Question

If $X$ and $Y$ are two non-empty sets, where $f:X\to Y$is function is defined such that $f\left(C\right)=\left\{f\left(x\right):x\in C\right\}$ for $C\subseteq X$ and $f^{-1}\left(D\right)=\left\{x:f\left(x\right)\in D\right\}$for $D\subseteq Y$, for any $A\subseteq Y$ and $B\subseteq Y$, then

• A. $f^{-1}\left(f\left(A\right)\right)=A$
• B. $f^{-1}\left(f\left(A\right)\right)=A$ only if $f\left(X\right)=Y$
• C. $f\left(f^{-1}\left(B\right)\right)=B$, only if $B\subseteq X$
• D. $f\left(f^{-1}\left(B\right)\right)=B$

Answer 1

The correct option is C

$f\left(f^{-1}\left(B\right)\right)=B$, only if $B\subseteq X$

Explanation for the correct answer:

It is given that $f:X\to Y$

$\Rightarrow$$f^{-1}:Y\to X$

It is also given that $A\subseteq Y$ and $B\subseteq Y$

Hence, only $f^{-1}\left(A\right)$ and $f^{-1}\left(B\right)$ are defined.

$f\left(A\right)$ and $f\left(B\right)$ cannot be defined as $A\subseteq Y$and $B\subseteq Y$

Now$f^{-1}\left(A\right)$ and $f^{-1}\left(B\right)$ will give a value each in the set $X$

Hence, $f\left(f^{-1}\left(A\right)\right)=A$ and $f\left(f^{-1}\left(B\right)\right)=B$ are defined only if $A,B\subseteq f\left(x\right)$.

Hence, option $\left(C\right)$ is the correct answer