68 practice questions on Relations and Functions (Class 12), sorted Easy → Hard. Try each one first, then open its answer page for the worked explanation. Want the full theory first? Read the Relations and Functions (Class 12) notes.
📐 Mathematics · Class 12 · JEE
Relations and Functions (Class 12) - Practice Questions with Answers
68 free MCQs on Relations and Functions (Class 12), each with its own worked answer and explanation. Types of relations, equivalence classes, one-one and onto functions, composition, and invertible functions
Take the timed Relations and Functions (Class 12) chapterwise test →Easy - 20 questions
Q1.
A relation R on a set A is reflexive if:
- A (a, a) ∈ R for every a ∈ A
- B (a, b) ∈ R implies (b, a) ∈ R
- C (a, b) and (b, c) ∈ R imply (a, c) ∈ R
- D R is the empty set
Q2.
A relation R is symmetric if:
- A (a, b) ∈ R implies a = b
- B (a, b) ∈ R implies (b, a) ∈ R
- C (a, a) ∈ R for all a
- D R contains every ordered pair
Q3.
A relation R is transitive if:
- A (a, a) ∈ R for all a
- B R has exactly three elements
- C (a, b) ∈ R and (b, c) ∈ R imply (a, c) ∈ R
- D (a, b) ∈ R implies (b, a) ∈ R
Q4.
An equivalence relation is one that is:
- A Reflexive and symmetric only
- B Symmetric and transitive only
- C One-one and onto
- D Reflexive, symmetric and transitive
Q5.
A function f : A → B is one-one (injective) if:
- A f(x₁) = f(x₂) implies x₁ = x₂
- B Every element of B has a preimage
- C The range equals the codomain
- D A and B have the same number of elements
Q6.
A function f : A → B is onto (surjective) if:
- A A has more elements than B
- B Range of f equals B
- C f(x₁) = f(x₂) implies x₁ = x₂
- D f is its own inverse
Q7.
A function that is both one-one and onto is called:
- A Transitive
- B Constant
- C Bijective
- D Reflexive
Q8.
The composite function (g∘f)(x) is defined as:
- A f(g(x))
- B g(x)·f(x)
- C g(x) + f(x)
- D g(f(x))
Q9.
A function f is invertible if and only if it is:
- A Bijective
- B One-one only
- C Onto only
- D Reflexive
Q10.
The identity function on a set A is defined by:
- A I(x) = −x for all x ∈ A
- B I(x) = x for all x ∈ A
- C I(x) = 0 for all x ∈ A
- D I(x) = 1 for all x ∈ A
Q11.
If f : A → B is invertible, then f⁻¹ is a function from:
- A A to A
- B B to B
- C B to A
- D A to B
Q12.
For a relation R on set A, the universal relation is:
- A R = ∅
- B R = {(a, a) : a ∈ A}
- C R = A
- D R = A × A
Q13.
The empty relation on a non-empty set A is:
- A Symmetric and transitive but not reflexive
- B Reflexive but not symmetric
- C An equivalence relation
- D Reflexive and transitive
Q15.
The number of elements in A × B when n(A) = 3 and n(B) = 4 is:
- A 81
- B 64
- C 12
- D 7
Q17.
The relation 'is equal to' on any set of numbers is:
- A An equivalence relation
- B Symmetric only
- C Transitive only
- D Not a relation at all
Q18.
If f : A → B and g : B → C, then g∘f is a function from:
- A A to B
- B A to C
- C C to A
- D B to C
Q19.
A constant function f(x) = c on a set with more than one element is:
- A Bijective
- B An equivalence relation
- C Not one-one
- D One-one but not onto
Medium - 20 questions
Q21.
The relation R = {(1,1), (2,2), (3,3), (1,2)} on A = {1,2,3} is:
- A Reflexive and transitive but not symmetric
- B An equivalence relation
- C Symmetric but not reflexive
- D Neither reflexive nor transitive
Q22.
On the set of lines in a plane, the relation 'is perpendicular to' is:
- A Transitive but not symmetric
- B Symmetric but neither reflexive nor transitive
- C An equivalence relation
- D Reflexive and transitive
Q23.
On ℤ, the relation aRb if a − b is divisible by 5 produces how many equivalence classes?
- A 10
- B Infinitely many
- C 5
- D 2
Q24.
The function f : ℝ → ℝ given by f(x) = x² is:
- A One-one but not onto
- B Onto but not one-one
- C Bijective
- D Neither one-one nor onto
Q25.
The function f : ℝ → ℝ given by f(x) = 3x + 5 is:
- A Bijective
- B One-one but not onto
- C Onto but not one-one
- D Neither one-one nor onto
Q28.
The number of one-one functions from a set with 3 elements to a set with 5 elements is:
- A 125
- B 243
- C 10
- D 60
Q30.
If f : ℝ → ℝ is f(x) = 2x − 3, then f⁻¹(x) is:
- A 1/(2x − 3)
- B (x + 3)/2
- C (x − 3)/2
- D 2x + 3
Q31.
The function f : ℕ → ℕ defined by f(n) = 2n is:
- A Bijective
- B Neither one-one nor onto
- C One-one but not onto
- D Onto but not one-one
Q32.
For an equivalence relation on a set A, the equivalence classes are:
- A Always of equal size in every case
- B Allowed to overlap partially
- C Always exactly two in number
- D Pairwise disjoint and their union is A
Q33.
If f : A → B is bijective with n(A) = 5, then n(B) equals:
- A 5
- B 10
- C 25
- D 1
Q34.
The relation R on A = {1,2,3} given by R = {(1,2), (2,1)} is:
- A Transitive only
- B Symmetric but not reflexive or transitive
- C An equivalence relation
- D Reflexive and symmetric
Q36.
The function f : ℝ → [0, ∞) given by f(x) = x² is:
- A One-one but not onto
- B Bijective
- C Neither one-one nor onto
- D Onto but not one-one
Q37.
If g∘f is defined, which condition must hold?
- A The range of f must be contained in the domain of g
- B The domain of f must equal the range of g
- C f and g must both be one-one functions
- D f and g must have identical formulas
Q38.
The number of equivalence relations on the set {1, 2} is:
- A 3
- B 2
- C 1
- D 4
Q39.
If f(x) = x³, then f : ℝ → ℝ is:
- A Onto but not one-one
- B Neither
- C Bijective
- D One-one but not onto
Q40.
The relation 'is a sibling of' on a set of people (excluding oneself) is:
- A Reflexive and symmetric
- B Transitive and reflexive
- C Not symmetric
- D Symmetric but not reflexive
Hard - 28 questions
Q41.
For a function between two finite sets of the same size, one-one implies onto. Why does this fail for infinite sets?
- A An infinite set can be placed in bijection with a proper subset of itself, leaving room for injective maps that miss elements
- B Infinite sets never admit one-one functions in the majority of documented cases
- C Onto functions cannot be defined on infinite sets under standard conventions
- D The pigeonhole principle applies only when the codomain is uncountable
Q42.
If g∘f is one-one, what can be concluded?
- A Neither f nor g need be one-one
- B f must be one-one, but g need not be
- C g must be one-one, but f need not be
- D Both f and g must be one-one
Q43.
For invertible functions f and g, (g∘f)⁻¹ equals:
- A f∘g
- B (f∘g)⁻¹
- C f⁻¹∘g⁻¹
- D g⁻¹∘f⁻¹
Q44.
Why is the empty relation on a non-empty set not reflexive, even though it is symmetric and transitive?
- A The empty relation is not a valid relation on any set by definition
- B Reflexivity holds vacuously as well, so the relation is in fact an equivalence relation
- C Reflexivity requires the set itself to be empty in all cases
- D Symmetry and transitivity hold vacuously with no pairs to check, but reflexivity actively demands that (a, a) be present
Q45.
The number of equivalence relations on the set {1, 2, 3} is:
- A 5
- B 3
- C 6
- D 8
Q46.
The number of onto functions from a set of 3 elements to a set of 2 elements is:
- A 9
- B 6
- C 8
- D 2
Q47.
The number of reflexive relations on a set with n elements is:
- A 2<sup>n</sup>
- B n²
- C 2<sup>n² − n</sup>
- D 2<sup>n²</sup>
Q48.
Why is restricting the domain of f(x) = x² to [0, ∞) significant?
- A It makes the function onto ℝ, which the unrestricted version fails to be
- B It converts the function into a linear map on that interval
- C It removes the need for the function to be continuous
- D It makes the function one-one, allowing the square root to be defined as its inverse
Q49.
If f : ℝ → ℝ is f(x) = x/(1 + |x|), then f is:
- A One-one with range (−1, 1), so not onto ℝ
- B Onto ℝ but not one-one
- C Bijective from ℝ to ℝ
- D Neither one-one nor onto
Q50.
A relation that is symmetric and transitive but fails reflexivity on some element a indicates that:
- A Symmetry and transitivity are incompatible properties
- B The element a is not related to anything at all
- C The relation must be the universal relation
- D The set must be infinite in such cases
Q51.
The number of bijective functions from a set with n elements to itself is:
- A 2<sup>n</sup>
- B n<sup>n</sup>
- C n!
- D n²
Q52.
If f : A → B and g : B → C are both onto, then g∘f is:
- A One-one but not necessarily onto
- B Neither one-one nor onto
- C Bijective in every case
- D Onto
Q53.
On the set A = {1, 2, 3}, how many relations are both reflexive and symmetric?
- A 8
- B 64
- C 512
- D 27
Q54.
Why does f invertible require f to be onto, not just one-one?
- A The inverse is only defined for continuous functions in most cases
- B f⁻¹ must be defined at every element of the codomain, which requires each to have a preimage
- C One-one functions never possess inverses in standard practice
- D Ontoness alone is sufficient for invertibility without injectivity
Q55.
The relation R on ℝ given by aRb if a ≤ b is:
- A Symmetric and transitive only
- B Neither reflexive nor transitive
- C Reflexive and transitive but not symmetric
- D An equivalence relation
Q56.
If f(x) = (x − 1)/(x + 1) for x ≠ −1, then f(f(x)) equals:
- A x
- B 1/x
- C (x + 1)/(x − 1)
- D −1/x
Q57.
The number of symmetric relations on a set with n elements is:
- A 2^(n(n+1)/2)
- B 2<sup>n² − n</sup>
- C 2<sup>n²</sup>
- D n!
Q58.
Composition of functions is associative but not commutative. This means:
- A Composition is undefined unless the functions commute
- B (h∘g)∘f = h∘(g∘f) always, while g∘f = f∘g may fail
- C g∘f = f∘g always, while grouping matters
- D Both grouping and order are irrelevant to the result
Q59.
If A has m elements and B has n elements with m > n, the number of one-one functions from A to B is:
- A mⁿ
- B nᵐ
- C 0
- D n!
Q60.
On ℤ, the relation aRb if a + b is even is:
- A Not reflexive, since a + a may be odd
- B Symmetric but not transitive
- C An equivalence relation with infinitely many classes
- D An equivalence relation with 2 classes
Q61.
The number of equivalence relations on a 3-element set is:
- A 3
- B 5
- C 8
- D 15
Q62.
The inverse of the bijection f(x) = 3x − 7 on R is:
- A (x + 7)/3
- B (x − 7)/3
- C 3x + 7
- D (7 − x)/3
Q63.
For the binary operation a * b = a + b − ab on R, the identity element is:
- A 1
- B 0
- C −1
- D 2
Q64.
The number of bijections from {1, 2, 3} to {1, 2, 3} is:
- A 3
- B 6
- C 9
- D 27
Q65.
The binary operation a * b = |a − b| on the reals is:
- A associative but not commutative
- B commutative but not associative
- C both associative and commutative
- D neither
Q66.
On the positive integers, the relation aRb defined by 'a divides b' is:
- A an equivalence relation
- B a partial order but not an equivalence relation
- C symmetric only
- D not transitive
Q67.
The number of onto functions from {1, 2, 3, 4} to {1, 2} is:
- A 8
- B 14
- C 16
- D 2
Q68.
For f(x) = x/(x − 1), x ≠ 1, the composition f(f(x)) equals:
- A x
- B 1/x
- C −x
- D (x − 1)/x
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