If $X$ is infinite and $A\subseteq X$ finite, then $X\backslash A$ and $X$ have the same cardinality.
Proof:First,it is easy to prove that $X$ has a countable subset using AC(AC ensure the existence of the choice function,then the construction of the choice function would be simple).Let this countable subset be $B$.Then it is easy to verify that
\begin{align*}
B\backslash A \equiv B
\end{align*}
So it is easy to see that
\begin{align*}
X\backslash A\equiv X
\end{align*}.