Relations and Functions Class 12 Mcq Test

The function f: Z → Z defined by f(x) = x³ is:
A) One-to-one but not onto
B) Onto but not one-to-one
C) Both one-to-one and onto
D) Neither one-to-one nor onto

Let R be a relation on a set A such that R = R⁻¹. Then the relation R is:
A) Reflexive
B) Symmetric
C) Transitive
D) Equivalence

The smallest equivalence relation on the set A = {a, b, c} is:
A) {(a, a), (b, b), (c, c)}
B) {(a, a), (b, b), (c, c), (a, b), (b, a)}
C) A × A
D) {(a, b), (b, c)}

Let A = {1, 2, 3}. Which of the following relations on A is reflexive?
A) {(1, 2), (2, 1), (1, 1)}
B) {(1, 1), (2, 2), (3, 3), (1, 3)}
C) {(1, 1), (2, 2)}
D) {(1, 2), (2, 3), (3, 1)}

If a relation R on the set {1, 2, 3} is defined by R = {(1, 2)}, then R is:
A) Reflexive
B) Symmetric
C) Transitive
D) None of the above

A relation R on the set of integers Z is defined by (x, y) ∈ R if |x - y| ≤ 1. The relation R is:
A) Reflexive and Symmetric
B) Reflexive and Transitive
C) Symmetric and Transitive
D) An Equivalence Relation

Let R be the relation on set A = {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Which statement is correct?
A) R is reflexive and symmetric but not transitive.
B) R is reflexive and transitive but not symmetric.
C) R is symmetric and transitive but not reflexive.
D) R is an equivalence relation.

Let the function f: N → N be defined by f(x) = 2x. Then f is:
A) One-to-one but not onto
B) Onto but not one-to-one
C) Both one-to-one and onto
D) Neither one-to-one nor onto

The greatest integer function f: R → R, defined by f(x) = [x], is:
A) One-to-one
B) Onto
C) Both one-to-one and onto
D) Neither one-to-one nor onto

The universal relation on a non-empty set A is always:
A) Not Symmetric
B) Not Transitive
C) An Equivalence Relation
D) None of the above

A function f from the set of natural numbers N to the set of integers Z defined by f(n)=(n-1)/2 when n is odd, and f(n)=-n/2 when n is even, is:
A) One-to-one but not onto
B) Onto but not one-to-one
C) Both one-to-one and onto
D) Neither one-to-one nor onto

The function f: R → R defined by f(x) = eˣ is:
A) One-to-one but not onto
B) Onto but not one-to-one
C) Both one-to-one and onto
D) Neither one-to-one nor onto

Let A = R - {3} and B = R - {1}. The function f: A → B defined by f(x)=(x-2)/(x-3) is:
A) One-to-one but not onto
B) Onto but not one-to-one
C) Both one-to-one and onto
D) Neither one-to-one nor onto

The relation R in the set of real numbers R defined as R = {(a, b) : a ≤ b²} is:
A) Reflexive
B) Symmetric
C) Transitive
D) None of the above

Let f: R → R be defined as f(x) = x⁴. Choose the correct answer.
A) f is one-one and onto
B) f is many-one and onto
C) f is one-one but not onto
D) f is neither one-one nor onto