# Pre-Image / Image proof

• Mar 8th 2012, 07:26 PM
monster
Pre-Image / Image proof
I'm new to topology and have a pretty basic proof to perform but haven't had much experience with proofs and could use some help,

Question asks, to prove that if f: A -> B and X is a subset of A

prove that X is a subset of f^-1( f(X)) i.e that X is a subset of the pre-image of the image of X,

this makes perfect sense to me that it's true but i'm unsuer how to construct the proof any help would be appreciated, cheers.
• Mar 9th 2012, 02:30 AM
FernandoRevilla
Re: Pre-Image / Image proof
Quote:

Originally Posted by monster
Question asks, to prove that if f: A -> B and X is a subset of A prove that X is a subset of f^-1( f(X)) i.e that X is a subset of the pre-image of the image of X,

If $\displaystyle x\in X$ then, $\displaystyle f(x)\in f(X)$ . But by definition, $\displaystyle f^{-1}(f(X))=\{y\in A:f(y)\in f(X)\}$ , then, $\displaystyle x$ satisfies the condition for being in $\displaystyle f^{-1}(f(X))$ . As a consequence, $\displaystyle X\subset f^{-1}(f(X))$ .