Page 28 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 1] NUMBERS 19

COUNTABILITY
1.58. (a)Prove that there is a one to one correspondence between the points of the interval 0 @ x @ 1 and
5 @ x @ 3. (b) What is the cardinal number of the sets in (a)?
Ans. (b) C,the cardinal number of the continuum.

1.59. (a)Prove that the set of all rational numbers is countable. (b) What is the cardinal number of the set in (a)?
Ans. (b) F o

1.60. Prove that the set of (a)all real numbers, (b)all irrational numbers is non-countable.
1.61. The intersection of two sets A and B,denoted by A \ B or AB,isthe set consisting of all elements belonging
to both A and B. Prove that if A and B are countable, so is their intersection.

1.62. Prove that a countable set of countable sets is countable.

1.63. Prove that the cardinal number of the set of points inside a square is equal to the cardinal number of the sets
of points on (a) one side, (b)all four sides. (c) What is the cardinal number in this case? (d) Does a
corresponding result hold for a cube?
Ans. (c) C
LIMIT POINTS, BOUNDS, BOLZANO–WEIERSTRASS THEOREM
1.64. Given the set of numbers 1; 1:1;:9; 1:01;:99; 1:001;:999; .. . .(a)Isthe set bounded? (b) Does the set have
a l.u.b. and g.l.b.? If so, determine them. (c) Does the set have any limit points? If so, determine them.
(d)Isthe set a closed set?
Ans. (a)Yes (b)l:u:b: ¼ 1:1; g:l:b: ¼ :9(c)1 (d)Yes

1.65. Give the set :9;:9; :99;:99; :999;:999 answer the questions of Problem 64.
Ans. (a)Yes (b)l:u:b: ¼ 1; g:l:b: ¼ 1(c)1; 1(d)No

1.66. Give an example of a set which has (a)3 limit points, (b)no limit points.

1.67. (a)Prove that every point of the interval 0 < x < 1is a limit point.
(b)Are there are limit points which do not belong to the set in (a)? Justify your answer.
n
1.68. Let S be the set of all rational numbers in ð0; 1Þ having denominator 2 , n ¼ 1; 2; 3; .. . .(a) Does S have
any limit points? (b)Is S closed?
1.69. (a)Give an example of a set which has limit points but which is not bounded. (b) Does this contradict the
Bolzano–Weierstrass theorem? Explain.
ALGEBRAIC AND TRANSCENDENTAL NUMBERS
p ffiffiffi p ffiffiffi
2 p ffiffiffi p ffiffiffi p ffiffiffi
1.70. Prove that (a) p 3 p ffiffiffi, (b) 2 þ 3 þ 5 are algebraic numbers.
ffiffiffi
2
3 þ
1.71. Prove that the set of transcendental numbers in ð0; 1Þ is not countable.
1.72. Prove that every rational number is algebraic but every irrational number is not necessarily algebraic.
COMPLEX NUMBERS, POLAR FORM
1.73. Perform each of the indicated operations: (a)2ð5 3iÞ 3ð 2 þ iÞþ 5ði 3Þ, (b) ð3 2iÞ 3
10 2
5 10 1 i 2 4i
; ; ; ð1 þ iÞð2 þ 3iÞð4 2iÞ :
ðcÞ þ ðdÞ ðeÞ ð f Þ 2
3 4i 4 þ 3i 1 þ i 5 þ 7i
ð1 þ 2iÞ ð1 iÞ
2
2
Ans. (a)1 4i, (b) 9 46i, (c) 11 i, (d) 1, (e) 10 , ( f ) 16 i.
5 5 37 5 5