Relations and Functions (Class 12) - Practice Questions with Answers

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

Q14.

If f(x) = 2x and g(x) = x + 3, then (f∘g)(1) equals:

• A 4
• B 8
• C 5
• D 7

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

Q16.

If f(x) = x + 5, then f⁻¹(x) equals:

• A 5 − x
• B 1/(x + 5)
• C 5x
• D x − 5

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

Q20.

If f(x) = 3x − 2, then f(2) equals:

• A 8
• B 1
• C 6
• D 4

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

Q26.

If f(x) = x + 1 and g(x) = x², then (g∘f)(2) equals:

• A 4
• B 9
• C 5
• D 6

Q27.

If f(x) = x + 1 and g(x) = x², then (f∘g)(2) equals:

• A 6
• B 4
• C 5
• D 9

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

Q29.

The number of relations on a set with 3 elements is:

• A 2⁹
• B 3³
• C 2³
• D 9

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

Q35.

If f(x) = 1/x for x ≠ 0, then (f∘f)(x) equals:

• A 1/x²
• B x²
• C x
• D 1/x

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

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 2ⁿ
• B n²
• C 2^{n² − n}
• D 2^n²

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 2ⁿ
• B nⁿ
• 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^{n² − n}
• C 2^n²
• 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