Karnataka 1st PUC Mathematics Sets MCQ with Answers

GOVERNMENT OF KARNATAKA
DEPARTMENT OF SCHOOL EDUCATION (PRE-UNIVERSITY)
18TH CROSS, MALLESHWARAM, BENGALURU – 560 012
CHAPTERWISE MULTIPLE CHOICE QUESTIONS FOR COMPETATIVE EXAM
SUBJECT:MATHEMATICS – I PUC
NAME OF THE CHAPTER: SETS

Definition:
 A set is a well-defined collection of objects..
 Object in a set is called its element or member.
 We denote sets by capital letters A, B, C. etc.
 The elements of a set are represented by small letters a, , c, x, y, zetc.
 If x is an element of a set A we write x ∈A, and read as ‘x belongs to A’.
 If x is not an element of a set A, we write x ∉ A, and read as ‘x does not belong to A.’
 For example, zero is a whole number but not a natural number.
∴ 0∈W and 0∉N
Sets and their Representations
1) Roster method or Listing method or Tabular method:
In the Roster method, we list all the elements of the set within the curly (flower) bracket and separate the
elements by commas.
Example: State the sets using Roster method.
i) A is the set of all days in a week.
A = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
ii) B is the set of all vowels in English alphabets.
B = {a, e, i, o, u}
iii) C represents all the Indian states that share a direct land border with Karnataka:
C = {Goa, Maharashtra, Telangana, Andhra Pradesh, Tamil Nadu, Kerala}
iv) D represents the distinct letters found in the capital of Karnataka.
In roster form, duplicate elements are only listed once:
D = {B, E, N, G, A, L, U, R}

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2) Rule method or Property or set builder method:
In the set builder method, we describe the elements of the set by specifying the property
which determines the elements of the set uniquely.
Example: State the sets using set-builder method.
i) A is the Bordering States of Karnataka
A = {x | x is a neighboring sate of Karnataka}
ii) B is the Vowels in the word "KARNATAKA"
B = {x | x is a vowel in the word KARNATAKA}
iii) C is the set of all months of a year.
C = {x | x is month of a year}
iv) D is the set of perfect squares of natural numbers.
D = {x ∈ N/x is perfect square}
Note: 1) In roster form, every element of the set is listed only once.
2) The order in which the elements are listed immaterial
e.g., the sets {1, 2, 3}, {3, 2, 1}, {1, 3, 2} are the same set.
SYMBOLS

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TYPES OF SETS:
Null Set Or Empty Set Or Void Set:
A set containing no element is called an empty or a null set and is denoted by the
symbol  or { } or void set.
e.g. A = {x / x ∈ N, 1<x < 2}, n(A) = 0
B = {x:x is a snow-capped mountain range in Karnataka}
Note:
1.  is never written within brackets.

2. n     0
3. A set consisting of atleast one element is called non-empty set.

Singleton Set:
A Set containing only one element is called a singleton set.
e.g. i) Let A be the set of all integers which are neither positive nor negative.
∴A = {0}, n (A) = 1
ii) A= {x:x is the current capital of Karnatak}
∴A = {Bengaluru}, n (A) = 1
Finite Set:
A set in which the process of counting of elements comes to an end is called a finite
set.
e.g. i) The set of letters in the word 'beautiful'.
A = {b, e, a, u, t, i, f, l}, n (A) = 8
ii) A= {Mysuru, Bengaluru, Badami, Shivamogga}
n(A)=4
Infinite Set:
A set which is not finite, is called an infinite set.
e.g. set of natural numbers.
Note:
o An empty set is a finite set
o N, Z, set of all points on a circle are infinite sets.
Equivalent sets:
Two finite sets A and B are said to be equivalent if n (A) = n (B)
A = {d, o, m, e} & B = {r, a, c, k}
Here n (A) = n (B)
There for A and B are equivalent sets

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Subset:
If A and B are sets such that every elements of A is also an elements of B, then A is said to be a subset of
B, symbolically we write A  B which is also read as “A is contained in B” we may also write this
relationship as B  A which is read as “B is a super set of A” or B contains A
Formula: If a set has “n” elements, then the number of subset of the given set is 2 n and
the number of proper subsets of the given subset is given by 2n  1 .
Note:   A i.e., Null set is a subset of every set.
A  A i.e., Every set is a subset of itself.
If A  B and B  C then A  C
Subsets are classified as
 Proper Subset
 Improper Subsets
A proper subset is one that contains a few elements of the original set whereas an improper subset, contains
every element of the original set along with the null set.
For example, if set A = {2, 4, 6}, then,
Number of subsets: {2}, {4}, {6}, {2,4}, {4,6}, {2,6}, {2,4,6} and Φ or { }.
Proper Subsets: { }, {2}, {4}, {6}, {2,4}, {4,6}, {2,6}
Improper Subset: {2,4,6}
Proper Subset Symbol
A proper subset is denoted by ⊂ and is read as ‘is a proper subset of’. Using this symbol, we can express a
proper subset for set A and set B as A ⊂ B
Improper Subset Symbol
A subset which contains all the elements of the original set is called an improper subset. It is denoted
by ⊆.
Note: The empty set is an improper subset of itself (since it is equal to itself) but it is a proper subset of any other
set.
Equality of sets:
Two sets are said to be equal if they contain the same elements i.e. if A⊆ B and B ⊆ A.
For example:
Let X be the set of letters in the word 'ABBA' and Y be the set of letters in the word 'BABA'.
∴ X = {A, B}, Y = {B, A}
Thus the sets X and Y are equal sets and we denote it by X = Y
Note: Two sets A and B are equal if 𝐴 ⊆ 𝐵 𝑎𝑛𝑑 𝐵 ⊆ 𝐴
Note: Equal sets are always equivalent but equivalent sets may or may not be equal.

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Superset:
If A ⊆ B, then B is called a superset of A and we write, B ⊇ A.
Proper Subset:
A nonempty set A is said to be a proper subset of the set B, if all elements of set A
are in set B and atleast one element of B is not in A.
i.e. If A ⊆ B and A ≠ B then A is called a proper subset of B and we write A ⊂ B.
e.g. Let A = {1,3,5} and B = {1,3,5,7}.
Then, every element of A is an element of B but A ≠ B.
∴ A⊂B, i.e. A is a proper subset of B.
Remark: If there exists even a single element in A which is not in B then A is not a
subset of B and we write A ⊄B
FACTS:
 Number of NONEMPTY proper subsets of a finite set consisting of n elements is 2n  1 (excluding null set)
 Number of proper subsets of set A containing n element are 2n  1 (excluding set itself)
 Number of non empty proper subsets of set A containing n elements are 2n  2
(excluding null set and set itself)
Intervals as subsets of R
Open Interval: Let a, b ∈R and a < b then the set {x / x ∈ R a <x < b} is called open
interval and is denoted by (a,b). All the numbers between a and b belong to the open interval (a,b) but a, b
themselves do not belong to this interval

∴ (a,b) = {x / x ∈ R, a <x < b}
Closed Interval: Let a, b∈R and a<b then the set {x / x ∈ R a ≤ x ≤ b} is called closed
interval and is denoted by [a,b]. All the numbers between a and b belong to the closed interval [a, b].
Also a and b belong to this interval

[a, b] = {x / x ∈ R, a ≤ x ≤ b}
Semi-closed Interval:
[a, b) = {x/x∈R, a ≤ x <b}

Note that a∈[a, b] and b  [a, b]
Semi-open Interval:
(a, b] = {x / x ∈ R, a <x ≤ b}

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(a, b] excludes a but includes b
Infinite Intervals:
The set of all real numbers greater than a
 a,     xR  : x  a

The set of all real numbers greater than or equal to a
 a,     xR  : x  a
The set of all real numbers less than b
 , b    xR  : x  b
The set of all real numbers less than or equal to b
 , b   xR  : x  b
The set of all real numbers i.e. (-∞, ∞)

∴ (-∞,∞) = {x/x∈R, -∞ <x<∞}
Remarks:
1. Four intervals  a, b  ,  a, b  ,(a, b], [a, b) are finite intervals as their length is b-a is real and finite.

2. The intervals  a,   ,[a, ), (, a), (, a] , are infinite intervals as their length is infinite.

3. R   ,   is the set of all reals.

Universal Set:
If in a particular discussion all sets under consideration are subsets of a set, say U,
then U is called the universal set for that discussion.
The set of natural numbers N, the set of integers Z are subsets of set of real numbers R.
Thus, for this discussion R is a universal set.
In general universal set is denoted by 'U' or 'X'
Union of sets
Union of sets A and B is the set of all elements which are in A or in B.
(Here 'or' is taken in the inclusive sense)
Thus, A ∪ B = {x | x∈A or x∈B}
The union of two sets A and B can be represented by a Venn-diagram

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A  B = {Set of all those elements which are in A or in B}

A  B =  x / x  A or x  B
n
A1   x : x  A1 for atleast one i
i 1

If x  A  B then x  A or x  B
Some Properties of the Operation of Union
i) A ∪ B = B ∪ A (Commutativity)
ii) (A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative Property)
iii) A ∪  = A (Identity of Union)
iv) A ∪ A = A (Idempotent law)
v) A∪ A' = U
vi) If A⊂B then A∪B = B
vii) U∪A = U
viii) A⊂(A∪B), B⊂(A∪B)
Ex.: let A = {x/x is a prime number less than 10} & B= {x | x is a factor of 8}. find A∪B.
Solution: We have A = {2,3,5,7} & B = {1,2,4,8}
∴ A∪B = {1, 2, 3, 4, 5, 7, 8}
Intersection of sets
The intersection of two sets A and B is the set of all elements which are both in A and B.
It is denoted by A∩B.
Thus A∩B = {x | x∈A and x∈B

A  B = {Set of all those elements which belong to A and B both}
A  B = {x / x  A and x  B}
n
Symbolically Ai   x : x  A1 for all i
i 1

Some Properties of Operation of Intersection
i) A∩B = B ∩A Commutativity
ii) (A ∩ B) ∩ C = A ∩ (B ∩ C) Associativity
iii)  ∩ A = 
iv) A ∩ A = A Idempotent law
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v) A ∩ A′ = 
vi) If A⊂B then A∩B =A
vii) U∩A = A
viii) (A∩B) ⊂A, (A∩B) ⊂B
Remark: If A∩B = φ, A and B are disjoint sets.
Distributive Property
a) A∩(B∪C) = (A∩B) ∪ (A∩C) b) A∪(B∩C) = (A∪B) ∩ (A∪C)
DeMorgan's law
If A and B are subsets of a universal set, then
i) (A∪B)′ = A′∩ B′ ii) (A∩B)′ = A′∪ B′
Difference of sets
Difference of Set A and Set B is the set of elements which are in A but not in B and is
denoted by A-B or A\B
Backslash: (A\B) Often read as A minus B" or "A set minus B,"
The shaded portion in fig. represents A-B.
Thus, A-B = {x|x∈A, x∉B}
Similar, B-A = {y|y∈B, y∉A}

Let's note:
i) A-B is a subset of A and B-A is a subset of B.
ii) The sets A-B, A∩B, B-A are mutually disjoint sets, i.e. the intersection of any of
these two sets is the null (empty) set.
iii) a) A-B = A∩B' b) B-A = A'∩B
iv) A∪B = (A-B)∪(A∩B) ∪ (B-A)
Shaded portion in the below fig. represents A∪B

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Complement of a set:
Let A be a subset of universal set. The complement of the set A is denoted by A' or AC . It is defined as
A' = {x / x ∈ U x ∉ A} = set of all elements in U which are not in A.
e.g. Let X = {0, 1, 2, 3, 4, 5, 6, 7, 8} be the Universal Set and
A = {2, 4, 6, 8}

∴ The complement of the set A is A' = {0, 1, 3, 5, 7}
Properties:
i) (A')' = A
ii)  ' = U (U is the universal set)
iii) U' = 
iv) If A ⊆ B then B′ ⊆ A′
Properties of Cardinality of Sets:
For given sets A, B
1) n (A∪B) = n(A) + n(B)-n(A∩B)
2) When A and B are disjoint sets
n(A∪B) = n (A) + n(B), as A∩B = φ,
∴n(A∩B) = 0
3) n(A∩B') + n(A∩B) = n(A)
4) n(A'∩B) + n(A∩B) = n(B)
5) n(A∩B') + n(A∩B) + n(A'∩B) = n(A∪B)
For any sets A, B, C.
6) n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B)-n(B∩C) - n(A∩C) + n(A∩B∩C)
7) If n(A) = m, n(P(A)) = 2m where P(A) is the power set of A
8) n(P') = n(x) - n(P)

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Set &their Representations
1. Which of the following is a set?
A) The collection of all the months of year beginning with the letter J
B) The collection of ten most talented writers of India.
C) A team of eleven best-cricket batsman of the world.
D) A collection of most dangerous animals of the world.
2. Which of the following is a not set?
A) The collection of months in a year.
B) The collection of some months in a year
C) The collection of first ten prime numbers.
D) The collection of days in a week.
3. Of the following concepts, the only one that is well defined in an axiomatic development of set theory
is
A) Set B) Element C) Belongs to D) Super set of
4. Which of the following options represents the complete set of states that directly share a land border
with Karnataka?
A) {Kerala, Andhra Pradesh, Telangana, Tamil Nadu, Odisha}
B) {Maharashtra, Goa, Kerala, Tamil Nadu, Madhya Pradesh}
C) {Goa, Maharashtra, Telangana, Andhra Pradesh, Tamil Nadu, Kerala}
D) {Maharashtra, Goa, Gujarat, Kerala, Telangana}
5. Write the set builder for of A = {-1,1}
A) A = {x: x is a natural number}
B) A = {x: x is a root of equation 𝑥 2 + 1 = 0}
C) A = {x: x is a real number}
D) A = {x: x is a root of equation 𝑥 2 = 1}
6. If set A = {1, 2, 3, 4} then it can be written in set builder form as
A) 𝐴 = {𝑥: 𝑥 ∈ 𝑁 𝑎𝑛𝑑 𝑥 ≤ 5} B) 𝐴 = {𝑥: 𝑥 ∈ 𝑁 𝑎𝑛𝑑 𝑥 < 5}
C) 𝐴 = {𝑥: 𝑥 ∈ 𝑁 𝑎𝑛𝑑 1 < 𝑥 < 5} D) 𝐴 = {𝑥: 𝑥 ∈ 𝑁 𝑎𝑛𝑑 𝑥 < 4}
1 2 3 4 5 6 
7. The set  , , , , ,  in the set builder form is
3 4 5 6 7 8 
 n   n 1 
A)  x : x  ,where n  N ,1  n  6  B)  x : x  ,n  N ,1  n  6 
 n 1   n2 
 n   n 
C)  x : x  ,n  N  D)  x : x  ,n  N ,1  n  6 
 n2   n2 

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8. The roster form of the set
A  {x | x is a positive integer less than 10 and 2x  1 is an odd number}is

A) A  1, 2,3, 4,5,6,7,8,9,10 B) A  1, 2,3, 4,5,6,7,8,9

C) A  1, 2,3, 4,5,6 D) A  1, 2,3, 4,5,6,7,8,9,10,11

9. Roster form of the set D  {t | t 3  t , t  R} is

A) 1,1 B) 0,1 C) 0,1, 2 D) 1,0,1

10. The set {𝑥 ∈ 𝑅: 𝑥 − 2 + 𝑥 2 = 0} is equal to
A) {–1, 2} B) {1, 2} C) {–1, –2} D) {1, –2}
11. There are three sets are given in column A and column B
Column A Column B
(i) 1, 2,3,6 (a) {x: x is a prime number and divisor of 6}

(ii) 2,3 (b) {x: x is odd natural number less than 10}

(iii) 1,3,5,7,9 (c) {x: x is a natural number and divisor of 6}

which one of the following matches is correct?
A) 1  a, 2  b, 3  c B) 1  b, 2  a, 3  c C) 1  c, 2  a, 3  b D) 1  a, 2  c, 3  b
12. There are three sets given in column A and column B
Column A Column B
(i)  x  R | 2 x  11  15 (p) {P, 1}

(ii) x | x  x, x  R
2 (q) {2}

(iii) {x | x is a positive factor of prime number P (r) {0, 1}
which one of the following matches is correct?
A) (i)  (p); (ii)  (r); (iii)  (q) B) (i)  (q); (ii)  (p); (iii)  (r)
C) (i)  (q); (ii)  (r); (iii)  (p) D) (i)  (p); (ii)  (q); (iii)  (r)
13. Each of the set on the left in the roster form with the same set on the right described in set builder
form is given below:
(i) 1, 2,3,6 (p) {x: x is a prime number and a divisor of 6}

(ii) 2,3 (q) {x : x is an odd natural number less than 10}

(iii) M , A, T , H , E , I , C , S  (r) {x: x is a natural number and divisor of 6}
(iv) 1,3,5,7,9 (s) {x: x is a letter of the word “MATHEMATICS”}

A) (i)(r);(ii)(q);(iii)(s);(iv)(p) B) (i)(r);(ii)(p);(iii)(s);(iv)(q)
C) (i)(r);(ii)(p);(iii)(q);(iv)(s) D) (i)(p);(ii)(r);(iii)(s);(iv)(q)

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14. Which of the following statement is true?
A) 37 {x | x has exactly two positive factors}
B) 28 { y | the sum of all positive factors of y is 2y}
C) 7,747 {t | t is a multiple of 37}
D) 128 { y | the sum of all the positive factors of y is 2y}

Set &their Representations
1 2 3 4 5 6 7 8 9 10
A B D C D B D B D D
11 12 13 14
C C B B

Types Of Sets (Empty set, Equal set, Finite & Infinite sets)
1. In rule method the null set is represented by
A)  x | x  x B)  x | x  x C)  D)  

2. Which of the following is an empty set?
A) A   x : x  N , 2 x  5  6 B) B   x : x  N ,1  x  2

C) C   x : x is an even prime D) D   x : x  Q,1  x  2

3. Which of the following is an empty set?
A) {x|x is a real number and 𝑥 2 − 1 = 0} B) {x|x is a real number and 𝑥 2 + 3 = 0}
C) {x|x is a real number and 𝑥 2 − 9 = 0} D) {x|x is a real number and 𝑥 2 = 𝑥 + 2}
4. Which of the following is null set?
A)  x : x  1,x  N  B)  x : x  5,x  Z 

C)  x : x 2  1,x  Z  D)  x : x 2  2 x  1  0,x  R

5. Which of the following is not an empty set?
A)  x :1  x  2; x is natural number B) { x : x 2  2  0, x is rational number}

C)  x : x 2  4, x is odd D) {x: x is a real number and x2 = 9 = 0}

6. Which of the following sets are equal?
A) A  a,b,c,d  ,B  d ,c,b,a B) A  4,8,12,16 B  8, 4,12,16

C) A   x : x is a multiple of 10 ,B  10,15, 20, 25.. D) Both A and B

7. Which of the following is a singleton set?
A)  x : x  5,x  N  B)  x : x  6,x  Z 

C)  x : x 2  7 ,x  N  D) Both A and B

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8. Which of the following is infinite set?
A)  x  N : 2.5  x  3.5 B)  x  R : x 2  2

C)  x  R : 2.5  x  3.5 D)  x  Z : 2.5  x  3.5

9. Which of the following is finite set?
A) Set of rational numbers in  0,1 B) Set of irrational numbers in  0,1

C) Set of natural numbers in  0,1 D) All the above

 
10. A set T  x  R |  x  3   x  4  ,x  2 . Which of these would be TRUE about the set T?
2

A) T is an empty set B) T is an singleton set
C) x  3 is a member of T D) x  4 is a member of T
11. Consider the following sets
I. The set of lines which are parallel to the X axis
II. The set of animals living on earth
III. The set of circles passing through the origin (0, 0)
Of these sets, the infinite sets are
A) I and II B) II and III C) only I D) I and III
12. Which of the following sets are equal?
A) A  2,3 & B  {x : x is solution of x 2  5 x  6  0 }

B) A  {x : x is a multiple of 10} & B  10,15, 20, 25,30...

C) A  {x : x is a letter in the word FOLLOW} & B  { y : y is a letter in the word WOLF}

D) A  {n : n  z and n 2  4} & B   x : x  R and x 2  3x  2  0

13. How many possible rational and irrational number between 0 and 1
A) Infinite B) 0 C) finite D) 1
Types Of Sets (Empty set, Equal set, Finite & Infinite sets)
1 2 3 4 5 6 7 8 9 10
A A B A D D A C C A
11 12 13
D C A

2026-27 MATHEMATICS CET MATERIAL Page 13 of 27

Subsets
1. All the subsets of the set 1,0,1 are

A) , 1 0 , 1 1,0 1,1 , 0,11,0,1

B) ,1 0 1 1,0 1,1 1,0,1

C) 1 0 1 1,0 1,1 0,1 1,0,1

C)  1 0 1 1,0 1,1 0,1 1,0,1

2. Consider the following statements:
1. The null set is a subset of every set
2. Every set is a subset of itself
3. If a set has “n” elements, then the number of subset of the given set is 2n  1
Which of the following statements is/are correct?
A) 1 only B) 2 only C) 1 and 2 D) 1, 2 and 3
3. If a set has n elements than the total number of subset subsets of A is
A) n B) n 2 C) 2 n D) 2n
4. The number of nonempty subsets of the set 1, 2,3, 4 is

A) 15 B) 14 C) 16 D) 17
5. Let A   x : x  4n  1, 2  n  5 then the number of subsets of A is

A) 8 B) 15 C) 4 D) 16
6. The number of proper subsets of a set having n+1 elements
A) 2n1 B) 2n1  1 C) 2n1  2 D) 2n2
7. If power set of A has 2k 1 elements then A has _____ elements

B) k  1
k1
A) k C) 22 D) 2
8. If A  1, 2,3, 4,5 , then the number of proper subsets of A is

A) 120 B) 30 C) 31 D) 32
9. Let S = {1, 2, 3, …, 10}. The number of subsets of S containing only odd numbers is
A) 15 B) 31 C) 63 D) 5
10. If A  1, 2,3, 4,5,6 , then the number of subsets of A which contains atleast two elements is

A) 63 B) 57 C) 58 D) 64
11. If A  B and A  B , then
A) A is called a proper subset of B B) A is called a super set of B
C) A is not a subset of B D) B is a subset of A

2026-27 MATHEMATICS CET MATERIAL Page 14 of 27

12. X  8n  7n  1| n  N  and Y  49  n  1 | n  N  then

A) X ⊂ Y B) Y ⊂ X C) X = Y D) X and Y are disjoint sets
13. 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 respectively are,
A) 4, 7 B) 7, 4 C) 4, 4 D) 7, 7
14. Two finite sets have m and n elements. The number of subsets of the first set is 48 more than that of
the second set. The values of m and n respectively are,
A) 7, 6 B) 6, 3 C) 6, 4 D) 7, 4
15. Suppose P, Q and R are three sets, each with two elements. The number of subsets of the set P × Q ×
R, that have at least 2 elements is
A) 247 B) 257 C) 255 D) 248
16. Consider the following statements:
i)    ii)   

which of the given statement is/are correct
A) 1 only B) 2 only C) Both 1 and 2 D) Neither 1 nor 2
17. Which of the following statements is/are correct?
1. 2,3, 4  1, 2,3, 4,5 2. a,b  a,b,c 3.  x : x is anisoceles traingale   x : x is a traingle

A) 1 and 3 B) 1 and 2 C) 2 and 3 D) 3 only
18. Which of the following is false
A) {x: x is even natural number less than 6} {x: x is a natural number which divides 36}
B) {a, e}  {x: x is a vowel in English alphabet}
C) 1, 2,3  {1,3,5} D) a  a, b, c

   
19. Let X  1, , 42, 2 ,1,3 . Which of the following statements is/are true

i) P :   X ii) Q : 1,3  X iii) R : 1,   X

A) P only B) Q only C) R only D) P and R only
20. If A  3,4,5 ,6 then which of the following statements are incorrect

A) 4,5  A B) 4,5  A C)   A D) 3,6  A

Subsets
1 2 3 4 5 6 7 8 9 10
A C C A D B B C B B
11 12 13 14 15 16 17 18 19 20
A A B C A A A C D A

2026-27 MATHEMATICS CET MATERIAL Page 15 of 27

Intervals as subsets of R
1. The subset of R as intervals {𝒙: 𝒙 ∈ 𝑹, − 𝟏𝟐 < 𝑥 < −10} is
A) (-12,10] B) [-12, -10] C) (-12, -10) D) [-12, -10)
2. The interval  3,5 in set builder form is

A)  x : 3  x  5 B)  x : 3  x  5 C)  x : 3  x  5 D)  x : 3  x  5

3. The set of all natural numbers ‘x’ such that 4 x  9  50 is equal to
A)  x  N : x  1,10 B)  x  N : x  1,10 

C)  x  N : 1  x  11 D)  x  N : 1  x  11

4. If A  3,5 and B   7 ,9 , then

A) B  A B) A  B C) A  B D) A  B
5. Which one of the following have same sense w.r.t. open and closed intervals?
A) A   2,3 and B   x : x  R, 2  x  3

B) A   3,5 and B   x : x  R, 3  x  5

C) A   15, 2 and B   x : x  R, 15  x  2

D) A  1, 2  and B   x : x  R, 1  x  2

Intervals as subsets of R
1 2 3 4 5
C A A B A
Venn Diagrams
1. The Venn diagram in the figure represents

A)  A  B  B)  A  B  C)  A  B    A  B  D)  A  B    B  A 
2. Consider the following with respect to the following diagram

(a)  A  B  (b)  A  B  (c) A  B

of these operations, which of the following is correct
A) only (a) B) both (a) and (c) C) both (a) and (b) D) only (c)

2026-27 MATHEMATICS CET MATERIAL Page 16 of 27

3. This Venn diagram shows information about the number of people who have brown hair and the
number of people who wear glasses.

How many people have brown hair and glasses?
A) 10 B) 11 C) 4 D) 15
4. Max asks 𝟒𝟎 people whether they own a cat or a dog. 𝟏𝟕 people own a dog, 𝟏𝟒 people own a cat
and 𝟕 people own a cat and a dog. Choose the correct representation of this information on a Venn
diagram.

5. The following Venn diagrams each show two sets, set 𝐀 and set 𝐁. On which Venn diagram
has 𝐀′ been shaded?

6. Some people visit the theatre. The Venn diagram shows the number of people who bought ice cream
and drinks in the interval.
Ice cream is sold for £𝟑 and drinks are sold for £𝟐. A total of £𝟐𝟔𝟐 is
spent. How manypeople bought both a drink and an ice cream?
A) 27 B) 48 C) 22 D) 30

2026-27 MATHEMATICS CET MATERIAL Page 17 of 27

7. If U={1,2,3,4,5,6,7,8,9,10,11,12}, A = {multiples of 2}, A∩B={2,4,6,12}, A  B  1, 2,3, 4,6,8,10,12 By

putting this information onto the following Venn diagram, list all the elements of B.

A) {1,2,3,4,6,12} B) {1,3} C) {5,7,9,11} D) 
8. Directions to Solve. Study the following figure and answer the questions given below

By which number, married but untrained nurses in the hospital are represented?
A) 4 B) 6 C) 7 D) 5
9. In the given diagram, the rectangle represents doctors, the triangle represents surgeons, the circle
represents researchers, and the square represents hospital staff. Identify the region representing
surgeons who are doctors but not researchers.

A) 12 B) 5 C) 17 D) 7
10. Directions to Solve In the following figure small square represents the persons who know English,
triangle to those who know Marathi, big square to those who know Telugu and circle to those who
know Hindi. In the different regions of the figures from 1 to 12 are given.

How many persons can speak Marathi and Telugu both?
A) 10 B) 11 C) 13 D) 7

2026-27 MATHEMATICS CET MATERIAL Page 18 of 27

11. In this diagram, a total number of 100 players play different games.

How many players play Football and Hockey but not Cricket?
A) 20 B) 25 C) 15 D) 5
12. Find out the number of people who do not play any game

A) 18 B) 9 C) 15 D) 24
13. In the given figure, circles represent students studying three different subjects. How many students
study all the three subjects?

A) 2 B) 4 C) 3 D) 1
14. The following diagram represents people who speak different languages.
(i) Kannada (ii) English (iii) Hindi (iv) Marathi
What does the shaded area represents?

A) People who speak English and Hindi.
B) People who speak Kannada, English and Hindi.
C) People who speak Kannada, English and Marathi
D) People who speak Kannada and English.
2026-27 MATHEMATICS CET MATERIAL Page 19 of 27

15. The shaded region in the given Venn diagram is

A) A  (B - C) B) A  (B  C) C) A  (B  C) D) A – (B  C)
16. The shaded region in the given figure is
A

C B

A) A  (B  C) B) A  (B  C) C) A  (B – C) D) A – (B  C)
Venn Diagrams
1 2 3 4 5 6 7 8 9 10
A B C B B D A C A D
11 12 13 14 15 16
C B C D C D

Operations on Sets & Complement of a set
1. A set A has 3 elements and another set B has 6 elements. Then
A) 3 ≤ 𝑛(𝐴 ∪ 𝐵) ≤ 6 B) 3 ≤ 𝑛(𝐴 ∪ 𝐵) ≤ 9
C)6 ≤ 𝑛(𝐴 ∪ 𝐵) ≤ 9 D) 0 ≤ 𝑛(𝐴 ∪ 𝐵) ≤ 9
2. Consider the following statements,
1. If n  A  3,n  B   6, then maximum number of elements of n  A  B   9

2. n  A  B  will be the maximum if n  A  B  is minimum

Which of the above statements is/are correct?
A) 1 only B) 2 only C) Both 1 and 2 D) Neither 1 nor 2
3. If A = {2, 3, 4, 8, 10}, B = {3, 4, 5, 10, 12} and C = {4, 5, 6, 12, 14}, then (𝑨 ∪ 𝑩) ∩ (𝑨 ∪ 𝑪) is equal to
A) {2, 3, 4, 10, 12} B) {2, 3, 4, 5, 8, 10, 12}
C) {2, 3, 4, 10, 12} D) {2, 3, 4, 5, 8, 10, 12, 14}
4. If 𝑨 = {𝒙: 𝒙𝟐 − 𝒙 + 𝟐 > 0} and 𝑩 = {𝒙: 𝒙𝟐 − 𝟒𝒙 + 𝟑 ≤ 𝟎}, then 𝑨 ∩ 𝑩 is
A) [1,3] B) (−∞, ∞) C) (1,3) D) (−∞, 1) ∪ (3, ∞)
𝒙
5. Let𝑨 = {𝒙 ∈ 𝒁: −𝟏 ≤ 𝒙 < 4} and let 𝑩 = {𝒙 ∈ 𝒁: 𝟎 < 𝟐 ≤ 𝟑}. Then 𝑨 ∩ 𝑩 is equal to

A) {1, 2, 3} B) {2, 3} C) {1, 2, 3, 4} D) {0, 1, 2, 3}

2026-27 MATHEMATICS CET MATERIAL Page 20 of 27

6. Universal set,
U   x | x5  6 x 4  11x3  6 x 2  0 , A   x | x 2  5 x  6  0 , B   x | x 2  3 x  2  0

What is (𝑨 ∩ 𝑩)′ equal to
A) {1, 3} B) {1, 2, 3} C) {0, 1, 3} D) {0, 1, 2, 3}
7. If A = {1, 2, 3, 4, 5} and B = {2, 4, 6}, then A – B =
A) {1, 3, 5, 6} B) {0, 1, 3, 5, 6} C) {1, 3, 5} D) {2, 4}
8. A and B are two sets, then A  B   , if and only if
A) A  B B) B  A C) A = B D) A  B  
9. If A\B = {a, b}, B\A = {c, d} and 𝑨 ∩ 𝑩 = {𝒆, 𝒇}, then the set B is equal to
A) {a, b, c, d} B) {e, f, c, d} C) {a, b, e, f} D) {d, e, a, b}
10. If A and B are any two given sets, then A   B  A  is
A) A  B B) A  B C) A D) B 
11. For any two sets A and B  A  B    A  B  is equal to

A) A B) B C) A D) B
12. For all sets A and B which among the following is false?
A) A   A  B   A  B B) A   B  A   A  B

C) A   A  B   A  B D)  A  B   B  A  B

13. Consider the following for any two sets A and B
(a)  A  B   B  A (b)  A  B   A  A (c)  A  B   B   (d) A  B  A  B  B

Which of these are correct?
A) (a), (b) and (c) B) (b), (c) and (d) C) (a), (c) and (d) D) (a), (b) and (c)
14. For any two sets A and B A   A  B  equals
A) B B) A  B C) A  B D) A  B
15. Let A and B be two sets such that A  B  A, then A  B is equal to
A)  B) B C) A D) A  B

16. If A and B are any two sets, then A   A  B  is equal to

A) A B) B C)  D) A  B

17. If A, B and C are any three sets, then A   B  C  is equal to

A)  A  B    A  C  B)  A  B    A  C  C)  A  B   C D)  A  B   C

18. The set  A  B  C    A  B  C    C  is equal to

A) B  C B) A  C C) B  C D) A  C

2026-27 MATHEMATICS CET MATERIAL Page 21 of 27

19. If A  B then B  A is equal to
A) A B) B  C) A  B D) 

20. If A and B are two sets then  A  B    B  A    A  B  is equal to
A) A  B B) A  B C) A D) B 

21. Let U be the universal set and A  B  C  U . Then  A  B    B  C    C  A is equal to

A) A  B  C B) A   B  C  C) A  B  C D) A   B  C 

22. The set (𝑨\𝑩) ∪ (𝑩\𝑨) is equal to
A) [𝐴\(𝐴 ∩ 𝐵)] ∩ [𝐵\(𝐴 ∩ 𝐵)] B) (𝐴 ∪ 𝐵)\(𝐴 ∩ 𝐵)
̅̅̅̅̅̅̅
C) (𝐴 ∪ 𝐵 )\(𝐴 ∪ 𝐵) D) (𝐴̅ ∪ 𝐵)\(𝐴
̅̅̅̅̅̅̅
∪ 𝐵)
23. Let X and Y be two non-empty sets such that 𝑿 ∩ 𝑨 = 𝒀 ∩ 𝑨 = ∅ and 𝑿 ∪ 𝑨 = 𝒀 ∪ 𝑨 forsome non-
empty set A. Then
A) X = Y B) 𝑌\𝐴 = ∅ C) 𝑋\𝐴 = ∅ D) 𝑋 ∪ 𝑌 = 𝐴

24. If A   x, y  : x 2  y 2  1, x, y  R & B   x, y  : x 2  y 2  4, x, y  R then

A) A – B = A B) B – A = B C) A – B = ∅ D) B – A = ∅
25. If A   x, y  : x 2  y 2  25 and B   x, y  : x 2  16 y 2  144 , then A  B contains

A) one point B) two points C) three points D) four points
26. If A and B are disjoint sets, then B ∩ A', where A' is complement of A is equal to
A) A B) B C) A' D) B'
27. If A = {1, 3, 5, 7, 9, 11, 13, 15, 16, 17}, B = {2, 4, 6, 8, 10, 12, 14, 16, 18} and
N = {1, 2, 3, 4, 5, ……18} is the universal set, then A'∪ ((A ∪B)∩ B′) is
A) A B) N C) B D) 𝐴 ∩ 𝐵
28. If X and Y are two sets, then X∩ (Y∪X)' equals
A) X B) Y C) ∅ D) X-Y
29. Which one of the following is false?
A) A    A B)   A  A C) U  A  A D)  A

30. Which one of the following pairs of sets are disjoint?
A) 1, 2,3, 4 and {x : x is a natural number and 4  x  6 }

B) a, e, i, o, u and c, d , e, f 

C) {x : x is an even integer} and {x : x is an odd integer}

D) a, e, i, o, u and a, b, c, d 

2026-27 MATHEMATICS CET MATERIAL Page 22 of 27

31. Let F1 be the set of parallelograms, F2 be the set of rectangles, F3 be the set of rhombus, F4 be the set
of squares and F5 be the set of trapeziums in a plane. Then, F1 may be equal to
A) F2 ∩ F3 B) F3 ∩ F4 C) F2∪ F5 D) F2∪ F3∪ F4∪ F1
32. Let S = set of points inside the square, T = set of points inside the triangle and
C = set of points inside the circle. If the triangle and circle intersect each other and are connected in a
square, then
A) S  T  C   B) S  T  C  C C) S  T  C  S D) S  T  S  C
33. Given sets A = {1, 2, 3}, B = {2, 3, 5} and C = {4, 5, 6} identify which of the following statement is
incorrect.
A) B – C = {2, 3} B) 𝑛[(𝐴 ∪ 𝐵) ∩ (𝐵 ∩ 𝐶)] = 1
C) (𝐴 ∩ 𝐵) ∪ 𝐶 = 𝐴 ∩ (𝐵 ∪ 𝐶) D) (𝐴 ∩ 𝐵) ∩ 𝐶 = ∅
𝒙
34. Let 𝑨 = {𝒙 ∈ 𝒁: −𝟏 ≤ 𝒙 < 4} and 𝑩 = {𝒙 ∈ 𝒁: 𝟎 < 𝒙 ≤ 𝟑}.Then A  B 

A) 1, 2,3 B) 0,1, 2,3 C) 2,3 D) 2,3, 4

35. The set expression 𝑨 ∪ (𝑩 ∩ (𝑨𝒍 ∪ 𝑩𝒍 )) is equivalent to

A) (𝐴𝑙 ∪ 𝐵 𝑙 )𝑙 B) 𝐴 ∩ 𝐵 C) 𝐴 ∪ 𝐵 D) 𝐴 ∩ 𝐵 𝑙
36. If N a  an : n  N  and Nb  N c  N d where a,b,c,d  N and b,c are relatively prime then

b
A) d  b  c B) d  bc C) d  b  c D) d 
c
37. Let N a  an : n  N  , then N 6  N8 is equal to

A) N2 B) N48 C) N8 D) N24
38. Let A = {All prime numbers less than 10}, B = {All odd numbers less than 10} find A   A  B 

A)  B) {1,2} C){2} D) {2,3,5,7}
 
39. If set A and B are defined as A   x, y  | y  , 0  x  R  ,B   x, y  | y   x,x  R then,
1
 x 
A) A  B  A B) A  B  B C) A  B   D) A  B  A
40. Let S   x | x is a postive multiple of 3 less than 100 , P   x | x is a prime less than 20 . Then

n  S   n  P  is

A) 34 B) 41 C) 33 D) 30
41. If A and B are two finite sets such that A  B   , then n  A  B  

A) n  A  n  B  B) n  A  n  B   n  A  B 

C) n  A  n  B   n  A  B  D) n  A .n  B 

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42. If A and B are finite sets and A  B then
A) n  A  B   n  A B) n  A  B   n  B 

C) n  A  B   n  B  D) n  A  B   n  A
100
43. If A1 , A2 ,....A100 are sets such that n  Ai   i  2, A1  A2  A3  ....  A100 and A  Ai then n  A 
i 3

A) 3 B) 4 C) 5 D) 6
 79 
44. If A1  A2  A3  ....  A79 and n  Ai   i  2, then n  Ai  
 i 1 
A) 80 B) 81 C) 90 D) 91
20 n
45. Each set Xr contains 5 elements and each set Yr contains 2 elements and Xr  S  Yr . If each
r 1 r 1

element of S belong to exactly 10 of the Xr ’s and to exactly 4 of the Yr ’s, then n is
A) 10 B) 20 C) 100 D) 50
46. Suppose A1 , A2 ,...A30 are thirty sets each having 5 elements and B1 ,B2 ,....Bn are n set seach with 3
30 n
elements. Let, Ai  B j  S and each elements of S belongs to exactly 10 of the Ai ' s and exactly
i 1 j 1

9 of the Bi ' s , then n =
A) 15 B) 3 C) 45 D) 35
47. Given n U   20,n  A  12,n  B   9,n  A  B   4, where U is the universal set, A and B are subsets of

U, then n  A  B  =
l

A) 17 B) 9 C) 11 D) 3
48. Let 𝑨 and 𝑩 be two subsets of a universal set 𝑼. If 𝒏(𝑼) = 𝟏𝟏𝟔, 𝒏(𝑨 ∪ 𝑩) = 𝟗𝟗,
𝒏(𝑩) = 𝟔𝟏 and 𝒏(𝑨 ∩ 𝑩) = 𝟐𝟖, then 𝒏(𝑨′ ) is equal to
A) 47 B) 45 C) 53 D) 50
49. Let A and B denote the sets of students in a school who play cricket and football
respectively. If 𝒏(𝑨) = 𝟒𝟓, 𝒏(𝑩) = 𝟑𝟓 and 𝒏(𝑨 ∩ 𝑩 = 𝟏𝟑), then 𝐧((𝑨 ∩ 𝑩)′ ∩ (𝑨 ∪ 𝑩)) =
A) 43 B) 48 C) 50 D) 54
50. Let 𝑿 = {𝒂, 𝒃, 𝒄, 𝒅, 𝒆} and 𝑨 = {𝒂, 𝒃, 𝒄, 𝒅,}. 𝑳𝒆𝒕 𝐏 = {𝑩: 𝑩 ⊆ 𝐗 and 𝐀 ∖ 𝐁 = {𝒅}} . Then the number of
elements in the set P, is
A) 1 B) 2 C) 4 D) 3
51. Let A and B be finite sets such that n  A  B   18,n  A  B   25 and n  A  B   70 .

Then n  B  is equal to

A) 52 B) 25 C) 27 D) 43
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52. If n  P   8,n  Q   10 and n  R   5 (‘n’ denotes cardinality) for three disjoint sets P,Q,R then

n  P  Q  R 

A) 23 B) 20 C) 18 D) 15
53. Consider the following relations:
i) A  B  A   A  B  ii) A   A  B    A  B  iii) A   B  C    A  B    A  C 

which of these is/are correct
A) 1 & 3 B) 2 Only C) 2 & 3 D) 1 & 2
54. If U   x : x  N and 2  x  12 , A   x : x is an even prime , B   x : x is a factor of 24 , then

Which of the following is not true

D)  A  B   Al  B l
l
A) A  B is an empty set B) A  B  B  Al C) Al  Bl  B  A

55. If A   x : x is an int eger and x 2  9  0 , B   x : x is a natural number and 2  x  5 and

C   x : x is a prime number  4 . Then  B  C   A is

A) 3,3, 4 B) 2,3, 4 C) 3, 4,5 D) 2,3,5

56. Let U=R. If A   x  R : 0  x  2 ,B   x  R : 1  x  3 , then which of the following is incorrect?

A) Al   x  R : x  0 or x  2 B) Bl   x  R : x  1 or x  3

C) A  B   x  R : 0  x  3 D) A  B   x  R : 1  x  2

57. Consider the following statements in respect of sets:
1. The union over intersection of sets is distributive.
2. The complement of union of two sets is equal to intersection of their complements.
3. If the difference of two sets is equal to empty set, then the two sets must be equal.
Which of the following is/are correct?
A) 1 and 2 B) 2 and 3 C) 1 and 3 D) 1, 2 and 3
58. If A and B are finite sets, then consider the following,
(a) n  A  B   n  A  B   n  A  B   n  B  A  (b) n  A  B   n  A   n  B   n  A  B 

(c) n  A  B   n  A   n  B 
of these which of the following is correct?
A) only (a) is correct B) only (b) is correct
C) both (a) and (b) are correct D) both (a) and (c) are correct

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59. In a group of 50 people, 35 speak Kannada, 25 speak both Kannada and English, all the people speak
one of the 2 languages, then which option is the correct match for the following
(a) number of people speak English (i) 10
(b) number of people speak only English (ii) 40
(c) number of people speak only Kannada (iii) 15
A) (a)  (ii), (b)  (i), (c)  (III) B) (a)  (ii), (b)  (i), (c)  (iii)
C) (a)  (ii), (b)  (iii), (c)  (i) D) (a) (iii), (b)  (ii), (c)  (i)
60. In a group of 50 students, the number of students studying French, English, Sanskrit were found as
follows French = 17, English = 13, Sanskrit = 15, French and English = 09, English and Sanskrit = 4,
French and Sanskrit = 5, English, French and Sanskrit = 3, the number of students who studies in
respective subject are in column B with respect to column A
Column A Column B
(a) French only (i) 20
(b) English and Sanskrit but not French (ii) 30
(c) Atleast one of three languages (iii) 6
(d) None of the three languages (iv) 1
which of the following is correct match
A) (a)  (i); (b)  (iii); (c)  (iv); (d)  (ii)
B) (a)  (iii); (b)  (iv); (c)  (ii); (d)  (i)
C) (a)  (iii); (b)  (iv); (c)  (i); (d)  (ii)
D) (a)  (i); (b)  (iv); (c)  (iii); (d)  (ii)
Operations on Sets & Complement of a set
1 2 3 4 5 6 7 8 9 10
C C B A A C C A B A
11 12 13 14 15 16 17 18 19 20
A B B C B D A A D A
21 22 23 24 25 26 27 28 29 30
C B A C D B B C B C
31 32 33 34 35 36 37 38 39 40
D C C A C B D D C B
41 42 43 44 45 46 47 48 49 50
C C C B B C D D D B
51 52 53 54 55 56 57 58 59 60
A A D B A C A C C B

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