Proving a known proof involving injection.

[grjmmr]

trying to learn how to do proofs. So I have A=> B which is injective and E $\subseteq$ B then prove f^-1(f(E)) = E.
So let x $\in$ f^-1(f(E)) => thus f(x) $\in$ f(E) => x$\in$ E

So I have proved that x is a point within E, a subset of A, to me I think I am missing something and have not proved f^-1(f(E)) = E.

any suggestions?

 

[Bacle2]

You need to do a similar argument in the opposite direction. Take x in E and show

it is in f^-1f(E). You have then showed, for the two sets:

E is contained in f^-1f(E).

and

f^-1f(E) is contained in E.

This is the standard way of showing equality of sets.