Higher Mathematics for Engineering Students, Part 1: Linear Algebra and Fundamentals of Mathematical Analysis [1]

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A.V. Efimov, B.P. Demidovich (Eds.)

HIGHER MATHEMATICS for Engineering Students Part 1. Linear Algebra and Fundamentals of Mathematical Analysis

MIR PUBLISHERS MOSCOW

CEOPHMK SARAH n o MATEMATHKE Ajiji BTyaoit

JlrnieimaH ajire6pa H OCHOBLI MaTeMaTHiiecKoro anajin3a no a peAaKi^Meii A. B. EtjaiMOBa, B. n . fleMHAOBiiqa

H3AaTejibCTBO «HayKa» MocKBa

HIGHER MATHEMATICS for E ngineering Students

Worked Examples and Problems with Elements of Theory Part 1. Linear Algebra and Fundamentals of Mathematical Analysis Edited by A.V. EFIMOV and B.P. DEMIDOVICH Translated from the Russian

by

LEONID LEVANT

MIR PUBLISHERS MOSCOW

Contributing Authors:

V.A. Bolgov, B.P. Demidovich, V.A. Efimenko, A.V. Efimov, A.F. Karakulin, S.M. Kogan, G.L. Lunts, E.F. Porshneva, A.S. Pospelov, S.V. Frolov, R.Ya. Shostak, A.R. Yanpolskii

First published 1984 Revised from the 1981 Russian edition

Ha amJiuucKOM statute

© TjiaBHaH peflaKmiH n0 (n0 some natural number) are usually not considered here due to the following equalities: [x].999... = [x] + 1, [x].xxx2 . . . Zn0_1999... = lx].x{x2 . . . (Xn0-1 + 1)1 («o > L xnQ- 1=^9). A real number x is rational, i.e. representable in the form of the ra tio —, m, n £ 1, if and’ only if decimal (l)'is periodic. Otherwise the number x is irrational. The absolute value or the modulus of a real number x is defined as the nonnegative number r x, if « > 0 . I —a?, if z < 0 . It is supposed that the rules for comparing real numbers as also the arithmetical operations on them are already known to the reader.

1.1. Prove that the number 0.1010010001. . .10. . .01. . . n

is irrational. Write the first three terms of either of the sequences of finite (or terminating) decimals approximating this number with deficit or excess. 1.2. Represent the following numbers in the form of proper rational fractions: (a) 1.(2); (b) 3.00(3); (c) 0.110(25). 1.3. Prove that the number log 5 is irrational.

Introduction to Analysis

10

Ch. 1.

Let us assume that log 5 is a rational number, i.e. l0g5 = — ; n

Then

m, n £ y , .

7n . n

10 —5, 10m = 571, 2m -5m = 5n. But the last equality is impossible, since the number 2 enters into the factorization of the left-hand member in simple factors, but does not enter into a similar factorization of the right-hand member which contradicts the uniqueness of presenting whole numbers in the form of prime factors. Therefore our assumption is false and, consequently, the number log 5 is irrational.

In problems 1.4 to 1.9 prove that the given numbers are irrational: 1.4. 1/3. 1.5. p is a prime number, 1. 1.6. 2 + Vs. 1.7. V 2 + 1/ 3. 1.8. log3 p, p is a prime number. 1.9. — f r e f ? , if it is known that jt is irrational. In Problems 1.10 to 1.13 compare the given numbers. 1.10. ]/2 — ]/~5~ and |/ 3 - 2. Suppose the below inequality is correct: 1 hen:

/2 -/5 < /3 -2 .

(2)

/ 2+ 2 < / 5 - f - / 3 , «+ 4 / 2 < 8 + 2 / l 5 , 2 / 2 < l + /l5 , 8 < 1 6 + 2 /1 5 . Since (lie last inequality is true, by virtue of Ihe equivalence of the transformations performed, the initial inequality (2) is also true.1 1 , , 1

Sec. 1.1.

Real Numbers. Sets. Logic Symbolism

11

In Problems 1.14 to 1.16 prove the indicated numerical inequalities without using tables. 1.14. log3 10 + 4 log 3 > 4. 1.15. +log2---f — — >2. n logs ji 1.16. log4 26 > log6 17. 1.17. Prove that the modulus of a real number possesses the following properties: (a) | x | = max {x , —x}; (b)

\x-y\ = \x\-\y\ and

(c) I x + y | < | x | + | y | and | x — y | > | | x | — | y | | (the triangle inequalities); (d) V x 2 = | x |. In Problems 1.18 to 1.22 solve the given equations: 1.18. 13.r —4 1—4 “ • 1.19. | d 2 + *3- 0 . 1.20. | —x2-\-2x —31= 1. 1.21. | ~ + - 1 . 1.22. V (x —2)2= - x + 2. In Problems 1.23 to 1.27 solve the given inequalities: 1.23. | x — 2 | ^ 1. 1.24. | x l — lx + 12 | > xl — lx -|+ 12. 1.25. x2 , 2 / ( l T F 3 p - 1 0 < 0 . 1.26. + < 4- j. 1.27. / ( x + l ) 2< - x - 1. 2. Sets and Set Operations. A set is understood as any well-defined collodion of objects called the elements or members of the set. The notation a £ A means that the object a is an element of the set A (belongs to the set A), otherwise we write a $ A, A special set that plays an important part in set theory is the empty set, sometimes called null set, which contains no elements. The notation used to denote the empty set is the symbol 0. The notation A cz B (read: "A is contained in B'\ or, equivalently, “B contains A”) means that every element in a set A is also an element of a set B, in this case A is called a subset of B. The sets A aad B are equal (written A — B) if A a B and B cz A . There are two basic ways of defining (describing) a set: (a) The set A is defined by a direct enumeration of all of its ele­ ments fij, a2, . . ., any i.e. written in the form

A = (flj,

• • •»

12

Introduction to Analysis

Ch. 1.

(b) The set A is defined as the totality of those and only those elements belonging to a certain basic set T which possess the general property a. In this case we use a shorter notation A = {x£T\a{x)}
o ) . 1.31. A = { x £ N | xz —3x —4 ^ 0 } . 1.32. A = - { z£ Z I —< 2 * < 5 } . 1.33. yl = { a:€ N |lo g 1/2 — < 2 } . 1.34. A = {x £ dt | cos2 2x^=1 and 2ji}. In Problems 1.35 to 1.42 represent the indicated sets on the coordinate plane. 1.35. {(x, y) 6 (R2 | x + y — 2 = 0}. 1.36. {(x, y) 6 R2 I s 2 - y2 > 0}.

Sec. 1.1.

1.37. 1.38. 1.39. 1.40. 1.41.

Real Numbers. Sets. Logic Symbolism

{(*» {(^» {(*, {(*, {(#,

13

if) € 312 I (*2 — 1) (y + 2) = 0}. y) € 3l2 \ y > V 2 x 1 and 2x + 1 > 0}. y) 6 fR2 I y2 > 2x + l}. y) 6 id2 I 2X+1 = y2 + 4 and 2x~l < y). «/) 6 I cos 2x = cos 2y).

1.42. { (* ,« /)€ ft2 -— > y , x=^ 0 ’ 1.43. Describe by enumerating all the elements the sets A [ } B , A f| B, A \ B and if A = {x 6 ill I x2 -f x — 20 = 0}, B = {x 6 R | a:2 — x + 12 = 0}. The notation m | n, where ra 6 Z7, means that the number m is the divisor of the number n. Describe the following sets (Problems 1.44 to 1.47). 1.44. {x d N| x | 8 and x =^= 1}. 1.45. {x £ Z I 8 | x}. 1.46. {x 6 N | x \ 12) n {x e N | * | 8>. 1.47. {x e N| 12 I z} n 6 N I 8 | x). 1.48. Prove that: (a) the equality A f) B = B is true if and only if B cz A\ (b) the equality A (J B — B is correct if and only if 4 c 5 . 1.49. Let A = (—1, 21 and B — [1, 4). Find the sets A U B, A D Bj A ^ B , B \ A and represent them on the number axis. Taking the line segment T = [0, 1] for a universal set, find and represent on the number axis the complements of the following sets (Problems 1.50 to 1.53): 1.50. {0, 1}. 1.51. (1/4, 1/2). 1.52. (0, 1/21. 1.53. {1/4} U [3/4, 1). 1.54. Prove that the operation of taking a complement possesses the reflexivity property (A) = A, and is also connected with the inclusive relation cz and

u

Ch. 1.

Introduction to Analysis

operations (J and f| by the following duality laws: if A cz B, then A zd B, A [)B = A D B and A (] B = A [} B. 1.55. Prove that the operations |J and f| are related by the distributivity laws:

(a u b) n c = (a n c) u (b n c), (a n b) u c = (a u c) n (b u cy Using the results obtained in Problems 1.54 and 1.55, prove the following equalities (Problems 1.56 to 1.59): 1.56. A ^ J S fl (A U B) = A. Since A [J 13 = A fl # the left-hand member of the equality in question takes the form

(1^ 5) n (JTTS)-Un^) u {A n b) = a 1.57. A \ B --- A fl B. 1.58. A 1.59. A n ( A \ B ) = A f]B.

- A U B.

The operations (J and f] are generalized in a natural way for the case of an arbitrary (finite or infinite) family of sets. Let, for example, there be given a family of sets An, n £ |\j. The union of sets belonging to this family is denoted by the symbol U An and is defined as the n£N

set of all those elements each of which belongs at least lo one of the sets of An. The intersection fl An is defined as the set of all the elenm

ments belonging to each of the sets of An.

In Problems 1.60 to 1.62 find

(J A n and

f| A n for n£N

the given families A n, n £ N: 1.60. A n = {x 6 1 | —n ^ x ^ n). 1.61. A n = {3n — 2, 3ra — 1}.

1.62. l » = { l , ± , ± ..........i - } . 1.63. Let A be the set of all the points in the plane form­ ing the sides of a triangle inscribed in a given circle. Describe the union and intersection of all such sets if (a) triangles are arbitrary; (b) triangles are regular; (c) triangles are right-angled,

Sec. 1.1.

Real Numbers. Sets. Logic Symbolism

15

A set Ar is said to be countable if there can be established a one-to-one correspondence between the elements of this set and those of the set of all natural numbers. Example 2. Show that the set rA of all integers is countable. Let us establish one-to-one correspondence between the elements of this set and natural numbers, for instance, by ordering the set '// in the following way: 0, 1, - 1 , 2, - 2 , 3, - 3 , . . ., and then associating each integer with its ordinal number in this sequence.

In Problems 1.64 to 1.66 prove that the given sets are countable. 1.64. {n 6 N | n = 2k, k £ N}. 1.65. {n £ N | n = k2, k £ M}, 1.66. {n 6 N | n = 2h, k £ N}. 1.67. Prove that if the set X is countable and A cz X is its infinite subset, then the set A is also countable. Using the obtained result, prove that the set {n e T I n = k2 — k + 1, k £ N} is also countable. 1.68. Let X 1, X 2, . . ., X n be countable sets. Prove that their union \j Xn|i s a countable set. nefcl Let Xn = { x n%1> xn (2, . . ., xn,i, • • .}.j Then the the set U Xn c^n be tabulated as follows:

elements of

^1,1* • • •»xi ,1 t ♦ • ^2.1» ^2.2* • * *♦ X2*l » • • •• x n.l» x n , 2» • • •» x n d » • • •»

To prove the countability of the set

U it now suffices to number n£N all elements of this table in an arbitrary way.

Taking advantage^of the result of Problem 1.68, prove that the following sets are countable: 1.69. Q — { x 6 Dl | x — — for some m , n =7^ 0 from rt ) is the set of all rational numbers.

Introduction to Analysis

16

Ch. 1.

1.70. The set of all points in the plane having rational coordinates. 1.71. The set of all polynomials with rational coefficients. 3. Upper and Lower Bounds. Let X be an arbitrary non-empty set of real numbers. The number M = max X is called the greatest (maximum) element of the set X if M £ X and for any x £ X the inequality x ^ M is fulfilled. The notion of the least (minimum) element m = min X of the set X is defined in a similar way. The set X is said to be bounded above if there is a real number a such that x ^ a for all x £ X. Any number possessing this property is termed the upper bound of the set X. For a given bounded above set X the set of all of its upper bounds has the least (minimum) ele­ ment which is called the least upper bound or supremum and is denoted by the symbol sup X. The notions of a set bounded below, lower bound and the greatest lower bound or infimum are defined in an analogous manner. The latter is denoted by the symbol inf X. A set X bounded above and below is said to be bounded. Example 3. Find the supremum and infimum of the set [0, 1). This set has no greatest element, since for any x £ [0, 1) there is y £ [0, 1) such that y > x. The set of upper bounds for the halfinterval [0, 1) is the set [1, oo) with the least element equal to 1. Therefore sup [0, 1) = 1, where 1 (J [0, 1). On the other hand, the least element of the set [0, 1) exists and is equal to zero. The set of lower bounds is the s e t(—oo, 0] with the greatest element equal to zero which is the infimum of the half-inter­ val [0, 1). Thus, min [0, 1) = inf [0, 1) = 0, and 0 £ [0 , 1).

1.72. Prove that the above formulated definition of the supremum is equivalent to the following: The number M is the supremum of the set X if and only if: (1) x ^ M for all x £ X \ (2) for any e > 0 there is an element x £ X such that x > M — e. 1.73. L e tX = ( l ,

1 , .... A ,

(a) Indicate the least and greatest elements of this set (if any). (b) What are the sets of upper and lower bounds for the set X? Find sup X and inf X .

Sec. 1.1.

Real Numbers. Sets. Logic Symbolism

17

In Problems 1.74 to 1.78 find max X , min X , sup X , and inf X (if any) for the indicated sets: 1.74. X = { * € 3 1 1x = — , ra£N}. 1.75. X = [ - 1 , lj. 1.76. X | — 5 < x < 0}. 1.77. X = {* € R I* < p means: “the statement a implies the state­ ment P” (=^ is the implication symbol). The notation a