Basic Set theory question

• Sep 16th 2008, 01:03 PM
Smancer
Basic Set theory question
Hi,

Let f:X ->Y be a function, Let A and B be subsets of X and let C and D be subsets of Y.

Prove that

f-1(C U D) = f-1(C) U f-1(D).

f-1(x) means the inverse function of some element x.

I know I need to start by saying,

Let x /in f-1(C U D)

and I think this means that there is some y /in f(C U D) such that f-1(y) = x.

But I am not sure on how to proceed. Any help is appreciated.
• Sep 16th 2008, 06:02 PM
Jhevon
Quote:

Originally Posted by Smancer
Hi,

Let f:X ->Y be a function, Let A and B be subsets of X and let C and D be subsets of Y.

Prove that

f-1(C U D) = f-1(C) U f-1(D).

f-1(x) means the inverse function of some element x.

I know I need to start by saying,

Let x /in f-1(C U D)

and I think this means that there is some y /in f(C U D) such that f-1(y) = x.

But I am not sure on how to proceed. Any help is appreciated.

so you have to prove $\displaystyle f^{-1}(C \cup D) \subseteq f^{-1}(C) \cup f^{-1}(D)$ and $\displaystyle f^{-1}(C) \cup f^{-1}(D) \subseteq f^{-1}(C \cup D)$

to start you off on the first one, let $\displaystyle x \in f^{-1}(C \cup D)$. then then there is some $\displaystyle y \in (C \cup D)$ such that $\displaystyle f^{-1}(y) = x$. But $\displaystyle y \in (C \cup D)$ means $\displaystyle y \in C$ or $\displaystyle y \in D$. thus.....?