Need help for proving: $f(f^{−1}(A)) ⊆ A$.

Question

Informations: $f:X \longrightarrow X$ and $A \subseteq X$.

How can i prove this statement: $f(f^{-1}(A)) \subseteq A$

This is my thoughts until now:

$f^1(A)=\{x\in X |f(x)\in A\} \subseteq X$.

$f(A)=\{f(x)|x\in A\}$

$f(f^1(A))=\{y\in A:\exists \in f^1(A):y=f(x)\} \in A$

How do you normally show something is a subset of something else? — pjs36, Jan 29 '17 at 19:20
You can assume that the domain is not $X$ but some other $Y$, and this still holds. — Wolfram, Jan 29 '17 at 19:27

score 1 · Answer 1 · answered Jan 29 '17 at 19:23

1

Hint: If $x \in f^{-1}(A)$, what does that say about $f(x)$?

answered Jan 29 '17 at 19:23

Robert Israel

448,999

score 1 · Answer 2 · answered Jan 29 '17 at 19:23

First off, welcome to Math SE! It is encouraged here to use LaTeX formatting in your posts; you can find a tutorial here.

Second, it is encouraged that when you make a post, you include some information about how you've already tried to solve this problem, and where you're stuck.

For instance, this is a very straightforward problem, and you just need to follow definitions. That is, let $x\in f^{-1}(A)$. What does that mean? If you write out what is immediately implied by the definition, you'll find your proof writes itself.

Thank you. I have edited it now. – Mathe Jan 29 '17 at 19:27 — Mathe, Jan 29 '17 at 19:27

score 0 · Answer 3 · answered Jan 29 '17 at 20:22

0

Take $y \in f[f^{-1}[A]]$. So $y = f(x)$, where $x \in f^{-1}[A]$, and the latter means that $f(x) \in A$, so $y \in A$ and the inclusion has been shown.

answered Jan 29 '17 at 20:22

Henno Brandsma

242,131

Need help for proving: $f(f^{−1}(A)) ⊆ A$.

3 Answers3