Relations and Functions

What you'll learn

A friendly, from-scratch tour of relations (reflexive, symmetric, transitive, equivalence) and functions (one-one, onto, bijective), plus how to compose functions and find inverses — with fully worked examples for CBSE Class 12.

This chapter has

Relations and Functions — solved practice questions

8 Class 12 Mathematics questions with step-by-step solutions. Attempt each, then reveal the answer.

1. Q1easy

   On the set A = {1, 2, 3}, the relation R = {(1,1),(2,2),(3,3),(1,2),(2,1)} is:

   • AReflexive and symmetric but not transitive
   • BAn equivalence relation
   • CSymmetric and transitive but not reflexive
   • DReflexive only
   Show answer & solution

   Correct answer: (B) An equivalence relation

   R contains all pairs (a,a) so it is reflexive; (1,2) and (2,1) are both present so it is symmetric; the only new composition (1,2)&(2,1) gives (1,1), already present, so it is transitive. Hence R is an equivalence relation.

2. Q2easy

   The function f: R -> R defined by f(x) = 3x + 5 is:

   • AOne-one but not onto
   • BOnto but not one-one
   • CNeither one-one nor onto
   • DBijective (both one-one and onto)
   Show answer & solution

   Correct answer: (D) Bijective (both one-one and onto)

   A non-constant linear function is one-one (distinct x give distinct outputs) and onto R (every real y is achieved at x = (y-5)/3). Therefore f is bijective.

3. Q3easy

   The number of one-one functions from a set containing 3 elements to itself is:

   • A3
   • B9
   • C27
   • D6
   Show answer & solution

   Correct answer: (D) 6

   A one-one function from a 3-element set to itself is a permutation, so the count is 3! = 6.

4. Q4easy

   The function f: R -> R given by f(x) = x^2 is:

   • AOne-one and onto
   • BOne-one but not onto
   • CNeither one-one nor onto
   • DOnto but not one-one
   Show answer & solution

   Correct answer: (C) Neither one-one nor onto

   f(-2) = f(2) = 4 shows it is not one-one, and no negative number is an image so it is not onto R. Hence f is neither one-one nor onto.

5. Q5medium

   The number of equivalence relations that can be defined on the set {1, 2, 3} is:

   • A4
   • B6
   • C8
   • D5
   Show answer & solution

   Correct answer: (D) 5

   Equivalence relations on a set correspond one-to-one with its partitions. The set {1,2,3} has 5 partitions (Bell number B3 = 5), so there are 5 equivalence relations.

6. Q6medium

   On the set of integers Z, the relation R defined by a R b if and only if (a - b) is divisible by 5 is:

   • AOnly reflexive
   • BOnly symmetric
   • CAn equivalence relation
   • DReflexive and symmetric but not transitive
   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. Q7medium

   The number of onto (surjective) functions from a set of 3 elements to a set of 2 elements is:

   • A6
   • B7
   • C8
   • D9
   Show answer & solution

   Correct answer: (A) 6

   Total functions = 2^3 = 8; subtract the 2 functions that map everything to a single element. So onto functions = 8 - 2 = 6.

8. Q8medium

   The function f: N -> N defined by f(x) = 2x is:

   • ABijective
   • BOne-one but not onto
   • COnto but not one-one
   • DNeither one-one nor onto
   Show answer & solution

   Correct answer: (B) One-one but not onto

   Distinct natural numbers give distinct even values, so f is one-one. But odd natural numbers (like 3) are never obtained, so f is not onto N.

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