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Prove that the following statements are equivalent.

a

$f$ is one-to-one on $S$.

b

$f(A\cap B) = f(A)\cap f(B)$ for all subsets $A,B$ of $S$.

c

$f^{-1}[f(A)] = A$ for every subset $A$ of $S$.

d

For all disjoint subsets $A$ and $B$ of $S$, the images $f(A)$ and $f(B)$ are disjoint.

Having a tough time with this one.