Sets, Relations and Functions

This chapter has

Sets, Relations and Functions — solved practice questions

8 JEE Mathematics questions with step-by-step solutions. Attempt each, then reveal the answer.

1. Q1easy

   If a set A has 4 elements, then the number of proper subsets of A (subsets other than A itself) is:

   • A16
   • B14
   • C15
   • D8
   Show answer & solution

   Correct answer: (C) 15

   A set with n elements has 2^n subsets in total. Proper subsets exclude A itself, giving 2^4 − 1 = 16 − 1 = 15.

2. Q2easy

   For two finite sets A and B, n(A) = 20, n(B) = 15 and n(A ∪ B) = 30. Then n(A ∩ B) equals:

   • A5
   • B10
   • C35
   • D0
   Show answer & solution

   Correct answer: (A) 5

   By inclusion–exclusion, n(A ∪ B) = n(A) + n(B) − n(A ∩ B), so n(A ∩ B) = 20 + 15 − 30 = 5.

3. Q3easy

   The domain of the real function f(x) = sqrt(4 − x^2) is:

   • A(−2, 2)
   • B[−2, 2]
   • C(−∞, 2]
   • D[0, 2]
   Show answer & solution

   Correct answer: (B) [−2, 2]

   The expression under the root must be non-negative: 4 − x^2 ≥ 0, i.e. x^2 ≤ 4, giving −2 ≤ x ≤ 2, i.e. [−2, 2].

4. Q4medium

   The number of functions from a set A with 3 elements to a set B with 4 elements is:

   • A12
   • B24
   • C81
   • D64
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   Correct answer: (D) 64

   Each of the 3 elements of A can map to any of the 4 elements of B independently, giving 4^3 = 64 functions.

5. Q5medium

   The number of onto (surjective) functions from a set with 4 elements onto a set with 2 elements is:

   • A16
   • B14
   • C8
   • D12
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   Correct answer: (B) 14

   Total functions to a 2-element set = 2^4 = 16. Subtract the 2 that miss an element (all elements to one target): 16 − 2 = 14.

6. Q6medium

   The relation R on the set of integers Z defined by a R b if and only if (a − b) is divisible by 5 is:

   • AReflexive and symmetric only
   • BSymmetric and transitive only
   • CAn equivalence relation
   • DReflexive and transitive only
   Show answer & solution

   Correct answer: (C) An equivalence relation

   a − a = 0 is divisible by 5 (reflexive); if 5 | (a − b) then 5 | (b − a) (symmetric); if 5 | (a − b) and 5 | (b − c) then 5 | (a − c) (transitive). Hence R is an equivalence relation.

7. Q7hard

   The number of equivalence relations that can be defined on a set with exactly 3 elements is:

   • A5
   • B3
   • C8
   • D6
   Show answer & solution

   Correct answer: (A) 5

   Equivalence relations correspond one-to-one with partitions of the set. The number of partitions of a 3-element set is the Bell number B(3) = 5.

8. Q8medium

   If f(x) = (x − 1)/(x + 1) for x ≠ −1, then f(f(x)) equals:

   • Ax
   • B1/x
   • C(x + 1)/(x − 1)
   • D−1/x
   Show answer & solution

   Correct answer: (D) −1/x

   f(f(x)) = (f(x) − 1)/(f(x) + 1). With f(x) = (x−1)/(x+1), numerator = ((x−1)−(x+1))/(x+1) = −2/(x+1) and denominator = 2x/(x+1), so the ratio is −1/x.

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