The GATE Grind

GATE 2025 CS (CS1) – Question 17

Engineering Mathematics · Discrete Mathematics: Sets, Relations, Functions, Partial Orders and Lattices · 1 mark · Multiple choice

$g(.)$ is a function from $A$ to $B$, $f(.)$ is a function from $B$ to $C$, and their composition defined as $f(g(.))$ is a mapping from $A$ to $C$.

If $f(.)$ and $f(g(.))$ are onto (surjective) functions, which ONE of the following is TRUE about the function $g(.)$?

  1. $g(.)$ must be an onto (surjective) function.
  2. $g(.)$ must be a one-to-one (injective) function.
  3. $g(.)$ must be a bijective function, that is, both one-to-one and onto.
  4. $g(.)$ is not required to be a one-to-one or onto function.

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Correct answer: (D) $g(.)$ is not required to be a one-to-one or onto function.

Explanation

Counterexamples exist. A={1,2}, B={1}, C={1} with g constant and f identity gives g not injective. B={1,2,3}, C={1} with f constant gives g not onto. Both f and f∘g are still onto.