Q. Let $X$ and $Y$ be two non-empty sets. Let $f: X \rightarrow Y$ be a function. For $A \subset X$ and $B \subset Y$, define $f(A)=\{f(x): x \in A\} ; f^{-1}(B)=\{x \in X: f(x) \in B\}$, then

Relations and Functions - Part 2 Report Error

A

$f\left(f^{-1}(B)\right)=B$

B

$f\left(f^{-1}(B)\right) \subset B$

C

$f^{-1}(f(A))=A$

D

$f^{-1}(f(A)) \subset A$

Solution:

$f\left(f^{-1}(B)\right)=\left\{f(x): x \in f^{-1}(B)\right\}$
$=\{f(x): f(x) \in B\}$

$\Rightarrow f\left(f^{-1}(B)\right) \subset B $
Now if $x \in B \subset Y \Rightarrow x \in Y$
It may happen that $f^{-1}(x)$ does not Exist in $x$ as function is not given to be subjective.
$f\left(f^{-1}(B)\right) \neq B $
Also, $ f^{-1}(f(A))=\{x \in X: f(x) \in f(A)\}$ but
$f(A)=\{f(x): x \in A\}$
From above, we can't conclude $f^{-1}(f(A) \subset A$
If the function is non-injective,
then it may happen that $x \notin A$ and $f(x) \in f(A)$.
$\Rightarrow f^{-1}\left(f(A) ⊄ A \Rightarrow f^{-1}(f(A)) \neq A\right.$

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