Proving f(f^-1(C)) = C ∩ Im(f) for sets and functions

[alphonsas]

Homework Statement

f: A -> B is a function with C a subset of B. Prove that
f(f^-1(C)) = [C intersection Im(f)]. (f^-1(c) = f inverse of C)

Homework Equations

The Attempt at a Solution

Please let me know how to approach to the solution (not using venn diagrams). Also if possible give me any link that gives tips on approaching any general sets and functions problem.
Thank you.

 

[vela]

What's K?

 

[Office_Shredder]

By K did you mean C?

Usually when you want to prove two sets are equal to each other, you prove each is a subset for the other. If $$y\in f(f^{-1}(C))$$ can you prove that $$y\in C\cap Im(f)$$? Then you would have that $$f(f^{-1}(C))\subset C\cap Im(f)$$. Do the other way also and you're done

 

[alphonsas]

I am sorry, that C.. how can I prove that y belongs to c intersection im(f)?

 

[Office_Shredder]

Well, you have some property that y satisfies because we know it belongs to $$f(f^{-1}(C))$$, which you should try to write down. Then use it