Question

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

• 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 ⊆ f(x)
• D. f(f^-1(B)) = B

Answer 1

The correct option is B f^-1(f(A))=A only if f(x) = Y

The option b satisfied the above definition of function f so option (b) is correct. The option c is not correct because we can not prove that b has a pre image in c itself