Let and . Which one of the following functions is bijective?

Explanation: A function is bijective if it is both injective (one-to-one) and surjective (onto). For finite sets and of the same size, this means every element in must map to a unique element in , and every element in must be covered. Let's evaluate the options: (a) and (c): These are not even functions, as the element (in a) or (in c) maps to multiple values in . (b): This is a function, but it is not injective because both and map to . It is also not surjective because is not mapped to. (d): . Here, each element in has a unique image in , and all elements in are covered. We can express the condition for a bijective function as: Since option (d) satisfies both conditions, it is the correct answer.

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TEZPUR 2025 — Mathematics PYQ

TEZPUR | Mathematics | 2025

Let $A = {1, 2, 3}$ and $B = {4, 5, 6}$ . Which one of the following functions is bijective?

Choose the correct answer:

B.

$f = {(1, 5), (2, 4), (3, 4)}$

C.

$f = {(1, 4), (1, 5), (1, 6)}$

D.

$f = {(1, 4), (2, 5), (3, 6)}$

(Correct Answer)

Explanation

A function $f : A \to B$ is bijective if it is both injective (one-to-one) and surjective (onto). For finite sets $A$ and $B$ of the same size, this means every element in $A$ must map to a unique element in $B$ , and every element in $B$ must be covered.

Let's evaluate the options:

(a) and (c): These are not even functions, as the element $1$ (in a) or $2$ (in c) maps to multiple values in $B$ .
(b): This is a function, but it is not injective because both $2$ and $3$ map to $4$ . It is also not surjective because $6$ is not mapped to.
(d): $f = {(1, 4), (2, 5), (3, 6)}$ . Here, each element in $A$ has a unique image in $B$ , and all elements in $B$ are covered.

We can express the condition for a bijective function $f$ as:

$\forall x_{1}, x_{2} \in A, f (x_{1}) = f (x_{2}) ⟹ x_{1} = x_{2} (Injective)$

$\forall y \in B, \exists x \in A such that f (x) = y (Surjective)$

Since option (d) satisfies both conditions, it is the correct answer.