Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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


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

but I can not.

share|cite|improve this question

2 Answers 2

up vote 4 down vote accepted

The claim is false. For example, let $X=\mathbb{Z}$, let $f(n)=n+1$, and let $A=\mathbb{N}$. Then $f(A)=\mathbb{N} \setminus \{ 0 \} \subsetneq A$.

share|cite|improve this answer
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.