I am notexactlysure what these functions mean:

$\displaystyle \overrightarrow{f}: \mathcal{P}(X) \rightarrow \mathcal{P}(Y) $ defined by $\displaystyle \overrightarrow{f}(A) = \{ f(x) | x \in A \} $ for $\displaystyle A \in \mathcal{P}(X) $ and

$\displaystyle \overleftarrow{f}: \mathcal{P}(Y) \rightarrow \mathcal{P}(X) $ defined by $\displaystyle \overleftarrow{f}(B) = \{x \in X | f(x) \in B \} $ for $\displaystyle B \in \mathcal{P}(Y) $.

These two functions really are $\displaystyle f $ and $\displaystyle f^{-1} $ right?

So if $\displaystyle f $ is a bijection with inverse $\displaystyle f^{-1} $, then $\displaystyle f(x) = y_0 $ if and only if $\displaystyle x = f^{-1}(y_0) $ so that $\displaystyle \overleftarrow{f}(\{y_0 \}) = \{f^{-1}(y_0) \} $. So this implies that $\displaystyle \overleftarrow{f}(\{y_0 \}) \neq \{f^{-1}(y_0) \} $ sometimes?