l.:tmn th.tt if c,cry di• agonal oí an n X n matri, oí O's and r~ contain\ a O, then 1here must cxi~t s row) and I co\umns. s + t n + \, su.:h that cach oí thc .u entrics l)ing a t thc intc~tion oí thcse ro""s and columns i'> O (\,.'C Thcorcm 4.2, S..--ctmn 2.4). lf v. c 11n1icipatc thi~ rc..ult no"" v,.c can arguc 1hat unk)~ S ha~ J diagonal of l's. c,cry di:tg\1nal mu~I contain a O. Thcn 1,1,c i:.in conduJc that thcrc C\l'>t , ro-.) and , columns of S, r + , • 11 r 1 - 4 ,.. 1 • 5, su-:h thal S has O's at thc intc=tions of these , ro"" lio and 1 ~·olumns. Sincc 1hc last

r~·,

c:olumn of S c:onsi~ls cntircly of 1·s. it fol]o,,.,s that , < 4 T hc possibi\ities are s - 2. 1 - 3; s • J. 1 • 2: s • 4, t I The conditions lhat any k p:nicnts c,hibit thc symptoms of k diseascs among them, k - 1, 2, 3, are unaltcrc 1 - [~ 1

(e} .-t •

["~ oo1 oO

"11

(b) 1 •

:j,

rn

o

¡l. -l~ ~l 1

o

(d) 1

1 O

1

(1

2:J /

199

Powenoffncid,nc, M11tm

4 Lct X be an n-elenltnt w:t aOO Jet R._,\ X \ be a rtla11on ,,.hich sauwtS thc íollo-w1n¡cond111on~: ¡11cn any x andy m .\, cu her (x, )') ._ R

.,)e

Of (y, x) E. R, but not both, and morco1·cr, (.r, R íor any ·" € X Pro1e that, íor an) y E \', 1herc c1w,u an x E \, t ,"- y. su,h that c11hcr (x,)) E R or, íor l>Olne : E ,\", (x, :) E" R anJ (:.)) · R. (H1nt: Thc mcidencc m.11n11; ÍOf R. sau'.>OCS A + 1r J. - J. Appl)· Thcorcm 3.1.) 5 In 'rhr Admlr,,h/¡, Cr1rl11m1, 0:.car W1lde c..tabh!,hc\ that m a g¡1cn cnnronmcnt thc follo-..1ng mtcrn:la11on~ must hold: roe an) pcoplc. ooe mu~! donunatc t hc 01hcr and nlOJ'W\'Cr, no pcrwn dom,natC'ii hmuc:lí 1•ro1'C thli.t a ..domm;mt" ch.1rac1cr must emerge, 1.c., gi1·cn any othcr charoctcr, 1hc domm:m1 th.1rader e11hcr dommates th1s cha.racter, or donunath M>meooe -..ho doc~

'""º

6 For c:1d1 oí thc followmg !>Cb ,\ aOO p,a1rs oí fon,t1ons / and R on ,\' to X con~tnKt thc tOfre,¡,onJmg mddencc mó'.ltm:c~ and l'Cr1í) lhc cquahty m (8): {a} X J(I)

(1,2, 3:, 2, /(2)

J, /(])

g(I) 1, g(2) 1, g(J) (b) X (.f.••,:,11,t·,

l, 2;

/(1,) - t·, /(t) - ·"'• ,ymptoms of d 1, d 2• and d1; ,md p 3 c,hibits the symp1om\ of d; and d 1. In how man)' ways is it po))lbk to choosc thrce out of the four d1:.o:a~s so th,ll cach paticnt c,hibits thc s)mptoms of a d1ffcrcnl one of th,:,,c threc d1~a~s. Wc analyzc this problcm b} Jdining sub\Ch oí thc ..et of di~a\Cs [d1, . . , d1! as follows: S. will be thc set oí di..ca~s cxh1b11t:d b) p,., • l. 2. J. Thcn an SOR for the sets S 1, Si. S3 i~ a sequcncc of thrcc d11fcrc11t di"Ca!>iously cn~urcs 1hat thc union oí any k oí thc seis S,, u S ,, u··· u has at \cast k ckmcn\S in 11:

s,.

11(S,, u S,1 u· · U S,,)

(8)

~

k,

for any 1 :S i1 < i2 < · · · < ii ~ m, k $ m. Wc shall subscquently pro,c that thc inequalities (8) are in fact cnough by 1hemschcs 10 guarantcc that the sets S 1, ••. , S,., ha,e an SOR. lnd«d, wc shall ob1ain an cstim.itc oí the numbcr oíSOR's. We n1ake our result depcnd on a simple and clcgant combinatoria! thcorem conccrning matrices. This result is know•n as the Frobenius• Komg thcorem, namcJ aftcr 1b c.h!>Co,crcrs. In ordcr 10 makc our statements somcwhat brieícr, we first introduce sorne standard Janguage. lf Sis a matrix. then a sutmatrix is just the matri'< that can be formed using thc entries that are situated flt the intcrsec1io11s of ctrtain specificd rows and columns of A. For c\amplc, if

o( A

A

['

4

25 3] 6

thcn

is the submatrix of A lying in rows 1 :ind 2 and eolumns I and 3. Similarly,

e - 12 JJ is thc submatri\ of A l)'lng in row 1 and columns 2 and 3. On thc othcr hand

is not a submatriJt of A bccaus.! D is not ~itu.tted at thc interscctions of any sets of rows and columns oí A \\c can now statc .tnd pro,c our cent mi thcortm

Theorem 4 2

FroMni111•KQnig /.et -'I b~un II X n 0,1 matrn:. Thtn 1·n-r.r d1ug01wl o/ ,t hm" O"' 11 ifuntl emir 1/A con1um1 an r X I rnbmam,;: i.h()tt' t"ntrin urt" u// O mid /or i.·hifh $ + 1 • n + 1

Z 4 ¡ lnttoduct,on to Comt,,nator, 1 M~rr,x Tt, 1fY

2:07

Proof Beforc v.c cmbark on a formal proof of this m;ult it is instructi\C to considcr wmc uamples. lf n - 3. thcn C\Cry diagonal

º'

oo [11 11

A

º] 1 1

has a O in it and A conlams thc 1 X 3 submatrh (O O O]

of O's. Morco,cr 1 .imrktal..c:

+3-

-'

A -

3+ 1

o [O1

n

+ 1. As anothcr

e:, of B must he 111 thc last n - t columns. Thu., J $ n - 1, ,md it follOY>1 that f 1 $ n. Out Y>C ha\e a.,sumed that 1 + 1 11 + 1 áning Jppc-Jr,1ncc:

+

"

(10)

C •

11-

1

H

Thc matri,r, .◄ 1 in (10) li..-~ in thc fim n - 1 roY> s and columm of C. Anydiagonalof A I c;111 oh\ iou,1)- bi: cc,nipktcd toa dt.igonJI of C by adjoining lhc I in 1hc (n. n) p..hition. Sin.:c C\CT)' diJglln,11 r,f C contJin, JO, it noY> follo,..., that C:\Cí) diagonal of ~ 1 contJm~ ,1 O. Thu, thc inúucti~,n hyl'("llhl·~1. tell, u, th.11 1 1 mu,1 c11nt.1in iln 11 X 11 .,ubmatri~ of for Y>-hi.:h .r1 t 11 (11 1) +. 1 11. IJ) an interchangc of H>Y>) ,md 1.:l1lumm in thc matri,; (10) w.:: mJ} put 1his zero submatri1t of A I in thc uprcr lcft corncr .so that thc matri1t can nOY> be as-.umcd w h,uc thc folloY>mg form:

o·s

11- I¡ •

t1

rº

(11)

'' t o

D

1 11 -

l1

'' t

lt Í!>Ckar from (11)th,ll \" isa t 1 X 1 1 matri-.; and 1 isan lt X s 1 111Jltl\. Supp1ll.. of o·s in E v.hich lic) in the fir-.t íi rov.!> of E and in 1h.- fir,t 1 1 + 1: columns of f.". tí v.c compute thc wm of thc numbcr of rnw~ ;md column~ in tlm zcro ~ubm,1tri:>... v.c ha\C S2

+ (11 + f2)

+ I¡

(l1

1" /1)

•

(s 1

+ I)+ 11

•

(11

•

11

•

+ 11) +)

+1

In other v.orJ~. the indicatcd \ubmatru of O\ h,1s thc properl} th,11 the . Gi\cn an II X II mJtri, A 11ro·, an v,1th 1 +, • 11 + l. Dy an inkh:hangc of rnv,, .ind n•lumn~. thc ~uhma1ri, may be hrought into 1hc upp,;r kfl•hJnd ctm11J.rl). jfv,c .idd thc cnlrlcl> 111 X, "-C obtJm l.1. (Concci\abl). if .\ l. thcn ) d1ll:~ not app,.-ar. or if t l. thcn X docs m1t appcar in ( 1,): hut thc .argumcnt rcmamlt unch,ingcd.) Thlh thc s11m of thc cm riel> in \' anJ ) i,

(16)

l.\+,., -1.(i + /) l.(,1+ \) Out. sincc C\CT) onc of 1hc II n,,,.,~ in ,-t Cllntains prcá,dy k onc:s. thc '>Um of,111 thc cntric'> m -t i, l.n, and thi., 1s 111 conílu.:t v,ith ( 16)

24

I

lnrrodvc11on to Co'l'lbin111or11,I M1tn1< Th.o,y

213

lt follows that A must posscss a diagonal of 1·s and h.:ncc a p.1iring inlo d;mcc partncrs does indttd cxist. Ex11mple 47

Show th:it if. in thc st,1tcmcnt ofthcdancc problcm, we assumc lhat cach of thc ,i girls has mct at fro:rt k boys :rnd cach of lile II boys has met u, leau k girls, thcn a pairing into dance partners does not ncccssarily exist f or. supposc ,i - 4 , i.c., thcrc are four boys and four gi rls. Assume that the follo.,,,ing introductions ha\c bccn made. Girl I has been introduced to all four boys; girls 2, ), :md 4 ha,c bccn introduccd to boy 1; boy I has bcen introduced lO al\ four girls; and boys 2, 3. and 4 ha,·e bccn introduccd to girl l. In this case, each boy has mct at lcast onc girl, and each girl has met at lcast onc boy. Thc incidencc matríx A has thc following form:

A •

[

1¡ 1 1

1 O O O

l

1 O O O O . O O

+

The matrix A contains a 3 X 3 subma1rix of zeros and 3 3 • > 4 1 (Of coursc, 1he matrix con1ains a 2 X 3 subma1rix of zeros if i1 con1ains a 3 X J submatnx of zcros.) Thus, according 10 Thcorcm 4.2, A docs not posscss a diagonal of l's and no pairing into dance panners is po'l~iblc.

6

Quiz

+

Answer true or íal.sc.

1 Any two d11Tcren1 non-cmply subscu of a 2-clemcnt sel SDR. 2

3

~

an

A 2 X l matrix has six diagonals

H A IS daagonals.

an " X

m matnx. then A and Ar haH: the san,e numbc:r of

lf ali corre~pond1na d1a¡onals (1.c., corrcsponJ1n11 to thc l3mc ptr• mutatioru.)oftv,o" X "matrices are tnt: !>3.mt, then thc rnamccsareequal

4

5

Thc matm: [I

21 contams no subm.:nn ces.

6 lf A is an" X m 0.1 matnx such that C\'U) d1asonal has a O m 1t, then A c.:ontam:. an s X , subrnatri¡¡ whosc entries are ali O and s + t can ne\'er oettd " 1 7

fa-cry diagonal of thc matnx

contam~ a O.

Chllf}I('( J

,'

214

L1Mar AltJ of thc matril A. Thc inequahtic) in ( 1) impl) for thc matril A that gi,cn ,iny k row) of A. there c,ist al lcJ\l 1,, columns in which r, appcar somewhere among thc k giH:n ro\\~, 1,, l. . m In 01hcr word'>, ali the 1·,. th.ll arre.ir in ,my 1,, to\\~ of I ilre not "crammed'º into fewer 1han J.. co\umn'>. For C\ilmple, 11(S 1 u S~) ~ 2 means thilt S 1 U S-. conl.iin~ ,11 ka\t two

C~·I 1! diagonah of ¡ ·.,._ , o.. , kl -t , be the (m - J..) X (11 - k) lo...,cr right bloc l.. in I indi..:atch,1111 that -t _ c\lnt,1m, a J1Jg11n.1l of ¡·,_ lf C\Cí}' di.1g111ul 11f .j I CC'lnl,LHh il 1hcn thc samc •~ true f,1r thc follo...,111g (11 k) X (n J..) matri, 1h.11 111e can

o·~

º·

25

/

S)'!:ems ol o,s1mct RfJPfeunra•:,,,

219

consuuct írom A 1 by adjoining n

m rov,.s oí¡·., to A 2.

Thcn thc mmrill B contains an s X r submatri"t oí o·s in which + r ~ n - l.. + l. By an intcrchJngc of rov,~ and columns, v,c m,1y aC 11 , Y.e are ,1~uming 1h,11 gi,cn any k ro11,i.. there are at lc,1 1 For, takmg k 1, Y.C \Ce th,u any ro"' mu\t C(lnlain at lea,t 2 onc.-.. lt i\ JU~t a matter of rdabclhng the elem.:ntS to bring an} ofthc I' m. Th1~ completes the proof. 1

,, ,,,)' diagonal~ of i's in A .

Example 5.1

The rf'ult oí Theorcm ~- 1 can be U\.Cd to 1mpro,c our ans-.-.cr to thc danc,: probkm. In [\;m1ple 4.6 1,1c sho1,1cd that if in a group of n boys and II g,rls, Calh hoy ha, bccn miroduccd IO ¡m..-ci\CI)' t g1rl.. and each girl has been mtrodU, then thcrc are al lcast t! d11Tcrcnt pairings into dance partners.

The re!iult of Thcorem 5.1 can be used to analpe probJbihty problcms -.-.hich hc rctoforc ha,e bttn inaccc\!iiblc.

Example 5.2

A superm:ukct carrics 6 brands of bread. Si, hous.cv.iH~S ~uippcd 1,11th ccrrnin prcJudiccs entcr the marl.et to purcha!,C bre,id. lt 1s C(lUÍ•probabk that u gi,cn hou~wife "'ill purcha'>C any onc cif prcci-cly 3 brands oí bre,1d but definitely v.i ll not purchasc thc othcr 3. Morcmer. any p;,1rticular brand Í\ among thc acccp1,1bk: choiccs of preci,;ely J of the houscv.-i,cs. Que..tion: \\ hat ¡., thc probJbihty that ;1116 brands 1,1ill be purchased by thc hom,c1,1.i,e-, 'l Aho etinct . For then the righ1-lrnnd siJc of (6) i~ JUSI thc probabil1t)' tha t houscwifc s purchascs brand)., .f l. • 6. and brandsj 1, ... . ),. con)titutc all 6 brand). In othcr word~. p(,l) i~ ;1 sum oftcm1,. cach of which is;\ produ..:t of thc entric inio onc of the n original typcs oí pJrticles. Thus,

(8)

P11

+ P:, + · ·+p•. •

1,

j • 1, ... , 11, for, (8) is thc MJtcmcnt that thc probab11ity that p;1rt1d es of type ¡ clJJngc into at teast onc of thc II types oíp.m1cksis l.

(íi) Thc foct that C\cry onc ofthc original t}J)CS oí pa rticlcs nm~t be prcscnl at thc cnd of thc rcaction is jusi thc Matcment that (9)

P,1

+ P,: + · · · + P,~ •

l.

. , n. For. (9) ;h-,crt~ that p.¡rt1cles of l)'f!C I are pre!,Cnt ,11 the cnd of thc re.1ction ami v.e know that thcy camc from particle,; OÍl)pc rca)onablc to a)~ign the probabilil)' (11)

to the clcmcntary cH:nt conlii)ting of the 11-tuple in (10). Again a) in E,ilmple 5.2, v.c can \CTify th.it the func11on pin {11) )IClds a probability mcasurc on the spacc U. Lct X be thc C\cnt in U conSi\ting of all n-tupl~ of thc form (10) such that ()1.... , ) ,.) i~ an n-pcrmutation oí{!, . . . • 11; . Thc set X corrcsponds 10 1hc totalit) ofways in which particlcs of typcs 1, . . , 11 can come from paruclcs of t))'C' l. ..• 11 in sorne ordcr. i.c., p:irticles of t)pe s come from particles of typc ¡., s • 1, .... n, and ali n intcgcrs appcar among ;,. . . , ) •. Thus thc probability of 1hc c,cnt X is gi,cn by

(12) in which t hc summation in {12) is o,cr all n-pcrmutations {j 1, ..• j,.) of !he intcgcrs l.. , n. Wc can rcwntc (12) !>Omcv,hal more ~uccinctly as fo\lov.s: (13)

p(X)

r:

r,

p .... ..

. ,.s. •- 1

Thc summation in (13) is O\Ct thc set of ali n-pcrmutJ.tions of {l .... , n}, dcnotcd hcrc by s•. Equi\alcntly, thc rig,ht-hand sidc of (13) is thc sun1 of all 111 products formed by mult1plying togcthcr thc clcmcnts in a gi\Cn di;1gonal of P. Thc formula (13) is thc rcquircd e~pr~ion for 1hc probabili1y that 1hc n dilfcrcnt t)pc~ of p.1rticlcs rcmaining af1cr thc reaction camc írom thc n original typcs of partick'\ in ,;orne ordcr. \\ e shov, that thcrc c,i\tS at le:m onc diagonal product in P v.h1ch is p(hitiH!. For. sup(l('l-.c that thc produc1 of thc clcmcn1s of c,cry diagonal in/' is O. lt fol\ows that c,cry diagonal of /> mu~I conlain a1 lcast onc O. 'Jow lct P 1 be thc matrii¡ obtaincd from P b) rcp\Jcing cach of thc ~lrictl) po!,1ti,c cnlrÍe\ b) 1 and lca,mg thc O cnltics un.1ltcrtJ Thcn C\Ct) di,1gonal of /1 1 contain~ d O anJ wc may appl) Thcorcm 4.2 to concludc 1hat P 1 conum, an 1 X 1 ,ubmatn~ v.h(t'>e cntriC'> are all O and for v.hich 1 + I • 11 + 1. lt i~ ob,,ou, that /' mu~t h;nc thi~ prnpert) lb v.cll. w tha1 by a rcarrangcment ofthc rov.) and column) v.c ma)· assumc

15

/

Sy.,;1ems of D11MCI R~p,e

nflfr,

225

th,11 the ~ X , submatri, ofO's appcan, in P a.!> íollow'>.

,¡[~ 1

(14)

p -

o .. o X

The submatri,; A h,1~ 1 column, and e,1ch column ,um oí P is 1 lt íollow.!I from (14) that cach column 1>um of Xi'> 1 and hence the 1>um oí 1hc cn1nes 111 ,\ is/. Similarl)', cach row sum of } is I and hencc thc :,um of thc cntnc) in Y i:. r. Thus the sum of the entnes m \ and Y together is s + 1 ~ 11 + l. But cach row sum of P is 1 ,ind h.:ncc the '>um of 11II the cntries in Pis 11. This i:, a contradiction and i1 follo...,, tlrnt P po-, J. t hcn A 1~ called n ro11 1todmrtiema1rh. l fthe ~um of1heentrie-. in e.1ch column of A is l. thcn A is called a column .\/()c/1, \1 1, , C 7 • ll owc\cr. it 1s known that cert;1in 111,~ifir.:atmn, of di~tribution policics are ~sible whieh wi\l rhult in J greJter nct

C.,

profit for the entire economy. Ai. a pariicufar cxamplc, v.c can summari,e the mforma11on in a 5 X 7 matri, A in v.hich thc (1,J) entr) is I if a change in thc di,tribution pohcy in either M, or C, re5ults in an incrca,;e ,n the nct profit for the ent1re economy; if no such changc is ~ible, the (i,j) enlf) ,s O. MorcO\er. 1t 1s a!.Sumed that if a,1 • 1 1hcn lhc mercase in net proflt is the same v.hcther M, or C1 afTcc1s a changc in its distribution polic}

(15)

C 1 C2 Ca o 1 1 o 1 O o o 1 1 1 1 O o

MT

J.1 2 A • Ms M4 Mr.

e, e, e, e, o o 1 o o o 1

o

l

f

The problem is to find the mínimum number of comp.1nies among the M, ande, which must modif) 1hcirdis1ribution policies so that t he profit for this cn1ire scgment of the cc:onomy will be ma,imum In tcrms of the matrit A. this question amounts to finding the mínimum number of rows and columns in A so that all the 1·s in the malri1l appc.u among thesc rows and columns. For thcn it v.·ould only be neccssary for the companics M, corre,;ponding to thc gi\en rov.~ and thc companies C1 corresponding to the gi\en columns to change thcir distribution practiccs. Obscnc that if four comp.inies, M 2 , M~. C 3, and e,. changc thcir di~tribution policies, thcn thc maximum profit for thc cconomy wil\ be nchit\Cd, ~incc all the rs in 1he m:nrix appear in rows 2 and 4 and columns J and 6. Thus thc mínimum nurnber of companies nccessary i, no greatcr 1han 4 Beforc ,,.,e can completely ;mswc:r thc question ~ d m faamplc 5.4, wc pr0\C a rcmarlublc thcorem v.hich fo\lov.:. dircctly from Thcorem 5.1. To faciht.11e the i,tatement v.e mtro tcrm r,ml f. To ')Ce thi'>. ddin~ f ldcr thc n X n matnx

3 Le\ A be a 2 X 2 doubly ~toch:lab1ht) \. Countr) 2 ht1m:ues that 11 w1ll he anad,ed by Country 1 \\llh rrot>Jb1!11) \ and b) Countl'}' J \\llh probal:ithty ,. C0\11111) 3 c~uniat,-s lhat 11 \\l!I t'C att;id,t-d by Ct>Unlry I w1th proh,1bthly anJ ti) Countr)· 2 11,1\h proh:ll'lhty j. 1-inJ t hc probubilny that ea.:h of

!

1

25

f

Systems of D,irmct R~$. mear,~~

231

the countrics .,.,.ill anacl. anothcr country m such a way that ali threc countnes are undcr anad, 1n sorne ordcr. (llmt Thc matra, for 1h1~ situauon 1S A

[

¡]

o ¡ l o l . ! i o

The requ1red probo.bH1t) 1) the sum of the products of thc entrie!. in all d1agonah m A.)

9 Threc arehers shoot arro.,,.., at thn:e mrgels. On the ba~h oí pre\10~ performance, 1t 1s~omated th.lt the first archer .....11 i;hoot hii;arro.,,.i; 1n lhe bulllC)ts m targcts 1, 2, and 3 w1th prob3b1IJUC!> ½. i, and ~. m.pc('IJ\1:ly; lhe Sttonc.l ard1tr will shoot his arrows m the bullse)b oí thc 1hn:e f. anJ l, m.pecu vely ; and the tlurd archef1argets 1111th probob1lmes .,,.111 shoot h1s IIITOlll'S m the bullsc)es of the thn:e targels lll'llh probabilmes i, and f'e)pectl\'cly. lf ali thrcc archen shoot their arro111'! Wmullancou~y. wha11s the probablhty that the bulbe)es ofall three 111.rgcl'i .,,.11! be hn?

J.

A.

l.

• 10 LelAbea.5 X 5matrixin .... hicheachrowandeachcolumncontains precisdy thrn: t·s and two O's. l'ro\e 1ha1

I: ¡'¡ ••" ~ •· , and .S girh, au;h oí the bo}) has been introduceums oí A. (llmt: Sec U.ere~ 10, Section 2.3.)

f 12

t 13 Let I be an

m X II ro11, stoch:ts11c matm: and O be an II X q row )10tiC nlJlnx. Show thJl . 10 ¡~ an "' X q rov- Slocha)UC ma\n,¡ PrO\'C a \Un1lar resuh for column stochasuc matrim. llro\·c lhat the produ,1 of any two doul:>ly ~tocha,tic matricetic matmt

f 14

l.ct A and 8 be m X " row \lochastic m.11nce-; and lct C" be a numhcr .sat1,í)"mg O < l' $ 1 Show that C'. I • (1 C')O 1\ a row ~tocha,uc mam,. Provc the '3me rt!>Ult for .:olumn stocha~llc and doubl)' ~to.:h.hlK

m.11nec:s •

15 In c.i.,h oíthc fol1o"ing doYN} ~tOtha!>U..: m:itncn;c,¡h1b1t a dmgonal colb1,nn1 or poi.iti,c cntrie\:

,., li il, !

lºt . oo 11 · j

(b)

(e)

r1

1 1 o

(J ¼ to

·1

½ .

! o 1 o ½ ()

t 16 An n X n ,wrmu1a11"n mairi:c is a nutn\ "1th preci1,el) onc I and n - 1 rcro a permut.allon matri:how lh.at if I i\ an n • n pcrmut:it,on m.itn,;, thcn A i~ thc mcidcnce m.-un'I( for r,rcc1lo(l) one pcrmuiationf: V~ 17

• N.

Wnte down ali J X 3 pcrmu1:11ion rmtnm.

t 18 Letc1,

.c,.bcnonnc11.1111cnumbcr,1uth>uml,1.c.. f e , • l.

1 et A1, , -~ .. be a ..et oí 11 X n pcrmutallon nl.11riefi Show th.at t1A1 t,.A,. 1!> a douMy Mochasllc m.1tmi:. (Hmt. AJ.• J .. ,1, - J .. Hcoct:.

19

(i:

J .. Arpl)· thc r~ult oí Excrme 12.) r .. l ) , , 1 Wme ca~h of the doubly ,10.:ha,11.: m1111r1~-e~ m E\cr,1-.c IS m thc

Smularly,

form A¡, •

":!:, e, ·1, "here f r, • 1, ~) S1m1\arl)', thc sum of all entnes 1n .\is,. {Wh>·~) Hcncc. the ~um of ali the cntnc:s m .\' and Y ,s s + 1 - n + 1. Out thc sum of the entnes m II doubl)' stochlhllc n X n matnt is c,sctly n, and thc:rdore wc M1c a c:ontradu;:uon.) 21

1-1nd the 1ern1 rnnk of cach of the followm¡ matr«:n:

(a)

[i

(b) [ ;

(Uj¡~c~1' thc formula11on of a \inular ¡,rohkm 111,ohmg ,~·10.-.. and matnce-.. Thu,. 1f . 1 l\ :i g1\cn m X II m,uri, and /t ¡., a JÍ\Cll 1'MC..:tor. t hc:n thc fundamcn1,d rroblcm in the theory of hnC',!\\ ,lbo\C:, J '>t\lUIIOn n...-cd not C\l~I.

26

J

lnmxlvcr,on 10 Lm••r Equat,Qfl

135

so that a major prob\em implidt in linding all i,olutions to (1) in• ,ohc~ detcrmining 'Au,h i>ulutions c·u ~t

Examp/e 6. 1

Let 11 (2, 3) .ind v - (- 1. 1). F1nd all numbcri> _,· 1 ami x 2 such that the linear combinaiion x 1 11 + :c 2r is thc ,cctor (1,0) Wc compute th,11

- ( 1. 0).

Equating components,

Y>C

h.i,e the two equJtio ns

2x1

(2)

3.ti

.T,

1,

+ "2

O.

From thc first equation 111 (2). Y> C sce tha t .t 2 2x1 - 1, and írom thc ~ond eq uation, x 2 - -3.t 1 . llcntt. if (2) is to be samlied . it follows 1hat 2x 1 - 1 -Jx1, 5x 1 - l.

x, - J.

and hence,

Example 6.2

\\'rite the íollowing \)"Mcm or eq u:ition~ in the form A, • b, where A IS a matrix and .\ and b are ,ector~; X¡+ t 1

(3)

+ .\ 3 X4 1, x2 +x, • -2. 2.t, -

x, •

5.

Let A be thc J X 4 matri,; oí cocfficient'> v,.hich appcar in thc system (J):

A-[~ l

~I

-:1. O

Let x (.\"1,x 2,.-.,. x 4 ) and let b ( 1, 2,5). T hen thc thrcc equalions m (3) can be Y> ritten as a !>ingle matri\•\CCtor cqu.111011: At - b.

Example 6.3

• , e,.. forma cl~J econo m~ unit in thc A set uf n cit ics, c 1, that thcy trade only among thcm~hcs accord,ng to the

kn!IC

Chapter 1 / L,,,_, Algebta

23e

follo"-'ing schemc: e, sells what it produces a t an a\erage price of x, dollan pcr un1t produccd, i - J••.• ,n. Also, cuy e, purchascs a ccrtain numbcr oí units from city e,. call it b,1 , i. J - 1, ... ,n. Thcn

is thc amount or moncy th.tt e, PJ}S to e, for its purchases from ci, i,j - l. ... , 11. Thus, thc total amount !>pcnt by e, is

for 101a l purchases o f

"f

,tb, b, 11 units

1

T hus thc a\cragc amount spent by e, pcr unit p u rchascd is

Set

Obsenc 1ha1 a,¡ repres.ents the fraction oí thc iota\ purcha,;es 1ha1 e, ma l.es from e¡ Then the a\erage amount spent by e, pcr unit purc h11sed is

(4) lf thc cconomy is to be M.tble, no city should ~pcnd o n thc 0\Cragc more o r kss pcr pur.,;hJscJ umt thJn thc pricc: pcr p roduccJ umt. In other words, 1he iota! 3\cragc e"tpcnditure ( 4) per itcm purchascd l •... , 11. llcncc, b) e, must be cqual to .x., ,

L a,,xi • ,_,

(5)

If wc Jet A be thc

x,,

i -

1, ..

, n.

X n matrix whm.e (i.J) emry is a,; a nd set x (.l' 1 •.t 1, .....t,.), thcn the conJ1t ions (S) for a stable economy can be summMizeJ m thc following single matri:1 Acc1or cqualion : (6)

II

A-
/IS.) 1

h1ppcd from cach of thc 11,-archou..c,, to eac.h of the oullcl.\ m arder thal thc ,h1ppmg c0tlid to be line O, v,h1ch conuuns t ~ A. rov,) is 1,nearly dependen!. Thcn, íor cxamrle, a l)J"C II ckmen1111; row operation o n A rerlat'fi the !>ti o í ro..,.~

U -

' -f1,,

. A1¡, A 1 .1-,

,A,.,,

by onc of the follov, 1n1 se~:

1o1-hen: 1 S , 5 4 and c1thcr 1 S

J $ J..

or j

> A: or

whett !.. + 1 ~ p ~ m and cilher I S j 5 J.. or / > 4. In cach ca'IC, cdublt II set oí!.. IJnearl)· independtnt rows. This v,,11 ~ w that p(B) IS alW3)'S al kasc as largt as p{A), i c., p(B) p(.-f) T hen, SH~ ·1 cun abo be olH.a1~,J from B b) un clemcntary rov, oper.ition, 11 fl'lllov,s 1ha1 p(B) ~ p(A). Hence p{B) p(.-1). The argumenh íor t)pe I and t)pe 111 clcmcn1ary rov, opcrntions att H'r)" n1uch \1m¡,lcr.)

t 13 Lt:t //hea ma1m m llcrmitc l'IOfmal formasde~bed m Deftnition 6.3. Show th.it p(H ) 1~ the number of non-zero ro..-.~ m H. l h:n.."t: p(/1) 1) ..-.ha1 v,-.:: c:1llcd the rov, ranl oí// m l)cflruuon 6.3. t 14 Show thal if A i1 rcduced to lletnuh! normal íorm //, !hcn p{ ..f) p(/1). (Hint: Appl) the 1"1::!oull oí [,eró~ 12 repcatedl).) t 15 fmd thc row rnnk oíe3ch of t hc fol10"'"1na matrices by rcducin¡ 1110 lkrmitc normal íorm 11.nd u...,n.11 L,crc1sc 14:

(•) [l

1

o 1 -1

~i[[ ¡j, 1 1 1 ) 1 1

~];

- 1

2 7

/

A~lt:>11$ of rhe Herm,1, No,m.,J F,:,, n,

, ) [l (d)

¡¡

253

4

o

o 4 4

' _, _, _, l 2

-:i_,

o .

¡]

Let A be an II X II malri)( and '>UPJ>O$C p(A) 11. Preve that A is row equ1valcn1 10 thc 1dcnt11y rrmtrix t •. ( Hmt: Let 11 be tht> lfermite normal form or A. Thcn b)' L)(crcisc 14, p(_JI) n Ottscn-e thc remark

f16

unmcdiatcl)' íollo11.-mg Cxamplc 6.11.) f1 7 Show that if A is an II X II n\3tri.11 and p(A) 11, thcn thc onl) '°lution to thc S}~tcm Ax•() 1s thc 11·\'eClor .'C • O. (Hmt: Accordmg lo Cxerci'>C 16, the '))tCm Ax - O is cquwaknt to thc S)")lcm ,~ O.) f 1 8 Let U • {111, •.. , 11~} be• set of\-u:tors in p_.,_ Show that if p > 111 thcn thc set of \ttl()f) U •~ hncarly dependen\. In othcr v,ords, if the numbcr of VCCIOl""S m a set U R• c-,¡oecds m, thcn thc \'CciOíS must be \incarly dcpcndent (lhnt: Let A be thc p XIII matrix v,ho;e tOIO..s are thcvcctonu1, .u,.1.c.,

Since p > 111 the nun1bcr oí row) oí A uric1ly C'(oetS a sy~tcmatic "' ªY of sohing line3.r equ:1tions Ul>ing thc Hcrmile normal form of the coellkienl matri\. Thui,, kt A be: an III X II ma1ri'(, an tha1 ea..:h of 1he~ operations rt:)ulh in an equi,alent \)'>lem of cquJtion~. llcncc (/1:cJ i~ tht: ;mgmented matri, for a \)'>lcm llÍ equatillll\ equi\aknt to the original l>)'~tem A\ h, that i~. for thc S)'>lem (4)

llx

('

11 / App/1cllt1oms , •f th Hemw Nomi« Form

2 55

Thc systcm (4) has c'(actly the sarne set oí solution ,cctors as docs A.-,; - b. Thc solutions to 1/x e. howe,·er, are casy to obtain, bccausc // is in a grcatly simplificd form. Rec:all from Ddlnition 6.3 that thc first r row!> of // are non-i.ero and thc remaining m - r rov,.-s are ali u:ro; thcrc are, columns of //, numbcrcd 11 1, •• , n, (1 $ 11 1 < 11 2 < · · · < 11, $ 11), such that colunm 111 oí JI is thc m-,cctor ~ •. i l. , r; thc fir¡,t n, - 1 cntrics in row i oí JI are zcro,, 1• •• ,r. Supposc that onc of ••+I•···••.. is not O, llay, ;,d O. Toen, sincc 1hc (r + IPt row of 1/ consist.s cntircly ofO's, it follov.s that thc (r + l ►.,t cquat ion in llx ...- e is

••+•

which ob,iously cannot be ~tisfied for any 11-,ector x. T hercforc /1,,: e (and hcncc thc cqui,alcnt ~)'Mcm Ax b) ha~ nollOlutiOn) On thc 01hcr hand, sup~ tha1 e,+ 1 ~ • lcm -1 'I: - O. thcn any ',()lution of thc Niginal $)._,lcm •h • h is of the form u 11 + i: v.here u, E R~ 1, Lldined in (5). Thu'I the probkm rcdu~·t:) to finding ;:ill soluti~•m t· h) -h O Fortunatcl)", 1he rcdui:1ion of •1 to llermite normal f~•rm 11 \Írtually O. For, 11 pro,idc-; m v.ith a compktc set of 'Kllutíons to l.t h.b the follov,.ing form

,,,

(6)

O .

O 1 •

•o• . •o•

.o•

o o

ooo o

O 1 •

• O • •

• O•

000 ••

0 1 • . • •O•

nw, r • O

•

O . •. O 1 •

(Thc a~h:n~ks 1nd1c;1t,: un~¡,,:c1ficd cntri~.) 1í we w rite out thc S} stcm Jl.t O C'lplicitly. wc ha\c

º· - º·

(7)

In thICm Ax O. In othcr y,or,.b, a \,:t,:lor t:an samíy the homolcneou~ sy,tcm A.\ • O ií anJ onl)· ií it !Jtisfics thc cqui\aknt S)Slt.'lll (7). But J'> Y,C h,ne k'Cn, x );ltl.,fi~ (7) ií and onl)' ií it i, of the fonn (10). \\elcm

27

Ar,pl. -:.11,,>11. -;fth, H tN ,1,-N,-,rma/ form

Exsmple 7.2

259

Find a11 solution'> (if any e·o~t) of thc S)Sttm of cquations

+ hi + .t,+

2.,, (20)

2.\¡

+

X ;i

+ x,. • '.!O• +.\'-4 - 12.

X:

•

7.

Thc augmentcd m,mi,¡ for 1he system (20) is

'ºj

'1 21 01 1 12

(21)

[

0

1

2

1 O O

1

1

14 •

7

Pcrform the following scqurnce of elemrntary row operations on the mutri:ii: in (21). (a) lntcrchunge rows I and 2.

(b) Add - 2 lmlt!i row I to row 2. (e) Add -2 times row ! to row 4. (d) lntcrchangc rows 2 and 3. (r) Add row 2 to row 4.

( f) Add -1 times row 2 to row l.

(g) Add -1 times row 3 to row 4. (h) Add - 1 tim~ row 3 to row 2. (i) Add row 3 to row J. T he rcsulting matri,: i~

(22)

O -12 O

In this case r - 3 but c 4 - "•+I - 1 cquations in (20) has no solution

Enmple 7.3

-•j 18

1 -1 -4 0 0 1 ,¡,t

O, and hcnce the systcm of

\\'e rcfcr to faample 6.3. Suppose that 1hrcc citics, c 1, c 2 • and l'J, form a doscd cconomic umt, tradmg only among thcmsehes. A~umc 1hat cily c, !,dh what 11 produces at an a\l:rage price of .t, dollars per unit produced, i • J. 2. 3. Also. e, buys a fraetional amount of its totul purcha~ from e, at an a\erage price of ;,;, dollars ~r umt. Lct this frnctional amount be a.,, i,j - 1, 2, 3. In faamplc 6.3 we disco,ercd that gi,cn the matrit A whosc (i,j}

cntry is a,,. lhc conditions on a pricc \CCIOr x - (.l"1, :c2, x 3) lhat in!>urc that thc ecomllll)' Í\ i.tablc can ~ expr~ by a i.y~tcm of

linear cquations (23)

A):c - O.

(/3 -

Suppo,;c thJt r 1 obt.1ins 3 of ils purchasc-s from itsclf and i each from c 1 and c 2 : ci obrnins 1 from itself. ½from ci, and 1 from c 3 ; and r 3 obtains ½from itsclí and ½from c 2. Thc sym:m of cquations (23) thcn be-comes

(24)

-½l -!¡ [ o -½

-11

-j

0.

X

½

Wc reduce thc cocílicient n111trix in (24) to Jlcrmitc normal form. (a) In ordcr to simplify thc computations. "e lim multiply row 1 by 6, rov.s 2 and 3 by 4 and 2, respecti\cly, to ob1:1in

[_; -; ::J. O -1

1

(b) Add row I to row 2 to obta in

(e) Now mult iply row 2 by ½, add row 2 to rov.:. 1 and 3 and muhiply row I by ½10 produce che matri(

o1

o

-•1o

-1 .

which is in l lcrmitc normal form. For this matrix, 11 1 • 1, n 2 2, r 2, and thc S)'Stcmcorrcsponding to (7) becomcs X¡ -

Xa - 0,

x, - x, - o. Thus thc solutions to (:!3) consi!>I of all \eclors

for any \'aluc assigncd to x,. In othcr v.ords, whatc\·er a\cragc pricc c1 chargcs for 1hc goods II produces, and c- 1 mU)t chargc thc samc.

e,

Example 7.4

Thc p are to be ,1uC ~ thal thc equation1> in (33) imply 1ha1 if B 1.!> any n•M(uarc matrix lklti.!>Í)-ing (34)

thcn B X. In othcr words, if a matriJt X c,ists Y>hich "'1.tidie uid to be noruing11lar, or regular, or a unit mlllrix. Thcrc are a number of

ekmcmary resuhs concerning nonsingul.ar matrices. Theorem 7.3

(a) Tlrt prod11c1 ◄ B o/ 1wo noming11Jor 111/l/rictf A and B ,r no11•

singular and (AB,- 1

(35) (b)

J/ A

•

n- 1 A- 1•

iJ noosin,r11/ar, tht•n soª" AT and A- 1, and

(36) (37)

Proof (a) Since Y>C are assuming 1h:11 both A and B are non~ingular and that thc product AB is defined, it follOY>.!> that both A anJ 8 are 11-square. (\\ hy?) Then (ABXB·· 1A- 1)

A(BB- 1)A- 1 Al,..1- 1

,..

A. I

·I

and similar])·, llencc, o- 1 -1 1 ~111\fie) thc e Herm,w N.:,,~ 1F,i,m

211

and 2 of A But E itsclf 1~ obt.imed frnm 11 b) mtcrd1:1ngmg roY.~ 1 and 2 \IÍ 1 lf Y.C intcrchangc íClY.~ 1 11nd 2 of EA Y.C of couN: ff'COH'T

A

(38)

E(EA) - A

By SClling A

t J in (38) Y.C obtain EE • 13

(39)

andheoccE • C'. (d) Lct

E

ºj

1 O O I O . [e O 1

\\ e compute thm for nny :l X n mntrii. (tnke n - 4 again).

11,us for any 3 X n matriJ. A. E4 is obtaincd from A by adCd con~lrut.:ti\dy 10 ,kcide v,,hether A is non$ingular and, if MI, to produce A 1

Theorem 7.6

ÚI .◄ M t1n m-,¡qt1are ma1rl.\ and supporr A is red11ingular; lhc proof1~complc1c \\ e 1.1.l..c a clo-..:r loo!.. at 1hc impliet of 4 ro..,,~ oí 18 are lmc.1rl) 1nJcpcmknt. lt follo..,,, that p(AB) · p(A). Now apply this 1nequaht) w1th the role of I being u\.en b) AB 1md thc role of B be1n¡ takcn b), B · 1 _)

e/,,.,,-,.,,,,,.

t 22

Define an t',,lumn orera1inn rrec1~ly as m Defirut1on 6.2, rcpbcin¡ cach occurrcncc oí thc wonl "row" by thc word "column." Define c~>lumn a¡un-.1/enu- of t..,,o matn ~ 1n prcci!>tl)' thc samc w:ay (a) Show that períom1ing 11n clcmcntury column o¡,eratlOn on thc 1dtn111y- matmc: rcsulb m 11n eJ.emcntat) matri,c ns defincd m Dcfim11on 7.2

(Jl mt: Obscr•c t h.ll a n clcmcntary column

opcfllt1on on •I 1.:. thc 1>amc a.:. an ckmcnt.ary row opcration on Ar.) (h) !-.how that 1í A i~ an m X II matri,c and E i, an 11-!K¡UJrc ckmen10.ry mam ,c. then AEi\ thc mam11 obuuned from A by pcñor01Jn¡ the clcn~nL.lt) column opcr.111on o n 1 ..,,h1,h cormrond~ to /;;_ (Colum n opcrallon~ mu\t be a,.;h1eH•d l>) mu\t1plu.:at1on on the n1ht.)

t 23 Sho..,, that 1f A 1) an 111 )( 11 matm. ;md p(.'11

r, 1hcn I can t,c rcduced by a ),Cqutnct oí el 1 and 11

opc:rnle, rc-\ptcli\el)'. Then, of cour.c-, .l 1 ~ O, x2 ~ O, and thc proc.lur.:tion rcquircmcnt\ lx-comc

10x 1 + 20xi 30x1 + :!0.1'2

(1)

~

800. 1600,

~

50.l¡ · 20xi 2: 2000. Thc cost of opcrating thc tv.o plunb is (2)

"'h,:re x is the :!-h!Ctor .\ ,.. (.\ 1..lJ). In othcr v.orJ\. the problcm Íl> to minimize thc function/. subJCCl to thc r~trir.:tions in (1). Thc incquahties in (1) define a rcgion in thc lirst quadrant {i.c .. :t 1 ~ O, XJ ;:: 0) of thc x 1 , Xrplanc, namt-ly, that rcg1on con,istmg oí J\I 2-\ecton x 11,hich ~ltil>fy ( 1). \\ e c,m \implify thc incqual111~ in(\} by di\iding cach of thcm b} 10 to obtain an ··~ui\alent" S}Stem of incquahti("): X¡

+

2.lJ ~

80,

3x 1 +'.!xi~ 160, !h1 + h1 :2:: WO.

(3)

Thc function/can alM> be modifictl b) di,iding by :!0,000, i.c., it is dcar that thc mínimum oí¡,,,.¡11 be L.noy,.n as !,OOn as v,c L.nov, thc mínimum of thc fum:tion (4)

\\,:: ,·an graph thc r.::gi\ln dcfinl'd by lhc inequahties in ()). COn\irJer th.:: linc .t 1 + 2,i 80. Thi\ 1;;111 be rev,ritkn J\ (5) Qb,, iou~ly .,· 1

+ !.,:

~

kO is cqui,,alcnt to

(6) Thc lmc (5) has lhc follov,ing gr.1ph

(7)

1.1 f l11uod1JC/J :in

w LIIN(t( Pfct¡,iJtn

fil

281

Thus the point\ delined by (6) are precise!) 1hoc;e 2-H.'Ctor-, x ...,hich he abo,e thc hne m (7). Sinl'C both .t 1 anJ x 1 are restrict,:d lO be nonncgati,e, ...,e are only imcrested in thci-.e points abo,c tht: linc ...,uh nonncg.1.1i,c componcnts, i.c., thc intcrscctiM oí 1he arca abmc lhe line \O.Íth thc lint quadrant. Ji we similarly graph thc second and third lines in (3). we ...,¡11 again obtain ccrt.iin rcgion~ in thc fir.t quadrant (IÍ tht! pi.me. Smcc all thrcc inequalit,cs must be satisfiOO, the required region ...,¡11 be thc intcrsc-ction of the three regions boundeJ by each of thc linc~ !.Cp,uatcly. T hc picture (x1, x2)

(8)

in (8) tell:. us that this croS-$hatched rcgion will consist of ali ,ccton in the unbounded rcgion abo\i~ or on Jll thrcCCt1tlll\ ...,e will dc,clop 1hc neccssary mcthods for minimizmg thc íundil,n g in (4) for tho:.c '.!-,cctor:. ...,hich lie in thc lll(fü·atcJ rcg1on. llo...,c,.:r, .111 ad hoc argumcnt can be gi,cn for thi:. problcm Considcr an arb1trary ,cctor u (u 1,ul) in thc rcgi,111. Thc lir1t incqualiti~ in (.') must be ~Jti\ficd h) o

'""º

t11+1Cli;?-80, 3a1 + 1,,~ ~ 160.

4(u,

+ 11,l > 240.

ChlptM J

/ Co,,ve¡,,ry

212

Now, g(a) - a 1 + ªi and g((40, :!O)) 40 + 10 60. Thcreforc ií u 1s ,my poinl in thc rcgion, thc \;iluc oí gata c\cecds (or cquals) thc ,alue oí g at 1hc point (40. 20). \\ e can conclude that the fonc. tion g (and hence thc original function /) assum~ its mínimum ,a1ue at the point (40, 10). This means that plant I should opcrnte 40 days and plant JI should opcrale :!O da}s in order that thc COSl oí producing thc required units be mínimum. The student might lind it Lntcrcs1ing that (40. 20) is thc only point in thc rcgion wherc g lakcs a mínimum ,aluc. (5cc Scction 3.3.)

The conditions in thc prcceding example impl) that the íunction/ achic,cs its minimum ,aluc on a ..cerner·· oí the region in (8). Funhcrmorc. it is ckM that thc region in diagram (8) has thc fol!O\\Íng propcrty: gi,en any two poinls in thc rcgion. thc cntirc linc M:gment J0ining thc 1v.o points is complc:tdy cont,uncd in 1he rcgion. \loreo,cr, thc region is boundcd by ~trnight linc scgmcnts \\ e shall dc\elop general mcthod5o for mimmiling an 2. \\'e motl\alc thc formal delinition with thc followmg examplc. Ex11mple 1.2

A,; we saw in Scction 2.1, ií II is a 2•\CClor and t is a numbcr, then tui~ a \CClor which points in t he same direction as u ií t is posiliH:. and m the oppositc d1rcct 1on ií 1 1s negatoe. \1 oreo,er, thc length oí w can be obrnincJ from thc lcnglh oí u by muhipl)ing the lcngth oí u b} lhc numerical ,aluc oí 1.

Bcforc proc...~Jing. the ~tudcnt \hould re\Ít:\\- the material on \l'Cton in Section 2. 1. Lct a be it \edor m R~ Toen for any \aJuc oí the numbcr t. thc sum of the tv.o h:ctors a and lu can be obtained

3 1 f

fntrodur:-r,vn to L1~ 11 Prag,,, ,..,. 1g

213

by íorming the diagonal oí 1he p:uallelogram constructcd by 1rans• lat1ng 111 parallel to itselí until ilío initial poin1 coincideío with thc endpoint of a. As , \aril:'5, it i,; clear that 1he '·head" of lhc \CClor

(9)

a + tu hits C\"ery point on the line f through the point a in the dircc1ion of u. This elcmcntary gcomctric examplc in R 1 suggcsts thc íollowing general defini1ion

Defínition 1. 1

(10)

SrraiRht li,u Lct 11 anda be füed \CClors in R". u .,_ O. Thcn 1hc line through a in the direc1ion of u is the totality of points x of thc íorm

.\" • a+

/U

as r takcs on ali real \alucs. Thc formula (IO) is called lhe puramerric N¡1K1tion o( thc line. (The ..:1.riable I is called thc parameU'r.)

Example 1.3

Find 1hc parame1ric equation of a linc 1n R 1 which goes through thc poinh {l. 1) and (J, 4). Consider this diagram

{3, 4)

(11)

Let a be the \CCIOr ( l . 1). lt i!o clear from the díagram (11) th:n the (u1, u 2) must ha,e lhc propcrty that u+" requircd ,ec:tor u or equi,akntly that its coordinat.:!o 11 1 und IIJ mu~t s.ati~fy the equat1ons

(]. 4),

!11

+ 1-

111+

3 1 - 4.

that is,

3- 1

U¡ -

lli - 4 -

Thus u

]

2 3.

(2, 3)and thcrcforc: any point xon the linc (can bcdenoted X

(1, !)

a+ Obs,ene that if t

O thcn .\

+ t(2,J)

111.

a, and if t - 1 thcn .t

• a + ((3. 4) - a ] (3, 4). llence the line through the poin1s (l. 1) and (3. 4).

Definition 1.2

•

a

+u

t docs mdecd go

Standard inMr prod11et Let u and v be ,cctors in R~. u . , L'~). Then the (mmdard) inner (11 1•• , u.,) and e (t" 1• prod11ct of u and v i!o the numbcr

Thc inner product is denotcd by (11, e). A ,ery 1mportant propcrt}' of thc inner prod1Kt is duc to the faet thatiti~~•mplya~umofnumbcn..SuppO!ICII {11 1, .w.)· R• T hcn

(u+ 11,c) ... ~(u,+ 11·,)c,

~ (u,c, + 11,c,)

,t

~ U,t· + 11·,c, + (11·,C).

(ll,C)

A sim1lar computation )ield.,

3 1 J /Nrvducr Cll'I

ro

.,- Pr, gr

q

zas

Also, if (.t is a nurnbcr. wc can chock that (tw.t:)

.~ («11,)·t:,

to,(11,1.·,)

and that (u, ov)

lí

11

(u 1 ,

U1. u1)u

112)

n(u. v).

is a \·octor in R 1 , thcn (u, u) 1

2

-

(u 1 • u 1

+

(u?+ u;)I 2.

1 1 11) is just thc length of 1hc \C'Ctor u Suppo!loC th,H II and ¡; are two \~to~ in R 1 \\e know from clcm t•f thc ~1Jc-., (u, u) (t-,t"), i-. cqu,d l\l thc squarc of thc length of thc h)polcnuse, (11 + e, u T t.'). Jf 1,1.c writc out (D) c,plicitl)' in lcrms of thl! componcnts of 11 (11 1, 11l) ,ind t· (l'i,l' 2), \\C ha\c a,¡ neccs\ary anJ sufficicnt cond11it1ns for 1hc pcrpcndicularity of II ande thJt

Expanding ( 14) and cancclhng common tcrms from bo1h siCe

+ X1

e, th,n thc equation oí thc l.

lt is clcar from thc geomctry that u - (1, 1, 1) is mdced perpendicular 10 thc pla.nc.

Definirion 1.4

lly¡H!rplanit Lct II be a lh:cd ,cctor in R" and lct e be a numbcr. Thcn thc totaluy oí ,cctors x in R' that satisfy (u, x) • e

is called a h)ptrplon~. Thc ,cc1or u is callcd the normal to !he hypcrplanc.

Example 1.7

Find thc dist:mcc from thc: point o - (a 1• a,) e R, 10 thc linc ( c. Thc ,ec:tor u is perpendicular to thc v.hosc cquation is (x, u) linc /, and thus thc equation of the linc l' through a perpendicular to / is gi,cn by \ a+ /11. Now, thc point of intcrscction of f and f' is dctcrmined by that 1;aJuc oíl for v.-hich a+ tu satislic:s thc cquation of thc linc (:

(a+ tu,u) - c.

(19) Wc chccl th:H

(u+

fll,

u) - (u, u)+ (tu, u) - (a, 11) t(u, u),

+

and hcncc (19) bcc:omcs

(a.u)+ 1(u,u) - e

Thus thc point of intc~tion oí t :md f' is (20)

;: a+(c:¡/:)"))u. Thc distance d from a to thc point in (20) is thc kngth oí thc ,cctor

Chapt,:t 3

C

v,: • f'/

290

ti,

lh,1t1s,

d2

•

(: -

-

(

u,: - u)

("inh x oí the íorm .t = (1 - ()a+ 1h. 11>here I takc-; on a\1 \a]uc c.

Sup~ that (u, :e') ¡3 < c. lt is obúous from thc diagram that 1he entire linc segmí'nt joining .t 0 and x' h~ on the ~me side of 11. f"in.t note that .in) point on the liní' ~gment can OC ·,Hitten (1 1)., 0 + n', and so (11,

(1 - t)t 0

+ 1.\1)

+ I(u, .,')

~ (1 - l)(u, .,.,) - (1- t)a+ Id.

Toen it is ea\y 10 scc that therc c~i)tS a numbcr t i.atilf)ing O$ t $ 1 such that (1 - 1)n 1d - c. for. if y,c sohc 1h1) cquation for t, .,.,e obtain

+

• -

anrJ sinee d

J

< e < o. 11 follov. that O $

:

=; $

I, In othcr .,.,ords,

\\C can find ;1 p.,int on thí' hnc .\ol'gmcnt joining ., ,, anJ .,• .,.,hich lies on the h~pc:rplane II But th1~ 1, 111 Cllnnict y,1th our omcnat1on that th..: i:ntirc lin..: ~gment J\lining , 0 and _,• li~ c;,n one ~ir.le of 11 Thu~. m RJ the p,.1~i1i,e l'p:n half-,pacc i, pre,;1-.c-l) the \et or ull pomts which he abo,e the h)perrlanc. Con~ider a hypcrplanc II c,rth\1gon,1l 10 thí' ,·n:wr u in R~· (11, .,·) • c. Lct , 0 be thc intc™--ction poinl of thc h)pcrplanc and th~· lin~

f 1hr 1, i c., x' hes ··farther out.. than x., on ( in 1hcd1rection of u. Then (u. x') (11. >..\"o) - >.(11. xr,) - >.e > c. Thus thc cquation of thc hypcrplanc through x' orthogonal 10 11 is of the form d.

(11•.\'.)

y,herc d > c. We can summilrizc thC'lC ob!.cr,alions in a somc...,hat diffcrcnl .... Jy. Thinl of thc hn>erplanc (u..\'.)• e a~ being mo\cd pa rallcl to 11 mO\cd in !he d1rcct1on of - 11, then, ofcour:.c, thc right side decrcak'\. Th~ rcmarls provide us with a gcomctric method for finding thc maximum (and/or mínimum) \alues of the function

""hc:rc x ,aries O\Ct !iOmc rc:gion in illustratc thís in thc nc~t C\'.ample. Example 1.8

R~

bounded by hn,erplanes. \\ e

Find the m,1\'.imum and mínimum ,alu~ of the func1ion /(x) • h

1

+ ?xi,

as x ,aries o,er thc rcgion defincd b)· thc incqualiti~ (25)

X¡+ .\"2 XJ -

4\'.1

\

.\¡

-7, 1,

X1

-J.

Ch4'p1cr 3

¡ Con,..• ,~

294

Wc fi~t grnph thc rcgion delincd by thc prcccding incqualitics.

(26)

11 is ckar from thc diag.ram that the incqualitics in {25) define the shadcd region Y.hose ,crticcs (i.e .. comer poinh) are imhcJtcd. Ne,t lct u (2, 7). so that/(x) can be Y..rittcn in the form/(.t) (11•.,'). \\ e reproduce thc shadcd rcgion in (:6) 1ogc1her Y.ith a l)pical linc (u, x) • c.

lf Y.e imagine thc linc (u,.,) .. e a~ moúng acro,s thc shild«I rcgion in the direction of II and con .. tantl}' raralld to 11,;elf, .... e knoy, from the di'loCUS)Íon prccct1rator) mou..c rcqu1r~ 7, 10, and 4 umn oí 1,1wmm\ 1·. üOd :, n.',l'IC\:ll\d). ,~nmc:rt 1.1l íood m1,1urC\ -1 und H,:onwm

r..,,.,

lhC\c Hl.lm1w, in the folh.1..,,mii amounts per •,;:i.:l, nu,ture I lOOlJm5 ~. 2. and I umb oí.,·, 1·, and :, rc(b1t1vc: (e) open nepu,·c and open pos.1t1ve; (d) closed neg.ati,·c and doscd pos111,-r halí-spaccs, rcspcct1vcly. Show that thc hnc segrncnt Jommg u and b mu:,t mterscct the h)J>Crplanc.

15

16 Find al\ \"ttton u - (u 1, u~. UJ) m RJ which are ortho¡onal to the vecton (1,0, 1) and (1, 1, -1). ( Hmt; l f u - (ui.uz,uJ) then thc followmg equatioru. must be sausficd; U\ +

111

U¡ -

+ u2 -

0,

113 •

O.)

17 Two hyr,c:rplanes are said to be pura/fe/ 1f thcy are orthogon:i\ to thc samc vector u . \..et u (2, 1, 3, - 4) and find thc equat1on oí the h)r,c:rplanc m R 1 parallcl to the hyperplanc (u,x) .. 5 and conta1nm¡ t hc point (1, 1.0, 1).

,e

Let u • (l. 1, 1), l' - {I, -1. 1), and ,., .., (-1, 1, 1). Show that the only ,-cctor 'f (x¡. 1'2, x.1) in R1 that 1) orthoaon:d to tach oí u, t·, andwbx-0

19

(a) Le! u - (1.1.1). e• (1, -1, 1), and ,., (-1, 1, 1). Show that thc thrcc h)perplancs (u, x) c1, (i;. x) • c2, and (w, x) c;s mtcn,cct m prcc,scly onc pornt, roe an:,, valucs oí t:1,CJ,l"l,

(b) bnd thc J)Olnt oí mtersccuon 1r c 1 - l. ,2 - 2, and ci - O

20 (a) Let u and b be two \'CCtOrs m R 1. Let 8 be a numbcr sa11~f)m11

O < 8 < I . fmd thc coord,nate! ora pomt wh1ch 1s 6 oí 1he di,lan.;-írorn u to b alon¡¡ the hnc se11mcnt )O!nm¡ u and h (b) Let " (1, 1) and b (2, J). hnd thc equation of thc lmc wh11:h 1s pcrpcnd1,ul 3. EKample 2.4

In Examplc 1.7, wc found a formula for thc di~tancc from t hc poinl a • (a 1, a 2 ) to 1hc line l whosc cquation is (x, u) - e in R 2. In deri\ing this formula, we us.ed the ob\ious gcometric fact in R2 that the distante (i.e., shortc.'St distancc) from a to l is along thc Jinc perpendicular to l passing through a . This fact is obvious 1n R 2 and R 3 , but requirCSJUSlification in R•. Thus, supposc thal (x, u) - e is the equation of á hyperplane in W, and Jc1 a R•. Le1,.,. be a \C-Clor in R • and con:.1der the parametric equalion of 1hc linc l through a in the dirtction of c:

x • a+tu. Toen l inters«h the hyperplane for the "alue of t so chosen that (a+ tv, u) • e, that is (a. u)+ t(v, u) • c. In order that there be a point ofintersect1on. assume that vis chosen so that (11, u)~ O(i.e., v and 11 are not orthogonal). \\e thcn obtain

,- "T/:>".>. and thus the point of interscction is

x

a+

c_T/:;">t1.

The distancc betwecn this point-' anda is (17)

(a + e

(u. u) )

·(v, u)

v - u

le - (a, u) • (e, u)I

Now, by the Cauchy-Schw·arz lnequality, we know that :(11, u). S ·v; :rnd hencc (18)

u

Cflilpter 3

I

C?l'l~MY

301

lt follow-. from (17) that thc distance from o 10 thc point oí intcr~'C.:tion oí the hnc ( and thc hyperplanc is al-...a~ al !casi

le -

(19)

(o, u)1

On thc other hand, we know from the cac;e ofequah1y in 1he Cauchy• Schwar;: lncquahty that ií u and v are lincarly dependent (i.e.. if i: is a mult iple oí u) then (18) bccomes equality. The numbcr (19) is now exactly thc dis1ancc from thc point o to the intcn.cction oí thc linc t and thc hypcrplanc. whcrc ( is in thc dircction oí thc normal to the hypcrplane A~ nn cumple, we find the distance from (1. O, - 1, 2) 10 the hypcrpl::me defined by

In this

C3!,C,

a - (1.0.

1.2). u - (2.J.

1.2). ande - - 1.

Th O, 1t íollows that

e>

O.

1

In order that y,e be able to makc 11 more systematie anal)sis of probkms in linear programming. we introduce an importan! classifkation of sets.

Definition 2.1

Conru set A set S ho.,., that íor any /, O 5 t 5 l. ((1 - t)a

+ th) E

S.

3.2

/ G80f'MlrrymR"

311

Smcc u - (1 - 1)11

((1 -

t)a

+ tb) -

+ m, we compute th::it u • ((1 - 1)(1 + lb) -

((1 - t)u • (1 - t)(a - u)+ t(b - u) $ (1 - 1)(a - u) t(b - 11) • (l - 1 ) " - u +1 b - 1 1 $ (1 - t)r+ tr

+ tu}

+

+

T hus ((1 - t")a lb) e S and Sis COO\'CX. (d) A hypcrplanc and a half-.spacc are always con\ U set.s in R•. For, suppose that (x, u) • e is the equation of ::i h)·pcrplanc TI and that a and b are two mcmbcn of 11. This mcans that

(t1, u) - e and

(b.u) - c. Jf0 $ t $ 1. thcn ((1 - 1)a

+ th, u) -

+ t(b. 11)

(1 - r)(t1, 11) • (1 - t)c+ te

"

hcncc ((1 - t)a + th) e TI Wc \·crify thnt thc ncgati\C open hnlfspacc of rl is cOn\CX and lea\ C the \Crification of the con\C.\ity o f the rcmaining types of half-~paces as exercise~ for the student (see E- O.

Example 2.6

\\'rite the sy.stem of incquahties

2x 1 +x, -

x3

S l.

+ 2x2 ~ J, + X ;¡~ 0, X2 + )X;¡ 5 -1 X1

X1

X1 -

in the form (26). The second incquali1y can be rewriHcn in the form

-x 1 - 2x 2 s; -3. lf "e define 1he 4 X J m:itrix

A

2 1 _,] - 1 -2 O ,

• [

1 -1 O 1

1 3

thc: 4-\0CIOr b • (1, -3, 0, -1), and the 3-\0CtOr x - (,,., Xz, ,l'. 3), thcn thc system of incqu:ilitics has the form Ax S h.

Theorem 2.4

.

(a) /f!í. is afamily of suhsets of R ~ anJ Md1 SE~ is com;e.t, rhen

,,ns

5)'5/em

b, x?. 0.

,1 mutrix und b

R"', ltt S be 1he ut o/ soh1tion1

b, ."f?. O

Ax

(37)

(a) 1/b - O 1hen x • O ir the only extreme poml o/ S. (b) // b ;,l O und x ES then .t > 0, SUJ' x,, > 0, j l. . . , p. und 1he rrmui11in,: n - p rompo,,en11 o/ x ort' :ero. A 11efe1rary und ~11fficir111 nmdaion thot thiJ x be cm extrt·mc ¡,oi11t o/ S if /h¡¡/ thecofwm1r A''11•. , A .,. o/ A be li11Mrly i11d¡-p.•11dent. (e) Vo extreml!poi111 ofS ay A'•, , A ••. there mayor may not be an ei:.trenie po1nt \: E Sv.ith precisely the J)Cbl11\e componcnb .v,,, ...• x,,. But if thcrc is. thcrc is at most onc. For if y e= S and lhc posi1i,c componcnts of J' are prcciscly )',,. , )',•. thcn b - A .\'

anJaho

b

y,,A

ll cnce, cquatmg thc two,

1

+

+ y,,A '•'

3..2 I

GeomewmR"

319

and the linear independcoce of A '•1. .. , ,4 1'•1 impli0)1· ti,c solution. and. moreo,cr, the two columns (- !, U 11nd (!. O are elcarly lincarl)· indcpcndcnt. llcncc. according to Thcorcm 2.5, (0, O. l. 2) is an e,trcmc point oí thc set oí solutions to (44) (ii) \'1 - X3 • O:

Thus x 4 - j and X:2 • !- Also, thc two column~ (0. 1) and (!, O are lincarly indcpcndcnt. and hcocc (O, !. O, f) is anothcr c,trcmc point (iii) x, ... x 4 O: by a sinular cakula1ion, "'e find that x 2 2 and ."f J ,... 3 and hcncc wc do not ha,c an c,treme poirll (sincc .t3 is ncga1i,c). (i,)

x 3 - O: in th1sca !xi,

X¡

2.11·

ll

hi

~ 1,

14 lfSandfare!lebof1eoon1nR"amJ0'1~anumbcr.dclincS [x

i·

lf:::5/\JE I , and of thc ~t of \CCh)h a'•... . aA is called the com;ex polJhedro11 spa1111rd by t1 1, .,,i". \\ e denote this lauer sel by //(a 1.. . . tr). (In Fxcrci,;c 6 the !>tudcnt is askcd 10 f)fO\C that 11 (u' ..•.. a!) is indeed a comcx set.) O~ne that /1(,1 1, a 2) 1s what ""e ha\C prc\iously refcrred to as the line segmcn1 joining t1 1 and o 2 • Thc main result of this scction is contained in the follo,,.,ing theorem. Theorem 3. 1

Le/ A be 011 m X 11 11101,fa. b mi 111-r.:.-ctor 011d x un n•r.:ecror.

s1.·1

S

lf thc

o/ so/uiions 10 the S}'smn

(8)

A.\ • b,

\ ;?: O,

it hoimdcd, 1Ji,·11 Sis 1l1e wm:,·x po/Jhedro11 Umption thesc: ,1\1 lie in lh of moi,t of thc íunc1ions "hich "e study are takcn on at C\lrcmc point~. Thc

Ct.aNor 3

t

ConveJ1,1y

330

question th111 rcmains is this: how do Y.e find thc muimum and mínimum \alue. oí 1he íunction/(t} - (11, x) on the se1 or solutions to the sy~tem oí tintar i11N¡11ali1ies ( 11)? \\ e aDSY.cr this qucstion in the nex1 tY.o 1hcorems

Theorem 3.2

• (u., .... u,.) he an ex1reme poinl o/ rhe Sel So/ soli11ionf o/ (11). Then there exists a,1 m-111ple (11,. 1, ••• , u,.,._.,)such that

Ut 11

(12)

is a11 extrrm,e poínt o/ the .fe/ S' o/ sol111ionf o/ the equali1y form o/ the sysu:m (11).

,,,¡~

Proof Thc equality form of the systcm (11) is

n

(13)

ª••

a.,.¡

wherc b - (b 1••

,

1 O O 0 1 0

o

o oo

b ..). Since

II

o o

º º ['•xx,... 1 0 O 1

l

• h.

+1

: .\~+-

sa1isfies (11). Y. e knov. th11t 1

l •..• 111.

Define the nonnegatiH: numbcrs 11,.+ 1,. . • 11,.+.. by

(14)

\\ cclaim 1ha11/ given in (12) is an e,ucmc point oí S' For, suppose lhat r/ (1 - l)w' + fil, whcre

,nd

3.3

/

Linur func11on:s on C0t111f/1C Pr lyhlldra

331

arcmS',0 < 1 < l.Lctw • (w 1, .. ,11•,.)andv • (0 1•.. ,v,.). Th'" (15)

,-

l, , .. ,m,

i -

1, .. ,m.

and since w,.+, 2: O, we know that

i.e., w E S. Simi!tlrly v E S, and from (14), it is clear thal (16)

U •

(1 - 1)11•

+ IV.

But u is an extreme point oí S and since O< 1 < 1, we conclude that w - 1./, i.e., u docs not he in the interior oí any line segmcnt in S. Sin~ w and tJ are thc same. il follow:. from (15) that 11•,.+• • L'•+", • l. ...• m; for.

k·• +• - b, • b, -

,fi

u0

w,

,f, a,,v, i • 1, . . ,m.

llcncc w' - v'. In other words, we hue pro\·ed tha1 ií 1/ can be written in the form (14) in which O< t < l. thcn kl • ti and hcnce u' cannot be in the interior oí any line scgmcnt in S'. i.e., ,/ is an extreme point of S'. This completes the proof. 1 Theorem 3.2 together with Theorem 2.5 in the pre\ious scction shows us how to lind ali the e' 2x 1 + x 2 ; thc amount of cornstareh arnilable is 50 pounds. He~

i.e., lhc amoun t of cornstarch used must not excttd the amount a,ailable. Simitarly, thc amount of cocoa uscd is x 1 + 2.\' 2 : thc amount a,ailablc is 80 pounds, and hencc

Finally. 1he amoun1 of sugar used is 3x 1 ablc is 90 pounds, so that

+ 2.t

2:

the amount a,·ail-

These threc mequalities forma systcm of linear incqualities togcthcr with the ob,ious requirements x 1 2: O, x 2 2: O. This system defines a con,ex rcgion in the planc with casily computed c'tlrcmc points \\'e !>Ohc thc problcm by 1.... 0 mcthods: first graphically, thcn using Thcorem 3.2. lf wc graph thc thrce line$

Ch,p/,n J

t

COOVMlf)"

334

in v.hich " ,

~

0,/ • 1, ... , k, and

,.t , "' • J. Thcn by (21),

1(t, •,a')

.

f(x)

,~1"1/(a'). Lc:t , 1• /(a'). J - l. . , k. Thcn v.-uhout Joss of gcncrali1y wc can assumc that , 1 is thc largcst ofthc ,, and ,, is 1hc smalles1. Wc compute that '1 -

'1'

.

1

. .

r1 · ~ ", 1 1

• ,~.'1"1 2:: ,~.,,", • /(x).

/(a 1), so that thc function/assumcs i1s maximum "aluc at the c,m:me poin1 a 1• Similarl)'. v.c scc that

In othcr words,f(;r) :5 , 1

•

. ,.,

.

- ,•. ,_, E"1

.

- ¿,.", -s,~.'1"1 /(.,), and thus the mínimum \·aluc of/is/(rl). Thcorc:m 3.2 tells us hoY. to compute ali the e,trcmc points of thc set S of solutions of thc syMcm (22) ,md Thcorcm :u shov..., us that if S b boundcd, thcn any linear function a~um~ 1b maximum and mínimum \;ilue-. at thc e-.:trcmc point~. In linear prl."gr.1mming probkm-'>. thc linear funct1on/ i~ usuall} called thc ol,Jafltt' function.

Ex11mple 3.3

,\ cam.l) company ha~ on hand 50. 80, and 90 pounllS of C{'lrnstarch, CCICOa, and sugar, TC\pe.O kmds of fudge llfC

3.3

J

Linear F11nct,rx1J on Convu Poly/ledr,

335

concoctcd. A six pound box oí thc ñrst kind of fudge requires 2, 1, and 3 pounds cach oí cornstarch, cocoa, and sugar. respect1\ely, while a fi,·e pound box ofthe sccond kind rcquires J. 2, and :! pounds of thc respecti,c ingredicnts. The first kind of fudgc sclls for SJ, box and the sccond kind sells for S2.50 bo:,.. lfow many boxcs of cach kind of fudgc should be madc so that t hc company maximizes i1s proñts'.' Lct , 1 and x 2 be thc numbcr oí boxes of thc ñrst and sccond t)pcs of fudgc, respecti,ely. Then thc amount of cornstarch u~ in prcparing the 1wo kinds of fudgc is 2x 1 + x 2 ; thc amount of cornstarch a\ailablc is 50 pounds. 1-k ~

i.c., 1hc amount of cornstarch used must not exettd 1hc amount a,ailabk. Similarly, the amount of cocoa uscd is x 1 + 2.t 2 : thc amount availablc is 80 pounds, and hcnee

Finally, thc amount ofsuga1 used is 3x 1 ablc is 90 pounds, so th,u

+

2x2 ; 1hc amount a,ail-

Thcsc thrce inequalitics forma system oflinear inequalities togcthcr with the ob,ious requirements ., 1 ~ O, x 2 ~ O. This system defin~ a con\C~ rcgion in tht plane with casily computed e~treme points We sohe thc problcm by 1~0 mc1hods: ñrst graphically, thcn using Theorcm 3.2. lf wc graph thc threc line5,

Ch1pter 3

J Con.-.Jtity

3ll

numbcrs of days that plants I and JI opcrate, rcspec'IÍ\cl)') cannot e~cecd 180, i.e., x 1 5 180 and x 2 5 180, Thu$ ,1re can rc\Oorite 1hc S)Stcm (27) in thc s1and.ud form: (28)

-.l' 1 - 2x2 5 -80, -3x1 - 2.x 2 5 - 160. -Sx1 - 2x 2 5 -200, XL 5 180, X2 5 180,

X1 X2

2: 0, 2:_ 0.

l f we graph the region defincd by this last sy~tem, we find that i1 is a polygon in the planc whosc ,crtices are (O, 180), (O. 100). (20, 50), (40, 20), (80, 0), (180, 0), (180,180).

'%.,, Thc region Sis ckarly boundcd and hcnce Theorcm 3.3 now applies. \\ e compute that g((O. 180)) O+ 180 - 180. g((O, 100)) - 100, g((20, ,O)) 70,

g((40. 20))

60.

g((80. 0)) 80, g((l80. 0)) 180, g((l80, 180)) " 360.

3.3

J

LmH t FunctJon!l on Convex Pol-,htN:Jr,

339

Thc mínimum of g is ckarly 60, which is achic\·cd al thc \·crtcx (40,20). \\'e concludc this scction by shov.-ing how the te'•J O unless i • a(j). in which case a,,)',¡• a.,, .J· But (36) isjust the sum down a diagonal of A • (o,i)- lt follow~ that 1he mínimum cost for opcrating this chain is thc lca~t sum obtaincd as follows. Add thc clcmcnts in cach of thc n' = N diagonah of A, call thcsc sums a 1•• • a_\·· Thc 1c~t of thc numbcrs .kl,. i • 1, ...v. is thc mínimum c~t of supplying the chain oí outlcts from thc warchouscs. (Rccall that wc moJificd 1hc original _.,,·s by d1\iding by s.) Thc only thing that remain~ to be resohed i\ thc proofthat anydoubl)' \lochastic matri1 is of t he form (33). \\ e lca,c thi\ a\ [\eTCi!,C 2J. which i\ accompanicd by a lcngth} hint

3.3 /

341

Lmur Fvrict,onJ Ofl COflvelt Polyhedr.t

Quiz

Ans"",er trut or false. 1

The \CCIOr (O, O) IS3 ton\·Cll combm.at1on oí(-1, - l)nnd (1, 1) m R 2.

2

U Sis a lmc sc¡mcnt in R• JOm1n¡ a 1 and a 1, then S - ll(a 1,a1)

3 The COn\'Cl( rol) hcdron m R 1 ~r,anned b) (1, O). (O. 1). (-1, O), and (O, -1) coMifü oí 1hc interior and boundary of a §(·n ww and cl1lumn. That i~. if R choo)C\ row i und C chlX'l',C\ cl1l unm1. thcn C pa)\ R thc amount u, l f u, ¡., ¡xhitiH.~. it rcprumof11 1d 1 dolla~foralloutcomcs 11 1 , and in gcncrol, C pa)S R a sum of 11,di dollnrs íor all outcomes u1, t .. 1, ••• , k, Thcn thc total amount that C pays R O\Cr 1hc N plays of thc gamc is thc sum

Sin~ therc are a total of A trials. the a\eragc or e.-..pectcd earmng for Ron any g1H•n trial i~

(5)

Now. ir 1,1.1;• set p, • ; , 1hcn (S) can be rcwrittcn

(6)

Thc numbcr p 1 i~ 1hefr,·q11mq of pla}Cí R. Then

10

thc amount

"'º" or l~t b)

E(f) • p((S:)/(S) + p(¡Fi)/(F)

- ¼·1+1·(-1) - -t

In othcr word), pla)'er R will lose an a,·eragt of t of a dollar on any gi,cn tria l. and hcncc in íhc trials he can C\J)CCI 10 lose l of a dollar Example 4.5

Rccall thc game matri~ in Enmplc 4.).

A- [-' Jl J

-4

This matri\ does not ~sess a saddle point, i.e., there is no elemcnt which is smallest in its row and largest in its column. We for an) :r.trateg) y 1 that C U!.CS can be compulcd from (9) by rcplacing x 1 b) 7, 12: - 12X1Y1

+ 7(x1 + Y1) -

+ 7(T'/ + )'1) -

4 - - 12 · 1'V'1 - - 7y1 H

+

+ 7)'1

4

- 4

- 1' This result is mdepcndent of y 1• 1.e., R can guarantee an C" for R for the 1:hoi.:e of ,1rategic.?1 x ;md J . Of cour~. thc qu~t1on arises as to precisely hov, R can play the game u~ing a !>lrategy A A ~tratcgy can be determincd in a number of "ª)'5. For uample, if 111 6, R can decide on a choi..:e of J row on an) gi,cn play of thc gamc by rolling a d ie and pla)ing row i if, appcars on the d ie, i - 1, .. , 6. \\ e mu~t anal)'ZC v,.hat con~t1tutes an oplim1un strategy for thc two playcrs. \\ hat R wants to do. of course. is to find a siratcg)· x• so that his ctpccted v,.,inning\ .1rc a\ l.1rge as pos,sible. rcgardless o f v,hat C docs. I n other v,ord;,.. he wani,. to find .a strategy and thc largest number ll'H (r('presenting v.rnnin~ for R) such that for an} stratcgy >' for C. (11)

\\e can state ( 11) in a somcv,.,hat difrerent v,.a)' · namely, R seeh a stralcg) x • such 1ha1

and ~ N is the large.

\\ e can similarly analFe the bc!>ingy 1 • O. Thus in an) optimum stratcgy, J'• O. \l. e now obtain a n opltmum stratcgy for C. 1,,n(W.ing that y 1 - O. Lct (O.y 2, J'a) lx a stratcgy for C. und suppose R choosn row l. Thcn the cxpectcd winnings for R are

Jy2

+ SY, -

+

JJ•, 5( 1 - Y2) -212

+ s.

Similarly, if R chooses row 2, 1hcn R can cxpcct to .,,.¡n

512 Lct

+ 2y_

- .Sy, + 2(1 • 3y, + 2.

y,)

Ch~ptflr 3

I

Conlfl',,(y

360

For .in) gl\en numbcr rz, thc mO!>t that C can e,pect tn lo~ i\ the 1Jrgc~I of the two numbcl'> g 1ú· 1 ) .-.nd KAJ'l). Since C w,ml'> to minim1¿c thi~ maximum lo!>S, he will choo~ y 2 so that

is kast. lf wc graph thc runction~ g I and g 2 , we ,;ce th.11 the po1nt of inle™=Ction, as well as n~~n g(y 1 }, occu~ for Yi ~

Now, )'2 + J's • 1 and hcncc )':, • f. Thus thc optimum stratcgy for C is .1•• .. (O, f, f). and wr • J./.

The r~ult of !he preceding e,ample is perfoclly general. There is a fundJmental thcorem duc lo ,on Neumann which can bc Mntc:d as follows Theo,em 4.1

lt'I I be mr m X n R'a 5 l. 11+ )'2+ }'i 151•

(32)

Thc systcm (32) v.ill ha\C a nonnegati\C S('llut1on if v.c can oblam a nonncgati\c i,olution to the ~)")tem af equatiom

2}'1+312+511- 1, + 5yl + 2Y, - 1

7)1 1 )'1

+

+

)'2

)'l

l~I•

If v.c reduce this S)Stcm to llcrmitc normal form. v,c find thilt o nonncgali\C solution docs indccd exi5l ond IS, in fact.

Thus thc optimum stra1cgy J• for C is

,.

(0,J, !),

which concidcs v, ith the result wc obt.iined in Example 4 6.

Quiz

An.,,v,·ertn.icOf fo\5e

1

1be Jov,cr \'3.lue oí 1he pme matrix

A•[~ ISR •

0.

-~J

2

The urper volue oí 1he matnx A in Q~uon I h ;3 - O.

3

The matnx .-f m Quc c~mg 1 1'.) 7 Show 1hat man} pmc dcfincd by a matri-.; A. nll thc recnM\'e rOW) and column,, of I can be d.:lcted 1111hou1 affa;tma lhc \·alue of 1hc ¡3mc 8

U)ina 1hc mult of 1hc rreccd1ng C\trci5C, dclcte thc: rttu\1ve ro"")

4

f

G11me Th :,,y

319

and column~ m each of the follov.mg matrices 1md sol,c the íC!iuh..1ng gamcs: (a)[I

(b)

(e)

2 ) );

_,

[f _,1 ~]-

¡-1

i].

~

-2 (d) [-~

_,

i];

(e) 4' 21 3 2 '3] . [

1 1 1 1

9 Lct A be an nr X " matri'I. Denote the inner products in R'" and R ~ by the samc notauon: (,).Show lhat for any \CCl on x • R• and >· E R-, ") • (x, O) - O. lle~ 11' • ~

n~n(x, Ay) $ O.}

11 Show Out if there exbb a ~tratel)' .'I"' ~uch that 1rx- - O e R\ thcn the ,-alue 11· of the aame dclincd b> thc: matm: A is at lea~! 1.rro, I C, .. ~o. ( H mt: ~,¡(:e.Ay) ('l"', . IJ) - (A'.IC"',y)•(O,y) - 0.

l-lencc "

m:n ~'!(.\', .·I)"}

~ O.)

12 Shov, that thc:re Cl(l~U a l•\'CCIOf 'I"' s.uch that AT"I"' •

º· v.hcre

A-[-~ ~ -:1, b

O

-e

1n v.hich u, b. and e are pa.iti\'C numbcr... Shov. that thc samc: 3-,-eaor SDll~A.I'"

O.(llmt:Tf).\"" • (

t1 t,

I

h

r

) ~tion. "e rccon,1der certam ~reci,11 d.1,lart ,,.,ith u, at '>lJE-C /., b :,,,,.', and the probabil1ty th:11 thi-. re'iulh in u, at stagc k I i-. thcn:fore

+

Ch~tet 3

1 Conv( >1I'(

374

+

p,, ,,11 • Moremcr, the e,cnt u, can occur at s1agc k 1 in any of n dimnct w:iys: u, can fol1011,• us, u, can foltow u 2 , ••• , u, can follow u•. Hencc the probab11it)· that c,ent u, occurs at Stdge k i-'JUSI the sum of the probabihtics p.,.t;•1• J • 1, ... , 11, i.e.. the right side of()). The systcm of equations (3) can be summarizcd m a single mntriK• ,cctor cquation,

+\

where P is thc II X n malfÍ!( whosc (1,¡) cntry is p,1 and x' 1' • (:,;'111• x'}' ....• x~1').

Obser,e that in the preccding e,ample. the probability tha l u, is thc rcs.ponsc at the (k l)st stage, gi,en that u1 1s thc rC!oponsc at the Ath stogc, does not depcnd on k, 1.C,, 1he probabilll)' P•J IS indc• pc-ndcnt of k. In this s«tion, we are in1cres1cd in thc ana lysis of C!(periments m which 1he number of outc('lmCS :11 each nial is the same. and in which the probabihty 1ha1 1he 1th outcomc occurs a1 a gi,en 1rial (knowing th111 thc11h outcomc occurrcd nt the prc\-ious tria!) dOCl. not depend on 1he numbcr of the 1rial

+

Definition 5.1

Afurko, chuln, statt spau, tra,uition mu,rix, probubilitJ' di:uri• bution Supposc that a scquencc of e~perimcnts i:, pc:rformcd in which there are 11 ~siblc outcomes, 111, • • , ""· al ench tnal. lf !he prob.lb1h1y p,1 thJ.I 11, occurs at thc (k + l )st tnal. gi,en that u, occurrcd on thc kth trial, docs not dcpcnd on k, thcn thc scqucncc of c,perimcnts is callcd a \farko'-: chain. Thc set {11 1, . . . 11.; is c.illcd the srtl/e spoce. and c.ich 11, 1s callcd a s1a1e. Furthcrmorc, thc " X "matri!( p ......,h (i,J)Cntry P•J IS called lhe lrOIU/1/0I/ ma/rt:c, and thc 11-\l.-ctor ,\ 1 , wh1r.,e 1th componen\ 1i. thc probabilit)' 11 that outcome 11, occurs a1 tria\ k. i O~r,c th,11 thc ~um of thc cntric') in cach column of !he trani.i11(111 matri, /' h 1, ,mJ c.i..:h entry t'f P 1, nonncga11,c. Thu\ P Í'> a column Mr>..:h;"tlC m.1tn, a, dcfincd in Dcfimt1on 5,1. ~tion 2.5. Thc (A ~ lht prnl,.ib1ht) d"tr1bu11on .-•~ 1i. obt,uncd from thc

3.5 J M,,~ovcn.,,ns

3 75

4th probabihty di?>1ribu1ion by thc fundamental cqu:i.tion v.c obtaincd abo\C, namcly.

(4) Thus, scuing k • O. 1, 2..... v.c obtJin :c' 1 '

(S)

-

_\•2• •

.\',31 ...

P.r.' 11 Px i • Px 1',

lfwe replaee .r .'P in the second equ:llion in (5) by its \aluc gi\cn in 1hc firsi cquation, v.c sec that

S1milarly,

x' 31

2 - P(P x'º') • plx'º',

nnd. in gcncr.il, (6) Thus thc probability that a giH·n C:\Cnt u, occurs at a gi\Cn stagc dcpends only on thc initial probab1lity distnbu1ion xq and powcN

of 1hc tramition matri~ P. Thcrc are tv.o general qu~tions wc: wi\l con~idc:r in ~tudying \farko\ chains.

(a) To wh:it C\tcnt d~ thc initial situation. i.c:., .\"º\alfe-et the outcomc of the J.th tria!, for large J." (b) \\ hat prcci-,dy is thc outnimc ;1fler k trü1ls when k i~ larg..:7

Exampfe 5.2

Considcr thc .'.! X .'.! tran\11ion matri\

Thc :r.tudcnt v.ill COO\ÍOCC him~lf lhJ.I if k h C\CO, pl - / !• .1nd if J. i, odd, P~ • P. Thus from (6) v.c l>.

Example 5.3

Assume that

!~ pJr -

A c,i~ts when

p - [~

1

~ b] •

O < b :5, 1. (lñat this i~ indeed thc case will be prmed in Theorem 5.3.) Determine A e~phcitly from Thcorcm 5.1 (b), i.c., from (12)

PA

1

'

A

and PA

(13)

11

A;.

To simplify thc notation, lct :e be cithcr A Por A 2' Then (12) and (O) h;ne the form

P.t

.t,

lf wc write out this cquation m tcrms of componcnts, ...,e obtain thc S}Stem -h." (21) is called afix,·d or ;t ch,iract,•mtll' cector for the matrix P. Thc onl)' ;unbigu11y thJt rcmams is whe1hcr therc is more than onc 1Ha:tor v.hi~h sat1~fie, (21) whcn Pis prinuti,c. Wc resohc this mmor

O.

)'¡

O. Suppo~

: is the zcro \CCIOr, Thcn

o- .~'· ~ (x, -

By,)

-t.., -.,t.>, 1- B,

i.c., B - l. But wc are assuming x '#- y, and hcnce that B < l. Thus:. has at kas1 onc posilÍ\C componen!. Now, Pis a prinuti\c matri~. and thus, for somc pos1t1\e rntcgcr m, p• - S h3S only positi\(! cntries. 11 follows immediatcl)' (scc Ex.crcisc 15) from Px • x that P"':,; " x, and similarly, that P•)· • y, Jlcncc

Sx - x

'"ª

Sy ' y.

\\'e compute 1ha1

(23)

S:. - S(.t - BJ) ... sx-BSy • .'I: By

... =· 1h111 is, :. i~ a fhcd ,cc1or oí S

(24)

P"' From (2]) 11 follows that

Ch,pter

1

COllvr;r,ry

38&

Thus thc right Mdc of (2-1) 1§ a .sum of nonncgati\c multiplcs of 1he po!,1t1,e 11-tupk!, S and not JII the coclf1eient!, are ,ero. so th:11

=i§ a p())!loe 11-\l-Ctor w11h no zero componcnts. Yct v.c lnov. that

x• thus .\- - y.

1

O. This i!. a contradi1,;t1on ,ind

XL Jt

Y•

\\ e are now in a posilion to gi,e a complete ansv.er 10 quest,ons (a) and (b) for regulu Mari.o, chain!>. In ansv.er to (a}. the 1mt1al dis1nbu1ion .\" º' h;1s no ciT«t on thc long tcrm outcomc. The ansv.er to qucs11on (b) is th.it 1hc long tcrm outcomc is prcá,dy thc \CCtor x which is thc common column of thc hmiting matn.\ A _For. from (6) we scc that ., 1 approachc.s t he ,cc1or ,h' 11 ' for !Jrgc k and A.t'º) -

L AJ .\"~ ,_,

111

Thus thc long tcrm oulcome is complelcl)' JetcrmineJ by the tr;msition m:itri\ P. MorcO\er, we can compute the ,ector x w1thout computmg thc powc~ pt of the tran~ition mat ri~. for v.c know from our results that thc common colunm .t of lim p• - A is the uniquc probab11i1y \CCtor x which SJmli~

-·

Px - x. Exampfe 5.5

Assumc lhat thc population of 11>-omcn is di, ided into threc cl,h~s· thO!>C who MC OH:rwcight JI 40, th®' who ttrc undcrwcight 111 40, and thosc who are normal at 40. Call t hcsc thrcc cond111ons 11 1, 112, and 11 3, re!>pccti,cly. SuppIÍC!, are transfcrrcd from mothcr 10 daughtcr as follows. Mothcr

u, (25)

D.1ughtcr

1CY-, 2CY', 10' ,

30''1 'iifé 20':(

15 are probab1llt) vt.-.:10~ m R~ and (J is a numbcr, O ~ B S 1, fJ).t + fJ> i~ a rr0Nb1li1y ,e..tor then (1

3.5

/

M,rlc.:,v Ch

391

9 Thcre e, i~b a nontm1.:tl linear combination of the \.'CC't~ (1, 2, l )and (4, 1, )) "'hich has nonllCgJIÍ\'t: componcnts and PI lea5t onc zcro componcnt

10 Thc Mtirkov cham 1>1th trt1M11ton matm:

is regular.

Exercises

1

ldentiíy the probab1hty ,ecton. amona the follo-wm¡: (ti) (0,0, l):

(b) (-1,0,2); (d) (j,j, I); (Q , 1 and 11 , contam two vamlla :ind threc chocolate cool,.ics, Jumor (rcmc:mhcr h1mJ) tale,; a cool1e from c.1ch J.ir, but bttau~ oí íear of bemg d1\.CO\.:rL-d he puts thern bad.., mten:h.angmg thcrn m h1.) ha.~te. W hJ.t l.) the: probJb1ht) that there a.re two chocolote cookie5 m thc fil'!>t jar aíter thrcc such mtm:hanges"! Aíter a lorge numbe:r of mter,;h.ingi'\, wlut i,; thc prohah1ht) that therc are t10o·o choo.:ola1t" cool1t"S in thc fir-.tJilr? (ll1n1: Thereure) .)t.tte-., 111, 111,andu ,. ror 1he dJ.)\nbution oí 1hc cooli1cs m the 1wo1an that can ame by su.ch mter.:hanges. 11!'\\,e:t;ll\cl) .

U¡

Jarl

_jar ll

1.11

k 2c,lo e, 2o

lt-,k 2c

111

Thc prob:'lbilit) oí ¡¡oin¡ from u¡ to 111 wah onc mterchange is 1, 1he w1th one mtcri:hange is O, thc proba• probab1lit) of gorn¡ from u1 to b1hty oí g01ng frorn ui to 111 1::, (Wh>?J, t'tc. Thu.) 1ht' tran~1l10n nutm is

4

u,

111

p

"'

! ,]. i

:; [~ 111

!

0

The miual probab1hty d~mbution b .'(

- (1, O, O).)

8 Assumt' lhat m a pmball maLh1nc, 1here are four replay bunon~, 1, 11, 11 1, nnd IV. lf tht' rmball 1s hit by 1, u w1ll go to 11 with a rrobJbthty of .J. 10 111 w1th a probabihty of .4, 10 IV w11h a rrobabiluy oí .1, and 11 Y.111 l"O lhat (1, x:) · _ R, I.C., (x2,)') E' R, \Oohich ,mpht!,) f' (,iJ. H enle (ti) ,_ (xi] (e) U pon d111d1n1 an m tcgc:r b) 3, thc:rc 1) c:1ther a rc:ma1nder of O, 1, or 2. llcncc an clc:mc:nt oí .\ l\ cuhc:r m [t.lJ, (1). oc 12), )O,\ {0JU{IIU(2J. Out s1nct [,) ,\ for iln) xE X, 1t b dcar 1ha1 (O) U [J J U (2J (. \ Hcrn:c X • {IIJ U [I) U (2] No"' ~up~ [OJ n {I] 1' ji; lct .( e (ll] n 11). Thc:n (U• .t) R and (x, 1) E R. v.-hKh m1pllt$ (O. 1) R, ""hu:h 1\ ckarly ridKulous Sllk'e O - 1 - -1 is not dl~blbk b) 3 Thu) i0J n(IJ !!, and ~1m1tarl>, (clJn(2] • (l)n(2) • íl

.r .\' - ,.

(d) Hy rar1 (b), (.r1l [1:J 1f and only if(ri, .r:) e R, 1.e ., q - ,.. 2 •~ d1HMblc by ] Now ~urJ'l(l',C .q - Jm1 , r1 ami x2 Jmz ~ ri, v.hcrc ri a.nd ri are O, l . or 2. Thcn .r 1 - x 2 ](m¡ - 1111) ! (r1 - ri). v.h1,h IS di\"l!>ll:olc bt ]. lltnce , 1 - , 1 mu.\t be d1,1\iblc by]. Uut ~1n..:c , 1 ;ind ,, are O, 1. or 2. ,, - ,1; i~ lru than ~. an dcfimtion, 4!

2

False. The brnom1:i1 coeilkient (:). 1

4· l ·2-1

(;) - (n

24.

~k ~

n, has thc \:aluc

-11~)!~! •

Evtr.lmuing 1hc tv.o cocfficients in this qucstion, wc act:

(~)

O!(~11~0)!

1 ~! - l . 11

(;) - J!(/~ 1)! • 1 ,-(11 ~ I)! - 11 Wc h:n'e used thc facts th:it O! - 1 and 1! • l. l t thcn bccomes clcar thal the original question is false smce 1t does not hold for any value of nENcx~pl 11 • 1

3

(O) O •

True.

O!

1

O!("il= O)! • 1 1 • l.

4 False. Thc corred. numbcr of l-sampks oí a 4-dcment set is 43. Th1s rcsult follows from a stra1ght·ÍOJV,'1,rd apphcauon ofThcorcm 4.2(a). 5 False. We know írom Thcorcm 4 2(c) that thc numbcr of subsc:ts of an n•set 1s 2•. S1nc:c in this c:asc X !u, b: is a 2·!.el, thc numbcr oí is zi - 4. Thcy are m fact O, {a ) , {b] , (a, b}.

sub.sets oí 6

r

True. Sce lñcorcm 4.2(c).

False. True. As in Question 2,

Abo,

(:)

,, 11!(; - 11)!

l knce

(:) (' t)

11~! - 1.

A nswen r" Ou,ulS ,nd Sat • KJ ExtrC

9

s

401

True. Wc ag:un appcal to Theorem .i 2. paru (e) and (d). From part

(e) v.e know 1ha1 the numbcr oí r-cornb1na11om, of :in ,..sel is (:).

Part (d) tells us that !he nun1bcr oí ,-sdectinns oí 110 1,-se1 is

(" . ',. - ') .

In order to venfy thot the stalemcnt is corr« I we necd ooly show 1hz11

· (n

Now,

+ l)(n)

Noticing 1hat (n · r - IX" + r - 2) . . (n 1)(,1) ~ n(lf - 1) . (n - , · I) ,,.,hen 1 5 , 5 n, the r~ult follo.,.,s. A more d1rttt bul posslbly lcss c\car ar¡umcn1 can be m.ide from Theofem 41. paru (b) and (e), In par! (b) wc noticc that none oí 1hc t;s may be equal, v.here:u 1n parl (e) S0ml;! may. From lhis 11 follo,,.,~ that there are more sequence~ tlut sausíy the cond111on1 oí (e) than sat~íy thc cond111ons of (b), and hcncc the numbcr oí r-sclecuons oí nn n-scl cx.cceds lhc numbcr of r-!i1ble

ra1nngs are either

An5w,"l't

I

1 4 [:,¡e,c

.$

409

13 This problcm becomcs more tran)parcnt if Y.C thmk oí hning up 4 bo}s and 4 girb so 1h.i1 they altc:rnatc. l f "' ~tan y,nh a bo)', v.c ha,'e four choiccs for the li~t JIOl,ilion, then four cho,cc, oí a g1rl for thc sccond pos1t1on. 3 chotceS oí a bo) for thc thmJ po,111on and J cho1,;cs of a girl for thc fourth pos11ion, cte. That 1s, thcrc are

4 4 · 3 · J 2 2 • 1 · 1 - 4'4! - 576 p()Wblt arrangcmcnts,

16 (a) The compos1t1on oí thc comm1ttc:c: rc:qu1r~ that no more !han onc person be th~n frorn arnc,n¡ th())c v.hod1\hke onc another. lf we choose no onc from 1h1s sub!.c1. therc ,s c~adl)' onc v.ay to choosc thc commmcc of fh·c from the rcmamm¡ lh'e. lf we choo:sc c:t that the diffcrtnce

/(k , 1) - /(i) 1s dwi,1ble by 8, where/(A.) 1/(k

+

3it -

1. Then the sum

1) - /{i)J

+ /(k) ~ /(k +

1)

w1II be divisible by 8. Now,

l(k

+ 1)-/(J..) -

3!1l•I, - 3Jt • 32t.2 _ 32• _ 31,32 - Jª

-

J24()J -1)

- 3u 8. and !he truth of 1hc imphcntion p(,I;) - • p(I,.

1) follo...,-s.

10 Let us denole the thr« ¡iris by ( h Gi. G:1, and the thrte boys by 81, Bi, B, Wc set up the 1ncidcncc n~tm; íor 1h1~ Mtu;111on .... here thc (í.¡) cntl) ,, O 1f G, and 8, han: not b«n introc.lueed an

1

+u1j +9"-+6 - -

- -

.1.'+Je+J"- + 1 J (/.. +l)l J

that is, p(.(.

+ 1) l'> true

An~,1-w.s100rJ,u~.s11ndS~ •,dExttrc,ie.s

1 .5

410

Ouiz 1 lruc. Ir p + q - n, thcn 2

T ruie.

( 6) 3 2 1

( ") - "'-,

p:q.1

p q

-

'(

"'

p. " -

,

p).

6! 6! (') - 3!2!1! - J!2!J! 1 2 J ,

J True.

!.'11 ¡i 4

11 ·2z. 31 - 101!

False.

'

~l

- , 1 + 2z +

J' - 12.

True. By Thcorem 5.1 we l.. now that thc numbcr of Wll)'S oí do¡n¡ 1h15 is 5

(,',) - (;) 6

False.

7

False. See Question 6.

e:,) -TI:;,! -

31 - 6.

8 True.

:i: 2-2+2+-· +2-34. , -1

- - - --,-

171imes

9 T~.

" 2l - !3. i_:: .2ª - iHI f_II 10 tunes 10 True

t:

2

1 .5

- t.; - 1 +2+3-6.

Exercises 1

(d) (1)

2

(2 ~ 0) 2!~~! ~ :-~:•

e

1

:

) _ 5! _ 120. 1 1

(i) Cxpanding by lhe mul11nom,al formula v.e gel

6.

A nsw orJ' /

1.5 Cx11rc1

• 11

1

The coefhnent oí

. ( ')'·

,I_I

x,

1s

problen1 rt thal p((,\'1 U X2)n \'.1) - p(X1 U X:)p(X );

p({X, U ,\z)n X.1) • p((.\'1 nX,)U(X2 n X,)) - r(.\1 nXJ) • r(X¡)p(X.i )

• (p(X1)

t r:

p(x; n x;>

n Xz)

p(X1)p(X1) imphcs 1hat

p(X; n Xi) - p((X¡ u Xz)')

- 1 - p(.\',uX,) - l - [p(X1) + p(X2) - p(X1)p{,\'z)I - [\ - p{X,)J - p(.\,¡){l - p(X1)l - [\ - p(X1)JII - p(Xi)l - p(X;)p(X~.

7

Fal~. Choose 1ndcpenden1 e,~nts X, and Xi such that p(X1) ,'

O~ p(Xz). Then p(X1

Bu1 if X1

n .\"2

n Xz) ,. p(X,)p(X,) ,' 0

- O wc 1.1.ould havc p(X1

8

n X2)

- p(O} - 0.

False. See Q~11on 7.

9 True. Thisisad1rcctapplicauonofThcorcm7.2w1th11 - J,J.., - 2, k2- l ,x1 -½,x2-½. 1 0 False. The proOObihty oí obtninm¡ no hcad1 in J tosses is 1hc probablhty of obU1n1ng J taili m 1hr«: tosses. A¡pin aprtyin1 Thcorcm J, xi ½, a nd xz ½, tlut number is 7.2 w11h 11 - J, J.. 1 - O, J..z ( 0 3J ) ( '2) ' ( '2) ' - "•'J! 23 - 8'·

1.7

Exercises 1

(e) Define !he samplc ~pace U to be

Now define n probabihl)' measurc p by the proOObihty at 1he cnd or t:ach bran.:h:

p(:(11', JI', 111:} :fr. ¡,(:(11, W,B)l) - 3\, ¡,(:(11', B, 11");) - 3':.• p(:(11,B, B):)

,\,

p(:{B, IJ, 11 ): )

u;,

¡,{:(B, 11, B)! )

3\,

l'(:(B, 8, 11'):J

/r,,

p(;(H, B, 8):)

3\,

Anlwtrs /

1.7 éltMCises

419

.. (d) Ltt Ubc defined as in part (c)and again ~ ll!i probabilily mtasure the probabthues li$ltd ai tht tnd of tach branch in the utt bel~.

w

,,

•..

w ;:,

Answ,11s to a -

,.,d

420

(WWWW)

.:.

/8 8.8. 8)

¡;¡

(RR. R . R) (R.R)

(W.WWW)

ij

(88.8.8)

1¡i

(RR.R.R)

1li

(RR)

In thii ucc d1agram "e ha\'C shov,;n only those bnin.:~ "h1ch maner He.nce thc probahlhty that all four marble.5; are oí t hc same. color 1s thc

,um oí the 11bo\'f: probab1litin. 8

11

By T hcorem 7.2 we havc p([k suc«sses m n tnab:)

(:)

:ct.-ci•-•.

9) Theorcm 6.1, r(X1

n X~)

p(X1

n Yí) •

• p(X1) - p(X¡

n Xi).

Tñcrcfore p(T1) -

p(.\',)p(,\'z)

• ¡,{.\'¡J(I - p(X~)J • p(\"1)/1( \"Í}

S1n11larly,

ímally,

p(x; n .rí) ..

r((X1 u \"i)') • I -p(X1U \'1)

- 1 p((X1n ,,>u(.r1n rnuc.r:n .\'ill - 1 - Ir i. i.e, U, i) E!

R. Sinularly,

i. (i,i)E! R

7 True. H \'¡ i~ 1he mother of .\'!, then .\'J cannol be the mothcr oí x1, and no one cnn be hb ov.n mothcr Hcnce tí (>r1, x:,) E R, then ( .\ '1, .\'¡) E! R, and (\',. l,) E! R íor all i 8 Fal~. The nutri~ ½J. - V" \allslies A notan mu

l..

An.hVNS ¡

2 5 OuJJ

433

13 No. Corbtruct an 11K1den,c mamx, 1 for lh1, e,cr.:1-.c a, m [ ,ample 4.6. \\e .,,ce 1n 1h11 ca">C that ca,1blt'. ror e\ample. \UPl'Q',C n - 4 nnd J.. - 2 Thc follov.mg m.11n, h,h prcn,d) !Y.O 1·s m each co\umn and at lea~t one I m taLh ro"' :

"'ª"'

1

_

O o

1 1 1'] 1 O O O 1 O O O

r HOY.C\l?r, chcd,cd.

1

h.l\ no dia¡orul con,bLmg enurcl) oí 1\,

11\

c;m be ca,il)-

14 D.!fine a 5 X 5 matm: .-1 ~u.:h thal 1hc (t,/1 cn1ry of A 1s 1 1f 1hcrc is a one-Y.J)- road from W) r, to CU) r, and O 1f thcrc i, no1. íhcn cach column and each row oí I l"Ontam, preci:.c:1) thrtt I'). \\ e are agam lool..ma for a diat1on¡¡I or 1·). We l..no"' b,1o r umplc 4.6, ho1.1c~cr. th,H any n X n rnatn-.; Y..llh r,rc.:1,-cl) J.. ono 1n cach row and cac:h colunm, J.. < n, ha, u di.lgonJl of l\. Hen.:c A ha~ wch u dta(!OIMI, '-'> corre,pondmg 10 thc 5,permutill1on •: (u,, 1, ,u;. ). In fact, 1 has )et ano1hcr d1ilgonal of 1·)! For )U¡ll"IO\,C \I.C rnn'>lder the n1J.trn B 1 - P, 1,1,hcre P i) thc 1ncidcnce mam-.; o( 11. Then n 1\ m cach ro"' ami each column. 11> hamplc 4 (,, nabo ha, a d1at1onal c:on,i-t1n¡: cntircl) oí 1\. But nny d,a¡;cinal of l's of ll 1, 11 diagonal of 1·, nf ·1, d1~1,n,1 from (111., 1 . . . • , e,•,.,·.). Hen.-c II i, p()\,1blc 10 1our all foc c111c,,. ,h1tmg each cit)" prtu)t'I) once, and 11 rnn tic done in more than one "'ª>

ha~'"º

2.5

Ouiz 1 True . Thc:) are (l. 2. J), (1, J. 2J. anJ (2, ). 1) 2

Libe. The ~um of !he cntrlC\ in thc: 1h1rd to1un1n i, O

3

True . In Thcorcm 5.1, kt"

4

True.lnlhcorcm5.l,ktn • 10,m

5

True. Each ro"" und t,Kh colunm \unl •~ 1

6

Fal,A".Lttn •2, 4 - [ :

5. m

3. and 1

4

4andt

7

gJ.andO•l~

)\O.:hohti, .inJ 8 i) rnlumn ,tocha\tic. hut !-1

~l !B •

lhc:n l"row

ll

1\ J 1i not

douhl~ ,1ochawc. 7

True. In the prc,mu) !.eJ . Hen.;c 1hc term rnnl.. oí , I'' "equ.il to thc krm ranl..

º'23

1

íhc: matn~ -1 mu~\ ha,·e c"nl:,- 1v.o I'!> m cach row :md m cach colmnn. \\ 1:' l..no1.1 that ~ur.:h a rt1.11r1, h.1, a di.1gon..1I of J\, and thu, 11 mu,1 ha,c tcrm rnnl.. n Ddinc r suh'>Ch r1. • I , oí 1he .,.;t of mlegch '.s -· l, ,m) íollQ1.1,; a n mtcgo:r i (_ I 1f :md onl) 1f the: {i, /) cntr) oí ~ 15 l. No T, can be empt), for 1í th" 1.1cn· the r.:aw, ali thc 1\ in A could be covcrcd b) thc lirst J ro1.1:. amJ the 1 - 1 columnl> numbcrcd l. 2, .j - 1, i · 1, ,1. lhenthcl'sm l couldhc:co,'Crcdbyst1- I f- 1 h ~ . contrad1ctm¡¡ thc m1mmaht) oí l. 1'ow suppo,.e no ~DR c,ht5 for the :i.eb T1, • f ,. T hen h> Theorem !U, therc cxi)t l. of thc '>tb, T .,. su,;h that 1he umon o f lhc~ sets contam~ fe1.1er 1han l.. elcmcnb. Hcn.:c m column,j1 , ,/(. 1hc !'\ 11ppcar 1n fc1.1cr t h.1n l.. ro1.1~. !io3Y r01.1\ ii, . . , i,, 1 :5, 1/ < l. . Ci>mider thc fo1lo1.1m¡ '>CI oí lrnes m .J: rov.-, l. . , .1, row\ i1, , ;., ond thc I l.. column\ obtained by dektmg column}ji, ,j, from thc lir~t t column) . T h1s !>Ct oí hnt.o, conum~ ali t hc ¡-~ in I But thcrc ore 26

3l,

~) r,.

,

J

+ct

t-1..

f

t - (l. -11)

( -

(l.. -

y)

CIChon~ oí column~ 2. ), 4. and 7 1.11th ro1.1, l. 4 , ~. :md 6 1, r1 4 ;.'. 4 ,ut-matrn o í o·,. 1.1hcre 4 4 8 7 1. llencc l'I} Thcorcm 4 2, crcry diagonal o í thc maln~ con131n~ a O. l"hcrcíore l.. < 7. Now . llen.:e l. 6 a nd 1.1c can co11t m QUC'il1on 7 •~ ot,1:1.1ocd b) mulc1pl)mg thc ,;cconJ row of thi: TT\J.tr1, m Que,tion 6 b) - 1

9 f al,-e. lí the leíO 11L1tri, 1) the lh:rm11e normal íorm oí a non-uro matrl\ 1, 1hcn the)" are roY. ,:,¡m,,1knt . But lhi~ i~ not ~)1bk. ~•nce ckm,:nt.:ar)· row o('t'rulion'I c:annot c:hani;c a ,ero ID3tn,,; lo a non-zcro 10 True. U)lrtg thc n..,ult of f ,cr.·1,-e 12 m thi:; ,-e,;11011. 1f ·1 anJ B are 111.0 2 2 111.11rkl..., and -1 i~ cqui1akn1 to R, chl·n thc r~ ranl.. oí I b c4u;1\ to thc roY. ranl.. c,í B. NoY., 1he roY. ranl.. oí A,, cuhn O (in Y.hi.:h ca.,._. l B - OJ!l, 2 (in Y.h1~h ,a..c r - B - l!l, or I lf tht! ro"' ranl.. oí r and B &\ 1, lhe) are oí thc íor m

,-[~ :;J.

o-f,~ ~l

!-.in~c I and B are ro"' c,.¡u1\alcnt. 11 h de.ir th.u thc) rnn hlc c4u1,aknt h>- a t)l'IC (111) or,:ratwn onl). 1e.. rnul11pl1,·atu:>n of ro\\- 1 b, a constant But 1n orJcr lo pr,..crH' the 1 1n tlk: (1 , 1) po-.1lion~ of 1 and B, that i:on,tJnt m1 ,\ be 11 1 Hcn.:c " h anJ I B

2.6

Exercises 2

le)

aml h

l0,0.0)

3

(i) Add -2, -J, 1md

4 \m,e,, ro.,., ! to ro.,.,~ 2, ~.and 4, re~r,cct111':I), to produ~t thc ma\mi:

A• O ' [

-5 4

O -10 O - IS

-S'

-_IO - IS

-S' ] -IO - IS

AdJ 2 times row 2 to row) arn:I - ) umcs r o.,., 2 10 ro.,., 4, and thcn mulllply row 2 b) to produce

!

o

: o:1 .

o o

o

Addmg - 4 tm"IC$ ro.,., 2 to row I reduces 1hc matrilt to Hcmute norm.,I form:

'

11 -

5

[~

o

1 1

- 1 l 1

o

o

o

o .

(11) Let ax1 · t3x2 lx.1 r he thc equ.111on of thc rcqu1red pl:ine . Thcn ca,h oí lhc lhn:c gwcn •"CClon !i-all~fics th1s cqua11on and we ~t the follo.,.,1ng S)"stem:

-, • r,

Addmg thc sec:orn:I and 1h1rd cqu:1t1ons, v.c ha,c

Thcn the thm.l equauon gi,-n

(b) The ,cctor O, (O.O.O) doc-. not ..au,f> thc cquation of the pl:inc m (a). llcn..e thc thrcc 1ett0f'> mía) 1oge1hcr .,.,11h the uro ,ei;tor do nol he m J. rlanc 6

U,m¡¡ lhc no1.1t1(>n of 1 \Bmr,lc 6.7, .,.,e h:nc U11

'1,

1t11

7,

un • 6,

lJl2 •

4,

h, • l.?, b1 • l l

.r1 •

IO,

,Jj

I S,

Amw,:,f'l' ¡

2.6 ExflfC, ,s

439

lf x ,, dcnotcs the amount sh1pped from h, to"•·,

1, 2, J

1, 2, then

we h:n't (1)

(2)

x11

!-

x12

JO,

x21

+ xu

15

JC11

+ JC21

12,

·"U +x2~ - 13

Thc total {'Q,I oí ~h1pp1ng IS (l)

TherefOfe thc problem 1mounb to finlime:

t- c • r t-

.i,

,ector 1f 11,e

A11.swcr.s 'º 01M1e'!J•nct SW. red

e~.,,

440

mttrchan~ an) !Y.O of !he \ummanJ,. Hen.:c the lmear dcpcn• dmce oc 1nderendencc oí U i, not affected b) on m1er,h:1ngc oí

(ii) \\1thou1 lo,;., oí 11Cncr.1ht). a,,un111: u1 ts muluptic:d b) a non-zcro con,tJnt. and kt

\\e LnoY. 1hat t.,· i~ hnc,1rl) depcmknt if and onl) if therc t,1,1 cl>n,t.int:. r1 • • , r •. not ali tero, )UCh thJt

r/ (thl> b Jl'IJnb ,r,. \uch that

But thcn tlk hncar mJcptn ro.... 2 l>) - l. (111) Add ro.,.. 2 to rO\\) J anJ J, and add -2 time) row 2 to row 1 (1\) \full1rl) TO'w J h) I (\) Add 11 time.lmc or n::ro. Thcn add su1table muluplC\ oí row 2 to the Ja\l (11 - 2) ro1,,s !.O that \he -.ccond entry 1n each of thc ro1.1--. J, . , n 1~ zero. The conunuation of th,~ procC'S!I reduces l to l • . íhercforc, by Theorem 7.6, A 1s non~mgular ro ob1am A ·' .,.e perfocm thc !>ame i,cqucncc oí dcmcntor} opcrauons on / .. , and

so the (1,i) cntry 17 Suppo:,.e A

o( A · 1 1s

ri,.. i •

1,2.

a no~mgul,r Jo.,.cr tri.1ngular matri\. Thc elen~nlary row opem11on\ wc ¡,erform on I to re m thc liN ro" e>.ecrt the ( 1, "1) entry e.¡ual 10 n. "'-c\l. b) t)pc 11 cltmcntJ.t) 1.olumn opcra11on) reduce C\UY

AnSwI the (2, 11i) cntl) to O, ond !>O forth t-mall). 1nt,r,;h;1nge column~ 1 and 11,. 2 and 111.. , r a nd 11. m ordcr. íhc rel.ult1ng matwc i~ D 24 Lct P be thc clcmcn1ar) matrn ~uch that P I B lb in Exero..e 2J Lct Q be the rrodll(t ofckmcntal) matrices co1Te-;r,ondmg 10 thl:' column optral!on~ ~ r1bced abo\e, m thl:' !.arnt ordc'r. íhen (PA)Q D 25

(b) The gi\cn rnatrn 1~ non~mgul.ir (.-.ee E.\Crose 4 (d)). Therefore 11s rnnl. 1~ S and hcncc 1ts r,qu1r,d form is f

chapter 3

3.1

Quiz 1 Fal.-.e. Ry Dcfinition 1.1 wc 1.now 1ha1 the equation oíthl:' 11"' through a m thc d1rec11on of u 1s x - u tu, 1 a real number 2 False. Con)idcr the 111,0 non-zero \tcton (l. O) and (O, 1), 111hose 1nner ¡>roduct is ztro 3

T rue. Thc1r mner rroduct i~ zero: ((2.)),(-),21) -

(:?)(-))

())(2)-

o.

Th1\ is thc requirement for orthogon.,ht) gwcn by Definmon l .l. 4

True. Sec thc d1'K"usi.ion rre«dmg and mduding equation (11)

5 íruc.G1\'Cnah)pcrplanellmR•dcfinl:'db)( 1·,u) e,1hc1111oopcn half-~pa~~ correl.pond to thc: mcquaht1c1 (,, u) > r 11nd (x, u) < c. Sm~e no elc11'1('nt x oí R• cun \:IU,fy hoth 1neqUilhtics Mmuhaneoo)i), thc mtcr.ccuon o í thc 11110 halí-spatcs must be the null 1-et

'""º

6 True. lf 11 1\ the h)pc-rpLlne 1n Q~t1on 5, thc do-.ed halí-~racn are (i:, u) ~ e and (.,, u) $ c. An cle11'1('nl .i: E R~ can s.at1\Íy both m• equahhts 1f and onl) ir (i:. u) ~ e; hcnte the ,ntcrsecuon oí(,, u) ~ r and (1, u) $ e 1~ thc hn,crplanc 11 7 True. Agam. 1í 11 1~ thc h)ptrp\Jne m Quernon ,. thc 1-.0-itive open half-space of 11 1i (x, u) > e anrJ thc J}0'1h,e dO'>Cd h.11í-~p.i.ce of 11 1s (x, u) ~ r. An elemcnt l E R' can -.ah,f) both mequahue.s ,r 11nd only 1í (.,, u) > e; hcncc thc mteN:chon oí \he 111,0 halí·,race\ I\ thc ¡>0\1t,,c open half-\¡>act 8 True. l í 11 1~ 1he h)per¡,tane 1n Que,1,on ,, thc ¡>01,1U\C oren halfspa~ oí II i~ (.,,u)> r ami thc ne¡µh\'C closed lulf-~.u:c of II u, (.>.·, u) :e:; r. SmC'C no elemcnt i: oí R· can '-ilh~) both 1nc:quallt1c!>. thc mter:,ccuon oí (x, u} > r and ( r, u) $ r i~ the null ~t

1 ;.11-.c. Com1(kr thc 11o.o hnc \Cgmcnt\ 11 and l ! J01nm11 ti~ ¡,oint)

9

(O. 0). ( 1. 1) und (O. 1). (O, 2). rcsp«:11,dy. 1 rc,m the graph bclO\I, 11 can be !.l~n that thl"l,(' ')(:g_m,:nt) do not 1nte™:.

Graph1n¡ 1he 1nequalit1el>, we find th.'lt 1he reg¡on deternuned b) them hat vert1ttS (40, 30), (SO. 40). and (60, 20)

Then, 1e~tm¡/ on each of the \·erti«~ oí the ~on: J((S(), 40)) • 100 - 40 • 460,

t

400

/((60, 20)) • 120 - 20

t

400

- ,00,

/((40, 30)) • 80 - 30

+- 400

- 4~).

"'-e Stt th3t the ka\t CO'>t oí the m1,111re h

$450 and that the ¡rea,~t cc,c;1

of the m1,ture 1~ SY.l).

14 (a) Let JI denote !he h)'pcrp1ane {11. \) "· 1( both a ami b he in the cl0$Cd p11•,lli\e h.alí-)fla,e oí 11. tht:n (11, a) • cr ► e and (11, h) • d -~ c. lf a pomt ;t l1n on 1he lme ~gmcnl Jommg a anJ b, , l can be "'T1ltcn x (1 - 1"),.l +- 1h, ~here O S t i l.

(u,:c)

(1 -

t)u 1}(u,a)

(1

'"' '

tb) t(u,b)

(u, (1 -

~ (1 -

,p

t)t'+ I('

that IS, ;r il thc closed ~111\C half-spacc of 11. Smcc X llr"U arbltrary, 111e know 1he enure hnc: segrncn1 JOmm¡¡: a and b hes m the clO§Cd l)OSIU\'C half•sr,acc 11 (b) Ir u and b are m thc open r,os11we half-spacc of 11. then (u, u) a> r: and (u, b) - {J > r:. Then 1f x is o n thc h nc: se¡¡:mcnl ,ommg a a nd b. there 1s a , .sat1sfy1n1 O S t S I such that x - ( 1 - ,)u + 1b. S1oce

º'

(u,x) - (u, ( 1 - t)a

+ tb)

+ t(u,b)

-

(1 - 1}{u,a)

-

1)o: +1fJ (1 - t)t'+tr:

>

(1 -

X IS in the open (lOSlll\'C halÍ·spacc oí JI . (e) The open negat1ve half•Spacc of 1111¡1\-en by (u, x) < r:; sur,pose (u,a) - a< r:and(u,b) - ¡J < r:. Thentf x - (1 - {Ja tb,

+

0 $ t $ 1,

(u,x) ... (u,(I - r)a + tb) (1 - ,)a+,~ < (1 -1)c+rr:

(d) The closed ne¡ali\'C half•spacc of II is ¡iven by (u. x) $ e; :,;urpose (11, u) • o $ r and (u, b) {J S c. lñen 1f x (1 - 1)11 + lb, 0 $ f $ l. (u,x) • (1 - 1)o

S ( 1 - 1)!'

+ tP + re

20 (a) Let u - (u1,a2) and b - (b1,bz). Wc know that any r,oint x on lhe hnc: segrncnt bth1ttn a and b .sa11WCS thc equauon x-(1-8)11 v. hcrc O

8b,

S 8 $ 1. Thcn the squarc oí the d1stance: !rom x 10

(x - u, .1" - a) • ((1 - 8).11 t 8b1 - a,)i · ((1 - 8)a2 (8b1 - Bu,)'+ (Bbi - 8u2)i

+ 8b2 -

a2) 2

Answe,, / 3.2 Ov,z

455

111) 2

• IJli(b1 -

+ (b2 -

112)1)

• B2 (b- 11,b-11) In other wOfds, lhc distana: from x to from b to a. Thus,

II

is B or tllc d1nancc

+ B(b1, b2) + Bb1,(I - B>ai + Bb2).

x • (1 - 6)(111, a 2) •

((1 - B)a1

(b) The point j of t hc distana: from tJ

!H + (j)2.(I - (! + ¡. ¡ + !l - (!,;,.

x - ((1 -

( 1, 1) to b -j)I

(2, l) is

+ (j)l)

FollOWin¡ thc procedurc m Examplc 1 4, we scc that the rcquired \me must ¡o t hrough (i, ½) m 1hc d1rccuon oí a vector" which 1s perpendicular 10 u - (2, 3) - (1, 1) - ( 1, 2). lf we pick 11 10 be (2. - 1), thcn thc cquat1on or the hnc is x - < !, l> + r(2, -1), whtre r n.ssumes ali rtal values.

3.2

Ouiz 1 Truc.SccproofofThcorcm2. I (e). 2

True. This follo"") immcd1atdy from formula (5).

3 True. l fS1 C{u u-a $r1) andS2C{11 u-b $ r2} thcn S1 nS2C{u u - a $ r ), whcrc r • b- a. +r1 + r2. 4

True. lí S 1, S1, and rareas m Q ues1ion 3, tllcn

5

False.Tal.te • -2.

S1US2C{11

·11-u $ ri ,

6 True. Jf t > O lhcn both c j and e "'ould be po51live and thcir sum ""ould be pos11wc. But this contrad1ch the fact that cl c O. Thercforc c 5 O

+

7 True. 5cc úamplc 2.2

8 ralse. 5cc Theorcm 2.2 9 True. Scc lbcorcm 2.).

10 Truc.Suprosca E

íl

.u;:ft

A,bE

íl

A.Thcna EAandb E Aíor

AE!f

all A € ~. a nd SlllCC each liCI A 1$ con\C"·,+ ,-1 f ,-1 tb,,.,) ,-1 ,-1

- m1,i:m0in("t

- m¡, m¡,

(f t x~.,,, + f: x, (t ,,)) k

, ,l 1-I

,-1

1-I

- "'-' + k. Also, if x • 1s an op11mum strategy for R for t he game dctcnru ncd by B,

'"'"

,..• ~ Ell'(x •, y)

- ,-1t ,_,t x:b,,y, f. t x:u.,y, + t t. s.x:,•, ·-· ·-· ·-· ,-1

i c., x • is also an opt1mum strate¡y ror the prne determmed by A Con,eri;ely, 1( x• 1s an 0pt1mum stratcgy for the gamc dctermmed by A, then

so that x• 1s also 11n op11mum stratc¡y íor 1he ¡µme dctermined by B. l lencc the opt1mum s trate&Jcs for R wuh rcspccl to the matrn: D are thc same as thO!i.e for thc mamx A By anal~oui. cakulauons 1t can be shown tha t 1hc op11111um strateg¡cs for C are the same for bo1h ¡ame:§.

18

Thc matrix for 1h1s aame IS

-.hcrc row I corresponds to R standmg m íro nt of thc ba'lehnc, row 2 to R standmg bchmd thc ba5ehne. column 1 10 C dchenng a lllt s,er,c, and column 2 to C deh,-crma a tw1st ser,"C. T hc entrics 1ndica1e the p,rn:entage of pomls R wms multipl~ b) 10. (Scc E.,i;crci5e 19.) Lct x - (.\'1, xi) be a s trnte&> for R _lf C chooscs column 1, the expcclallon for R 1s

and 1f C c hoos,e,; column 2, 1t 1S

2.t,

+ Sx2

S - Jx,.

Thc highest point on the ¡raph of thc íunc1ion mm¡) + x1, S - lx1) 1s at

O), 1.e., x• is an 0plimum slraleB) íor R in thc gamc dc:tenruned b}' A. Similarly, ,r x • is an oplimum ~tralel)' fot R 11oith respe,;:t to A, thcn

Answers /

3.5 Q¡¡,z

479

•~ t,

x~(ka,,)y,

- E,,(."•,y), and x • is an optimum stratcgy for R with respcct to B. Thus the optimum s1rateg1ts dctermmcd by A and 8 íor R a re cqual and 1hc s.,mc can be shO\lSiti\"C powcrs of tht matrix, includmg the first ¡,owcr (i.c., the matn x 11selO w1\I ha,'t: posmve cnmcs. 3

False. T he columns do not sum to l . False. A ¡,robabdity vector has nonncgam·c componcnts only. False, Each enlry oí lhc matri:.: must be nonnegAh\'C

6 T rue. fa-ery enu-y of

IS ¡)OSl\l\'C,

7 True. We can check that PA· 1' • A' 1'. By Thcorem S.J we know that if

t~

pl •

A

C:l:lSIS,

then PA' ,, - A

1'

Sincc P is J'rllTllll\'C column

stochast1c, Theorem S.J tells us that A does e,1s1. and b) Theorem 5.4, A is unique. l-lence 1t n1ust be that f~~ pl - A. 8 True.Silltex,~O.,-.~O,i 1, .. ,n,v,'t:kno... 1hat(l - l).l',+ ly, ~O,/ - 1, .• n. furthmnore,

~ [(I

- B)x,

+ 8y,] •

(J -

6)

~ x, + I

t

J',

(1 - 6)+ 8

- l. 9

True. u~m¡ 1he mcthod m the ¡,roo! oí Theorcm S.4, we mu~t find 8,

1he smallc\l of the rallos

J't

o r x., J.

·"• ,-.

1, 2, J, "hcrt x

y (4, I, )). Clcarl} 8 ¼1s .wnallcl.t. TI.en the ,ectot": ha\c a zcro comronent. I n fo.el, :

(1,2, 1) - ( 1.t.

t>

(O,!,U 10 T rue. S1ncc t \ t f ) entry oí P~ is pos1t1,-e. P 1s ¡,r1m1ti\·C

(1, 2, 1) and

" - B,· v.111

AMwen

1e1 Q¡

,u,.

,d

3 .5

1, d l n.•t

480

Exercises "Yes" will mdicate 1hc matnll 1s a 1ransi11on mamx, "no" will mdicatc

2

not (u) no. no (1) no (m)no.

ll IS

(b) oo. ff) no.

(e)

3

(d) no. (h) )CS. (1) no

(e))~(g) no. (k) no.

(¡) no. (n) no.

(o)

)ts.

..ytlo·· ,,,111 mdicatc ~~IT!_ ~ cxi~ts, ..no.. w1II mdicatc it docs not fo) no,

(b) no,

(e) no,

[!o o t] !j

(d)>·cs:A • /J. (e) )·es;

A -

(í))·es:

A -

{g) )CS;

A

¡ l!

[ l\ lj j ! ! !

.

l

! ! !J ¡ . •• t

~!

4 Smce P is pnmime and column stochas11c, P' A ClliSts and satisfics PA 1 • A , i • l. 2. \\-herc, m ía.a M

l,Of>. •

1

a+ b

+

HcnccA isthemamx

A_[•!

b

:r].

u ·b a+b 9

Ltt the 1ransit1on matr1x be p

[:

:!l

where column~ onc and hlO correspond to rrctt) and homely mothers,

AtlS>\'t'ff

J 3.'i [:,, •

481

rC'>f)t"CIL\d), and ro,..s Ol'K" and t,..o co rrcll)' und homcl) dJuB,ht,:ri.. Thcn (.t¡, x J) 1~ the miti,11 dhmbu11on of thc tir.t b-encrauoo of che poflulallon mio prctt) j.lrh (q) 11nd homcly ¡tirl~ (1'~). thcn f't,\ 1s the d1\mbouon of thc (l. + l ht gcncra11on mto prl'tt) and homcl) g1rh. Thc rrol-..1b1ht)' oí a pr,:uy ,..om.m bcms the sreat gran.11 .:,>"'· 6~

•

C::t>mm11um~,t>! (;t>mmut.111,~I.,,., PI

c.·on•rl~••- ~ , !~ CU111r,.»icn!, 1~~

11cmcot,>r) '"" npc-r41,on, : u

l mri,..,1.11 H [er.al rropo111,o,u. 10 Coni>cdcJ rcl.mon. 137 Connc,u..-~, l Con1rap011\1\C,I C:on,cr,c,,I Con~n com~,,,11>1111.HI fon..uon,H7 hull. llt.r),)43 rol)hedron. 311 pol)hedrnn1r,anMdb)a'. , ... , ll(a', ....1.3:1

r .......

1 ""''• 11,

1 ,,-1cnu•lq""t11,r..,. J . 11 l•r•:.••'"'"\ l1~..,-1. '"• 1'4. Hó

,ty oí l ,,.en f.

E,p,.·,!Nlutkncµllan, 8 Ltt1¡tlh o! 11 •~~""• N • }00 Ltnil!hofthc rtv)t,Uon,,f 1hc •c.:1ort ,ri 1h< d,u,uonOfN,

"'-"""'ª ~- ~•. 79 1 ,nuc andv,,;t1on, ltJ

1

1 ""1.:rr-1h.ot,,l11y •r•sc, 111

IM.11.¾M

l m,1c,h... h.o.i,.,,rroc~s. l)-I l l\cJ•c,lor,}H trc.¡ucn,, ofo.:.:urrcn.:c, ll-1 1 ,.,b,:m11,.t..om¡ lho:or,m,:06 run.:11,,n, / · ,l -~ 11

r.

c,,.,,r,,,.,1ionof,60,ltl

Lunn.)71, Luw ,n • nuu ... 216 l .u11..-1•n,;o1rlJJq,cnJcnt,l~ L,,,.,Jrl) anJq,.•n.,k,>1, 1XI 1..,,.c,111.a,q;11IMm.otr",lH tn...., ••lucol •pmI 1cf'-' uf•º"'""• A '.16J

\ lullu'M)mul tocffi,;>mt. ( ,., ,.,"

"

\lulurh.;:auon rr,n.;,¡>k, 11 \l .,h,f'lk..1l1o.>nth«,rC m,n11,, )~0 P,in,,J)le loÍ m•tlwm,11"'•11nJ11,11on,

, .. _76 p,.,._,t,,luy d..rr,r>tltu,.,,17" mt.>>111,Kma, !:.97 s,,m,l\,;ld. 114 Sl•d.•.111•blc,)U

n

"ir,1,u .. ,n oí" •r~c,..., >ícqu,11,ona, i-11 Snh,,,,,.s-,mc, )61 S13nd~•d 1nnc,r rroJ""t. 1-., 100 St.11.:•r•c-,;,374

S1ocb,,u, ....,,,,., n, S1fWCI""St-1,11,1-ril,t,c,111,41 s,mm