Relations and Functions Class 12 Mcq Test

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 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 function f: R → R defined by f(x) = x² + 1 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: 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

A function f is said to be one-to-one (injective) if:
A) f(x_1) = f(x_2 ) implies x_1 ≠ x_2
B) x_1 = x_2 implies f(x_1) ≠ f(x_2)
C) f(x_1) = f(x_2) implies x_1=x_2
D) x_1≠x_2 implies f(x_1)≠f(x_2)

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

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 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

A function f: A → B is said to be onto (surjective) if:
A) The range of f is a proper subset of B
B) The range of f is equal to B
C) For every y ∈ B, there exists at most one x ∈ A such that f(x) = y
D) The domain of f is equal to A

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 signum function, f: R → R, given by f(x) = 1 if x > 0, 0 if x = 0, and -1 if x < 0, 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

A relation R is defined on the set of natural numbers N as (x, y) ∈ R if 'x is a divisor of y'. The relation R is:
A) Symmetric and Transitive
B) Reflexive and Symmetric
C) Reflexive and Transitive
D) An Equivalence Relation

The function f: R → R defined by f(x) = 3x + 5 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

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