SETS 21

Let Y be the set of letters in “ TRACT”. Then

Y = { T, R, A, C, T } = { T, R, A, C }
Since every element in X is in Y and every element in Y is in X. It follows that X = Y.
Example 24 List all the subsets of the set { –1, 0, 1 }.
Solution Let A = { –1, 0, 1 }. The subset of A having no element is the empty
set φ. The subsets of A having one element are { –1 }, { 0 }, { 1 }. The subsets of
A having two elements are {–1, 0}, {–1, 1} ,{0, 1}. The subset of A having three
elements of A is A itself. So, all the subsets of A are φ, {–1}, {0}, {1}, {–1, 0}, {–1, 1},
{0, 1} and {–1, 0, 1}.
Example 25 Show that A ∪ B = A ∩ B implies A = B
Solution Let a ∈ A. Then a ∈ A ∪ B. Since A ∪ B = A ∩ B , a ∈ A ∩ B. So a ∈ B.
Therefore, A ⊂ B. Similarly, if b ∈ B, then b ∈ A ∪ B. Since
A ∪ B = A ∩ B, b ∈ A ∩ B. So, b ∈ A. Therefore, B ⊂ A. Thus, A = B
Miscellaneous Exercise on Chapter 1
1. Decide, among the following sets, which sets are subsets of one and another:
A = { x : x ∈ R and x satisfy x2 – 8x + 12 = 0 },
B = { 2, 4, 6 }, C = { 2, 4, 6, 8, . . . }, D = { 6 }.
2. In each of the following, determine whether the statement is true or false. If it is
true, prove it. If it is false, give an example.
(i) If x ∈ A and A ∈ B , then x ∈ B
(ii) If A ⊂ B and B ∈ C , then A ∈ C
(iii) If A ⊂ B and B ⊂ C , then A ⊂ C
(iv) If A ⊄ B and B ⊄ C , then A ⊄ C
(v) If x ∈ A and A ⊄ B , then x ∈ B
(vi) If A ⊂ B and x ∉ B , then x ∉ A
3. Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show
that B = C.
4. Show that the following four conditions are equivalent :
(i) A ⊂ B(ii) A – B = φ (iii) A ∪ B = B (iv) A ∩ B = A
5. Show that if A ⊂ B, then C – B ⊂ C – A.
6. Show that for any sets A and B,
A = ( A ∩ B ) ∪ ( A – B ) and A ∪ ( B – A ) = ( A ∪ B )
7. Using properties of sets, show that
(i) A ∪ ( A ∩ B ) = A (ii) A ∩ ( A ∪ B ) = A.
8. Show that A ∩ B = A ∩ C need not imply B = C.

2024-25
22 MATHEMATICS

9. Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set

X, show that A = B.
(Hints A = A ∩ ( A ∪ X ) , B = B ∩ ( B ∪ X ) and use Distributive law )
10. Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty
sets and A ∩ B ∩ C = φ.

Summary
This chapter deals with some basic definitions and operations involving sets. These
are summarised below:
Æ A set is a well-defined collection of objects.
Æ A set which does not contain any element is called empty set.
Æ A set which consists of a definite number of elements is called finite set,
otherwise, the set is called infinite set.
Æ Two sets A and B are said to be equal if they have exactly the same elements.
Æ A set A is said to be subset of a set B, if every element of A is also an element
of B. Intervals are subsets of R.
Æ The union of two sets A and B is the set of all those elements which are either
in A or in B.
Æ The intersection of two sets A and B is the set of all elements which are
common. The difference of two sets A and B in this order is the set of elements
which belong to A but not to B.
Æ The complement of a subset A of universal set U is the set of all elements of U
which are not the elements of A.
Æ For any two sets A and B, (A ∪ B)′ = A′ ∩ B′ and ( A ∩ B )′ = A′ ∪ B′

Historical Note
The modern theory of sets is considered to have been originated largely by the
German mathematician Georg Cantor (1845-1918). His papers on set theory
appeared sometimes during 1874 to 1897. His study of set theory came when he
was studying trigonometric series of the form a1 sin x + a2 sin 2x + a3 sin 3x + ...
He published in a paper in 1874 that the set of real numbers could not be put into
one-to-one correspondence wih the integers. From 1879 onwards, he publishd
several papers showing various properties of abstract sets.

2024-25
Sets
AUB
AnB
AAB=(AUB)-(AnB)

A B
B

A' AnB' A'nB

B A
A B B

AUB' A'UB= (An B)' A'nB'=(AU B)'

B A B

1. Which of the following collection of objects is not a set?

a) The collection of all even integers
b) Thecollection of all months ofa year beginning with letter J
The collection of most talented writers of India
C)
d) The collection of all prime numbers less than 20

2. If A = {1,2,3,4,5), then which of the following is not true?

a) 0A b) 3E A c) {3}EA d) {3} cA

3. Which of the following is a null set?

a) {x:x E N, 2x - 1 = 3}
b {x:xE N, x' < 20}
c) {x:x is an even prime greater than 2}
d) {x:xEI,3x +7= 1}
4. Which of the following is a finite set?
a) {x:x E 2n, nE N} b) {x:x is a prime number}
c) {x:x E N, xisa factor of 128} d) {x:xEI,x<7}

5. Which of the following is not correct? NcT

a) NcR b) NcQ c) QcR d)

6. On real axis if A = [1,5] and B = (3,9,then A -B is

d) [1,3)
a) (5,9) b) (1,3) c) [5,9)
 7. On real axis if A= (1,5] and B =
(3,9),then A UB is
a) (5,9) b) (1,3) (1,9)
c) [1,9] d)
8. On real axis if A= [1,5] and B = (3,9)],then An B
is
a) (1,3) b) [3,5] (3,5)
c) (1,31 d)
9. On real axis if A= [1,5]and B= (3,9],then B- Ais
a) (5,9) b) [5,9| c) [5,9) d) (5,9]
10. If the number of non-cmpty subsets of a set is 4095, the number of elements of the set 1s
a) 10 b) 11 c) 12 d) 13

11. IfA, Band Care three sets, such that, AU B= AUC and An B = AnC, then
a) A= B b) B=C c) A= C d) A=B C

12. Number of clements in power set of A = (1,2,3} are

a) 8 b) 10 c) 6 d) 12

13. IfX and Y are two sets, such that n(X) = 17, n(Y) = 23, n(X UY = 38, Find n(X n )
a) 3 b) 4 c) 2 d)

14. If0 denotes the empty set, then which of the following iscorrect?
a) b) c) (0} E (0} d)

15. The shaded region in the given figure is.

a) BN (A UC) b) BU (A N C) c) BO(A-C) d) B-(AUC)

16. IfA, Band Care any three sets, then A- (BUC) is equal to
a) (A- B) U(A C) b) (A- B) UC
c) (A B) n C d (A-B) n(A-C)
17. Which of the following sets are null sets?
A= (x: |x|< -4,x ER)
B= Set of all prime numbers between 15 and 19
C= (x:x<5 andx > 6, xE R)
b) A and B c) A and C d) All
a) Only A

18, Which of the following is a null set?

a) (x<x ER,4x?-1= 0) b) (x<x ¬ N, ris odd, x+3 is even)
c) (xlx E R,x<3) d) (x<x ¬R.x²<0)
19. Which of thefollowing statement is true?
a {1} ¬ {1,2} b) 1{1,2}
d) {1,2) = (2,1)
c) {1,3} ¬ (x|x is apositive integer}
20. Total no. of elements in (0} is
a) 1 b) 2 c) None

IfP ={1,2, [1,2,3),Q = (1,2,3}, then which ofthe following is true?

21. b) {1,2} c P
a {1,2,3) EP All of Above
c) {1,2} c Q d) J a n

subsets of A is
22. When A = 0, then number of 1 None
2 b) 0 c)
a)

23. Consider the following statements:

primes}
I: {xx is collection of all positive odd
II: {xx is a collection of allpositive integers} element
a Iand II will have at least one common
b) Iand II will have no common element
element
c) Iand II will have exactly one common
None of these
d)

24. Consider the following statements

I: Any set A is comparable with itself.
II: {0} is a singleton set.
III: {0} is an empty set. of these statements
Iand II are correct b) Iand II are correct
a)
d) allare correct
c) IIand Ilare correct

a factor of 18 and x EN}

25. If A= (x:x is a factor of 12 and x E N} and B = {x: x is
then A n B is
b) {1,2,3,6,12,18}
a) {1,2,3,4,6,9,12,18}
{2,3,6} d) {1,2,3,6}
c)
power set of fTst set
26. Two finite sets have mn and n elements. The number of elements in the
set of second set. Then the value of
is 384 more than the total number of elements in power
(m - n) is
b) 3 c) 7 d)
a) 2

27. Choose the finite set from the following

a) the set of numbers which are multiples of 7
b) set of animals living on earth
c) set of circles drawn in a plane
the number of rational numbers lying in between 2 rational numbers
 28. IfA = {1, 5),B= (2, 6}, C= (2,4)then the number of elements of Ax(BUC)
a) 5 b) 6 c) 7 d) 8

29. In a class of 50students, 10did not opt for Mathematics, 15did not opt for science and 2 did
not opt for either Mathematics or Science. Then number of students in the class opted for
both Mathematics and Science are
a) 24 b) 25 c) 26 d) 27

30. If P = (1,2) find n(P x P x P)

a) 2 b) 6 c) 8 16

31. Given that B=(x:4<x<5, xE N) and C= (x:x' =25 and xis an integer}, then Bn
Cis
a) (5,-5) b) c) (5} d) (4,5)

32. Let P= {x:x is apositive prime number less than 20}. Then the number of proper
subsets are
a) 28 b) 20-1 c) 2 -2 29- 2
33. Which of the folowing statement is false?
a) A-B= AnB' b) A-B= A-(4nB)
c) A-B= A-B' d) A-B-(4UB) -B
34. The number of proper subsets of a set containing n element is
a) b) 2-1 c) n d) 20

35. If Ahas m elements and Bhas n elements and Aand Bare disjoint, then n(AUB)is:
a) m + n b) 7m - c) T X n d) m/n

36. It ACB and A #B then

a A is called a proper subset ofB.
b) A is called a super set of B.
A is not subset of B.
d) B is a subject of A.
37. Let A, Band Cbe three sets. IfA e B, and B c C, then
a) AcC b) AeC c) AEC d) hone ot these

38. The given real number line represents various intervals as subsets of real numbers, tind the
correct option to the representation:

a) (a.b].[a,b).[a.b]. (a.b) b)

c) (a,b),[a,b].[a.b).(a.6] d)
[a.b]. (a.b).(a8].[ab)
39. Given U=[-5,5] and A=(-3,5],then A' is:
a) [-5-3) b) (4,5] c) [5-3] d) [4.5]
elements is:
40. The number of subsets of a set containing 2n 2n d) 22n
a) 2n b) 22n-1 c)

41. If AcB then value of AUB is:

B b) A c) d) none of these
a)

42. For set A, AUA=A.This is

a) Law of universal element b) Law of identity element
Idempotent law d) Commutative law
c)
43. In set builder method the nullset is represented by
a) b) c) {x:x#x} d) {x:x =x}
44. Suppose A, Az, . . .A30 are thirty sets each having 5 elements and B,, B2,B, are n sets each
with 3elements, let UA, = U-1 B, = S and each element of Sbelongs to exactly 10 of
the A' and exactly 9 of the B,'. Then n is equal to
a) 15 b) 3 c) 45 d) 35

45. Two finite sets have m and n elements. The number of subsets of the first set is 112 more
than that of the second set. The values of m and n are, respectively
a) 4,7 b) 7, 4 c) 4, 4 d) 7, 7
46. Let F be the set of parallelograms, F, the set of rectangles, Fz the set of rhombuses, F, the
set of squares and Fs the set of trapeziums in a plane. Then, F may be equal to
a) b)
c) Fz UFs d) Fz U
Fg UF UE
47. Let S = set of points inside the square, T = set of points inside the
noints inside the circle. If the triangle and circle intersect each triangle and C = set of
other and are contained in a
square. Then,
a) SnTOC=0 b) SUTUC= C
c) SUTUC=S d) SUT =SnC
48. If R be the set of points inside a rectangle of sides a and b (a, b> 1) with two sides alono
the positive direction of X- axis and Y- axis, Then,
a R= (x,y) : 0Sxsa,0sys b)
R= (x,y) :0<x<a, 0sysb)
R= (x,y):0sxsa,0 <y< b)
R= («,y) :0 <x<a, 0<y <b)
 49. In a town of 840persons, 450 persons read Hindi, 300 read English and 200 read both. Then,
the number of persons who read neither, is
a) 210 b) 290 c) 180 d) 260

50. If X= (8" - 7n -1]n E N) and y = (49n 49|n E N). Then,

a) XcY b) YcX c) X=Y d)

51. Asurvey shows that 63% of the people watch anews channel whereas 76% watch another
channel. If x% of the people watch both channel, then
a) x=35 b) X=63 c) 39 <x < 63 d) X=39

52. If sets A and B are defined as

A=(y) y=,0*xER,B =(,y)ly =-x,xE R), Then,

a) AnB= A b) AnB = B c) AnB=0 d) AUB =A

53. If Aand B are two sets, then A n (AUB) equals to

a) A b) B c) d) AnB

54. IfS = xlx is a positive multiple of 3less than 100}and

P={x\x is aprime number less than 20} Then, n(S) + n(P) is equal to
a) 34 b) 31 c) 33 d) 41

55. The set {x E R: 1 <x<2} can be written as

56. When A = 0, then number of elements in P (A) is
57. If A and B are finite sets, such that A c B, then n(AUB) is equal to
58. Power set of the set A = {1, 2} is
59. If the sets A= {1,3,5), B ={2, 4,6} and C= (0,2,4,6, 8}. Then, the universal set of all the
three sets A, B and C can be

State True or False (Q 60 Q65)

60. If Ais any set, then A A.
61.IfM = {1,2, 3,4, 5, 6, 7,8, 9} and B= {1,2,3, 4, 5, 6,7, 8,9} then Bc M
62. The sets {1, 2, 3, 4} and {3, 4, 5, 6} are equal
63. QUZ=Q, where Qis the set of rational numbers and Zis the set of integers.
64. Let sets R and T be defined as
R= xE Z]x is divisible by 2}
T= fxEZ|x is divisible by 6}. Then, T c R
65. Given A= {0,1,2), B= (xE R|0 < x < 2). Then, A= B
66. Aand Bare two sets such that n(A- B) = 14 + x,n(B- A) =3x and n(AnB)=x.
Draw aVenn diagram to illustrate information and if n(4) = n(B) then find the value of x.
67. On the real axis, ifA = [0,3] and B[2,6), then find A-B.
68. A set A has 50 elements and another set B has 20 elements. Write the minimum and
maximum number elements in AUB.
 n(AUB).
69. If Aand Bare two disjoint sets, then write the value of

70. Write the power set of A=0.

71. n{P (P (P (O))} =??

set-builder form.
72. Write the set B = (3,9,27,81) in
A = {X:X EN and 3 <X<4} Is A single ton set, give reason? h J a i n
73.
set, give reasons?
74. B = {X: XE N and X' = X} Is B null
that A c B, then find
75. If A and B are two sets such AUB
i) AnB ii)
nCare non-empty sets and
76. Find setsA, Band C such that AnB, B^Cand A
AnBOC=0

and 70play
77. In a survey of 450people, it was found that 110play cricket, 160 play tennis
tennis?
both cricket as well as tennis. How many play neither cricket nor

78. In a town of 10, 000 families it was found that 40 % families buy newspaper A, 20% buy
newspaper B and 10% families by newspaper C. 5% families buy A and B, 3% buy B and C
and 4% buy A andC. If 2%families buy all the three newspapers, find the no. families which
buy
(2) B only (3) none ofA, B andC
(1) A only
(4) exactly two newspaper (5) exactly one newspaper
(6) A and C but not B (7) at least one of A, B, C.

79. If A is the set of all divisors of the number 15. B is the set of prime numbers smaller than 10
and Cis the set of even number smaller than 9, then find the value of (A UC)n B.

Case Studies
80. Agroup of 100 students of class 1llth went on atrip. Out of which 55 students took burger,
40 students took red bull and 15 students took both burger and red bull
i) Find the number of students who took only burger.
d) 15
a) 80 b) 40 c) 25
ii) Find the number of students who took red bull only.
a) 25 b) 40 c) 55 d) 15
ii) Find the number of students who took not red bull or not burger.
a) 85 b) 60 c) 25 d) 15
iv) Find the number of students who took at least one of burger or red bull.
a) 80 b) 40 c) 15 d) 20
V) Find the number of students who neither took burger nor red bul.
a) 80 b) 50 c) 20 d) 35
 81. There are 2000 students in aschool.Out of these 600 read novels, 550 reads comicsand 1000
read magazines. 120 read novels and magazines. 80 read novels and comics, 150 read
magazines and comics. Also 45 read all three types books. Answer the following question
i) Number of students who read only magazines
a) 700 b) 725 c) 750 d) 775
ii) Number of students do not read any of the books
a) 155 b) 180 c) 190 d) 175
iii) Number of students visited the library
a) 2000 b) 1851 c) 1845 ) 1810
iv) Number of students who read only comics
a) 360 b) 365 c) 370 375
Number of students who read novels or magazines.
a) 1480 b) 1350 c) 1400 d) 1300

82. In acity school during the admissions to class XI, 18 students took English, 23 Students took
Hindiand 24 students took Sanskrit. Ofthese, 13 took both Hindi and Sanskrit, 12 took both
English and Hindi and 1ltook both English and Sanskrit Due to the request made by some
students,the school authorities decided that 6 students willbe offered all the three languages.
Based on the above information answer the following:
i) How many students took Sanskrit but not Hindi?
11 d) 9
a) 6 b) 19 c)
ii) How many students took exactly one of these subjects?
b) 11 c) 20 d) 21
a) 25
ii) How many students took exactly two of the three subjects?
11 b) 21 c) 8 d) 18
a)
iv) How many students took Hindi but not Sanskrit?
b) 10 c) 19 d) 13
a)
) How many students took atleast one of the subjects?
37 d) 40
a) 35 b) 36 c)

a t h
 1. C 42. C
2. C 43. C
3. C 44. C
4. C 45. B
46. D
5. D
6. D
47. C

7. C
48. D ish
Jan
49. B
8. B
9. D
50. A
51. C
10. C
52. C
11. B
53. A
12. A
54. D
13. C
55. (1,2)
14. B
56. 1
15. D 57. n(B)
16. D 58. {0.(1), (2),(1,2)}
17. C
59. {0,1,2,3,4,5,6,8}
18. D 60. T
19. D 61. F
20. A 62. F
21. D 63. T
22. C 64. T
23. A 65. F
24. A 66. 7
25. D 67. [0,2)
26. A 68. 50& 70
27. B 69. n(A) + n(B)
28. B 70. {(0),. o)
29. D 71. 4
30. C 72. B = x:x=3", n E Nand1sns
31. B 4)
32. B
73. No, it is an empty set
33. C 74. No, B = (1)
75. i) A i) B
34. B
35. A 76. A= (1,2), B= (1,3), C= (2,3)
77. 250
36. A
37, B 78. i) 3300 i) 1400

38, C ii) 4000 iv) 600

39, C V) S200 vi) 200

40. D vii) 6000
79. (2,3,5)
41. A 80. B, A, A, A, C
81. D, A, C, B, A
82. C, B, D, B, A