GATE CS 2023 Q.49: Sets, Relations, Functions, Partial Orders and Lattices

Let $f : A \rightarrow B$ be an onto (or surjective) function, where $A$ and $B$ are nonempty sets. Define an equivalence relation $\sim$ on the set $A$ as

a_1 \sim a_2 \iff f(a_1) = f(a_2),

where $a_1, a_2 \in A$ . Let $E = \{[x] : x \in A\}$ be the set of all the equivalence classes under $\sim$ . Define a new mapping $F : E \rightarrow B$ as

F([x]) = f(x), \quad \text{for all the equivalence classes } [x] \text{ in } E.

Which of the following statements is/are TRUE?

$F$ is NOT well-defined.

$F$ is an onto (or surjective) function.

$F$ is a one-to-one (or injective) function.

$F$ is a bijective function.

Official answer and solution

Official answer: B, C and D
From: Official GATE 2023 answer key

Solution

Apply the First Isomorphism Theorem for sets on the quotient mapping.

Given $f: A \to B$ is surjective, and $a_1 \sim a_2 \iff f(a_1) = f(a_2)$ .

Well-definedness: If $[x_1] = [x_2]$ , then $x_1 \sim x_2 \implies f(x_1) = f(x_2) \implies F([x_1]) = F([x_2])$ . Thus $F$ is well-defined. (Statement A is FALSE)
Injective (One-to-One): If $F([x_1]) = F([x_2])$ , then $f(x_1) = f(x_2) \implies x_1 \sim x_2 \implies [x_1] = [x_2]$ . Thus $F$ is injective. (Statement C is TRUE)
Surjective (Onto): Since $f$ is onto $B$ , for any $b \in B$ there exists $x \in A$ such that $f(x) = b$ . Then $[x] \in E$ and $F([x]) = f(x) = b$ . Thus $F$ is surjective. (Statement B is TRUE)
Bijective: Since $F$ is both injective and surjective, $F$ is a bijection. (Statement D is TRUE)

Therefore, statements B, C, and D are TRUE.

Final answer

Options (B, C, D)

Common trap

Believing $F$ is not well-defined because different representatives $x$ can denote the same equivalence class $[x]$ .

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2023 Computer Science and Information Technology (CS)
Question: 49 of 65
Section: Computer Science and Information Technology

Report an error in this question

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.

Question

Solution

Source