question_answer

If X and Y are two non- empty sets where \[f:X\to Y\]is function  is defined such that \[f(c)=\left\{ f(x):x\in C \right\}\]for \[C\subseteq X\]and \[{{f}^{-1}}(D)=\{x:f(x)\in D\}\]for \[D\subseteq Y\] for any \[A\subseteq X\] and \[B\subseteq Y,\]then                                                                     [IIT Screening 2005]

A)                    \[{{f}^{-1}}(f(A))=A\]

B)                    \[{{f}^{-1}}(f(A))=A\]only if \[f(x)=Y\]

C)            \[f({{f}^{-1}}(B))=B\] only if \[B\subseteq f(X)\]

D)            \[f({{f}^{-1}}(B))=B\]