Search any question & find its solution
Question:
Answered & Verified by Expert
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,
Options:
Solution:
2531 Upvotes
Verified Answer
The correct answer is:
$g$ is surjective, $f$ may not be so
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.
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.
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.