Relations and Functions — Class 12 MCQs with Answers

Relations and Functions — important questions & MCQs with answers (Class 12 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   Relation R = {(a,a) : a ∈ A} is:

   ASymmetric onlyBReflexiveCTransitive onlyDEmpty
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   Answer: Reflexive

   Explanation: R = {(a,a) : a ∈ A} relates every element only to itself, matching the definition of reflexive exactly; it makes no claim about symmetry or transitivity.

2. Q2 of 6Easy

   f(x) = x is:

   ANot functionBMany-oneCConstantDBijective on R
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   Answer: Bijective on R

   Explanation: f(x)=x is the identity function: distinct inputs give distinct outputs (one-one) and every real y is hit by x=y (onto), so it is bijective on R.

3. Q3 of 6Medium

   R on N defined by aRb iff a = b is:

   ANot equivalenceBSymmetric onlyCTransitive onlyDEquivalence
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   Answer: Equivalence

   Explanation: Equality aRb: a=b is reflexive (a=a), symmetric (a=b⇒b=a), and transitive (a=b,b=c⇒a=c) — satisfying all three makes it an equivalence relation, not just symmetric alone.

4. Q4 of 6Medium

   Assertion (A): The function f : R → R given by f(x) = x² is not one-one. Reason (R): The function f : R → R given by f(x) = x² is not onto, because −2 is not the image of any real number.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true but R is not the correct explanation of A

   Explanation: A is true: f(−1) = f(1) = 1 although −1 ≠ 1. R is true: x² ≥ 0 for every real x, so x² = −2 has no solution. R is about f failing to be onto, so it does not explain why f is not one-one.

5. Q5 of 6Hard

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

   A−1/xBxC1/xD−x
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   Answer: −1/x

   Explanation: Substituting f(x)=(1+x)/(1−x) into itself gives [2/(1−x)] ÷ [−2x/(1−x)] = −1/x — the negative sign comes from the −x term picked up in the second substitution.

6. Q6 of 6Hard

   Assertion (A): The relation S = {(a, b) : a ≤ b³} in the set R of real numbers is not symmetric. Reason (R): The relation S = {(a, b) : a ≤ b³} in the set R of real numbers is reflexive.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: A is true but R is false

   Explanation: A is true: (1, 2) ∈ S since 1 ≤ 8, but (2, 1) ∉ S since 2 ≤ 1 is false. R is false: for a = 1/2, a³ = 1/8 and 1/2 ≤ 1/8 is false, so (1/2, 1/2) ∉ S.

Relations and Functions notes: key concepts (Class 12 Mathematics)

Class 12 opens by classifying relations as reflexive, symmetric or transitive — an equivalence relation being all three and splitting a set into disjoint classes — and functions as one-one, onto or bijective. It then builds composition, the inverse of a bijection, and binary operations with identity and inverse elements.

Reflexive relation

A relation R on A where (a, a) ∈ R for every a ∈ A; every element is related to itself, e.g. 'is equal to' on any set.

Symmetric relation

Whenever (a, b) ∈ R it forces (b, a) ∈ R; the order of a related pair can be reversed, e.g. 'is parallel to'.

Transitive relation

If (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R; relatedness chains through, e.g. 'is a divisor of' or 'less than'.

Equivalence relation

A relation that is reflexive, symmetric AND transitive simultaneously; 'a − b divisible by n' on integers is the standard example.

Equivalence classes

An equivalence relation partitions the set into disjoint non-empty classes [a] whose union is the whole set; classes either coincide or are disjoint.

One-one (injective)

Distinct inputs give distinct outputs: f(x₁) = f(x₂) ⇒ x₁ = x₂. A strictly monotonic function on R, like 3x + 2, is one-one.

Onto (surjective)

Every element of the codomain is an image, so range equals codomain. f(x) = x³ from R to R is onto.

Bijective function

A function that is both one-one and onto; it pairs the two sets exactly and is the only kind that is invertible.

Composition of functions

(g∘f)(x) = g(f(x)); apply f first, then g. The range of f must lie in the domain of g, and generally g∘f ≠ f∘g.

Invertible function

f⁻¹ exists precisely when f is bijective; then f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. Find it by solving y = f(x) for x.

Binary operation

A rule ∗ : A × A → A assigning each ordered pair a single element of A (closure), such as addition or multiplication on R.

Identity and inverse element

e is identity if a∗e = e∗a = a for all a; b is the inverse of a if a∗b = b∗a = e. For + on R, e = 0 and inverse of a is −a.

Commutative and associative

∗ is commutative if a∗b = b∗a and associative if (a∗b)∗c = a∗(b∗c); both must be checked separately.