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**pankaj** Suppose $\displaystyle X$ and $\displaystyle Y$ are two sets and $\displaystyle f:X\rightarrow Y$ is a function. For a subset $\displaystyle A$ of $\displaystyle X$ ,define $\displaystyle f(A)$ to be the subset {$\displaystyle {f(a):a\in A}$} of $\displaystyle Y$ .For a subset $\displaystyle B$ of $\displaystyle Y$ ,define $\displaystyle f^{-1}(B)$ to be the subset {$\displaystyle x\in X$:$\displaystyle f(x)\in B$} of $\displaystyle X$.Then which of the following statements is true?

(A)$\displaystyle f^{-1}(f(A))=A$ for every $\displaystyle A\subset X$

(B)$\displaystyle f^{-1}(f(A))=A$ for every $\displaystyle A\subset X$ if and only if $\displaystyle f(X)=Y$