Q. If $X$ and $Y$ are two non-empty sets where $f:X \rightarrow Y$ is function is defined such that $f(C) = \left\{ f(x):x \in C \right\} for\ C \subseteq X$ And $f^{- 1}(D) = \left\{ x:f(x) \in D \right\} for\ D \subseteq Y$, For any $A \subseteq X$ and $B \subseteq Y$, then
A) $f^{- 1}\left( f(A) \right) = A$
B) $f^{- 1}\left( f(A) \right) = A\ only\ if\ f(X) = Y$
C) $f\left( f^{- 1}(B) \right) = B\ only\ if\ B \subseteq f(x)\$
D) $f\left( f^{- 1}(B) \right) = B$
Explanation
(c)
The given data is shown in the figure below
[image:2_image11.emf]
Since, $f^{- 1}(D) = x$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f(x) = D$
Now, if $B \subset X,\ f(B) \subset D$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ f^{- 1}\left( f(B) \right) = B$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.