A={1,2,3,4,5}
f :P(A)→P(A), f (X ) =A\X.
how can i prove that this function is Surjective???
Suppose that $\displaystyle M \in P(A)$.
Now note that:
$\displaystyle \begin{array}{rcl}
{A\backslash \left( {A\backslash M} \right)} & = & {A \cap \left( {A \cap M^c } \right)^c } \\
{} & = & {A \cap \left( {A^c \cup M} \right)} \\
{} & = & {A \cap M} \\
{} & = & M \\
\end{array} $.