Q99 / 155 · WBJEE 2019 · Mathematics

Let $S, T, U$ be three non-void sets and $f: S \rightarrow T, g: T \rightarrow U$ be so that $g o f: s \rightarrow U$ is surjective. Then,

  • A
    $g$ and $f$ are both surjective
  • Correct answer
    B
    $g$ is surjective, $f$ may not be so
  • C
    $f$ is surjective, $g$ may not be so
  • D
    $f$ and $g$ both may not be surjective

Solution

We have, gof : $S \rightarrow U$ is an onto function. Let $Z$ be any arbitrary element such that $Z \in U$. Now, $g o f: S \rightarrow U$ is onto $\Rightarrow g_{0} f(x)=Z,$ for $x \in S$ $\Rightarrow g(f(x))=Z$ $\Rightarrow g(y)=Z,$ where $y=f(x) \in T$ for all $Z \in U$ $\therefore$ For all $Z \in U,$ there exists $y=f(x) \in T$ such that $g(y)=z$ $\therefore g: T \rightarrow U$ is an onto function.

Hence, the correct answer is option: 'B'.

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