Elementary Question

Let be $f:X\to X$ a bijection, an $A\subset X$ a invariant subset of $X$, i.e $f(A)\subset A.$ How can see that

$$f(A)=A$$

I'm trying to show that $$f(A^{c})\subset A^c$$

but I can not.

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The claim is false. For example, let $X=\mathbb{Z}$, let $f(n)=n+1$, and let $A=\mathbb{N}$. Then $f(A)=\mathbb{N} \setminus \{ 0 \} \subsetneq A$.
Snap! :-) $\qquad$ –  Brian M. Scott Apr 29 '12 at 18:16
The statement is false. Let $X=\Bbb Z$ and $f(n)=n+1$ for $n\in X$. Then $\Bbb N$ is $f$-invariant, but its complement is not.