Sets – Important Questions with Solutions (Class XI Mathematics), Ideal for CBSE & State Board Preparation

JEE - MATHS
EAT – (Easy, Average, Tough)
Chapter Name: Relations
Medium
1. Let 𝐴 = {1,2,3}. The total number of distinct relations that can be defined over 𝐴
is
(a) 29
(b) 6
(c) 8
(d) None of these
Solution:
(a) 𝑛(𝐴 × 𝐴) = 𝑛(𝐴) ⋅ 𝑛(𝐴) = 32 = 9
So, the total number of subsets of 𝐴 × 𝐴 is 29 and a subset of 𝐴 × 𝐴 is a
relation over the set 𝐴.

2. Let 𝑋 = {1,2,3,4,5} and 𝑌 = {1,3,5,7,9}. Which of the following is/are relations from
𝑋 to 𝑌
(a) 𝑅1 = {(𝑥, 𝑦) ∣ 𝑦 = 2 + 𝑥, 𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌}
(b) 𝑅2 = {(1,1), (2,1), (3,3), (4,3), (5,5)}
(c) 𝑅3 = {(1,1), (1,3)(3,5), (3,7), (5,7)}
(d) 𝑅4 = {(1,3), (2,5), (2,4), (7,9)}
Solution:
(a,b,c) 𝑅4 is not a relation from 𝑋 to 𝑌, because (7,9) ∈ 𝑅4 but (7,9) ∉ 𝑋 × 𝑌.
3. Let 𝑅1 and 𝑅2 be relations on the set {1,2, … … .50} such that
𝑅1 = {(𝑝, 𝑝𝑛 ) : 𝑝 is a prime and 𝑛 ≥ 0 is an integer } and
𝑅2 = {(𝑝, 𝑝𝑛 ) : 𝑝 is a prime and 𝑛 = 0 or 1}.
Then, the number of elements in R1 − R 2 is
(a) 9
(b) 6
(c) 8
(d) None of these
Solution:
𝑅1 − 𝑅2 = {(2, 22 ), (2, 23 ), (2, 24 ), (2, 25 ), (3, 32 ), (3, 33 ), (5, 52 ), (7, 72 )}
So number of elements = 8
4. Given two finite sets 𝐴 and 𝐵 such that 𝑛(𝐴) = 2, 𝑛(𝐵) = 3. Then total number of
relations from 𝐴 to 𝐵 is
(a) 4
(b) 8
(c) 64
(d) None of these
Solution:

, (c) Here 𝑛(𝐴 × 𝐵) = 2 × 3 = 6
Since every subset of 𝐴 × 𝐵 defines a relation from 𝐴 to 𝐵, number of relation
from 𝐴 to 𝐵 is ec number of subsets of 𝐴 × 𝐵 = 26 = 64, which is given in (c).
5. The relation 𝑅 defined on the set of natural numbers as {(𝑎, 𝑏): 𝑎 differs from 𝑏 by
3}, is given
(a) {(1,4, (2,5), (3,6), … … }
(b) {(4,1), (5,2), (6,3), ….
(c) {(1,3), (2,6)
Solution:
(b) 𝑅 = {(𝑎, 𝑏): 𝑎, 𝑏 ∈ 𝑁, 𝑎 − 𝑏 = 3} = {((𝑛 + 3), 𝑛): 𝑛 ∈ 𝑁} = {(4,1), (5,2), (6,3) …..
6. Let 𝐴 = {1,2,3}, 𝐵 = {1,3,5}. A relation 𝑅: 𝐴 → 𝐵 is defined by 𝑅 =
{(1,3), (1,5), (2,1)}. Then 𝑅−1 is defined by
(a) {(1,2), (3,1) , (1,3), (1,5)}
(b) {(1,2), (3,1) , (2,1)}
(c) {(1,2), (5,1), (3,1)}
(d) {(4,1), (5,2) , (6,3)}
Solution:
(c) (𝑥, 𝑦) ∈ 𝑅 ⇔ (𝑦, 𝑥) ∈ 𝑅 −1 , ∴ 𝑅 −1 = {(3,1), (5,1), (1,2)}.

7. The relation 𝑅 is defined on the set of natural numbers as {(𝑎, 𝑏 ): 𝑎 = 2𝑏 }. Then
𝑅 −1 is given by
(a) {(2,1), (4,2), (6,3) …..
(b) {(1,2), (2,4) , (3,6) … }.
(c)𝑅 −1 is not defined
(d) None of these
Solution:
(b) 𝑅 = {(2,1), (4,2), (6,3), … … } So, 𝑅 −1 = {(1,2), (2,4), (3,6), …..}.
8. Given the relation 𝑅 = {(1,2), (2,3)} on the set 𝐴 = {1,2,3}, the minimum number
of ordered pairs which when added to 𝑅 make it an equivalence relation is
(a) 5
(b) 6
(c) 7
(d) 8
Solution:
(c) 𝑅 is reflexive if it contains (1,1), (2,2), (3,3)
∵ (1,2) ∈ 𝑅, (2,3) ∈ 𝑅
∴ 𝑅 is symmetric if (2,1), (3,2) ∈ 𝑅. Now, 𝑅 =
{(1,1), (2,2), (3,3), (2,1), (3,2), (2,3), (1,2)}
𝑅 will be transitive if (3,1); (1,3) ∈ 𝑅. Thus, 𝑅 becomes an equivalence relation
by adding (1,1)(2,2)(3,3)(2,1)(3,2)(1,3)(3,1). Hence, the total number of
ordered pairs is 7 .
9. Let 𝑅 be the relation on the set 𝑅 of all real numbers defined by a 𝑅𝑏 iff |𝑎 − 𝑏| ≤
1. Then 𝑅 is