Proof on functions of an intersection of sets

[jeffreydk]

I'm working out of Abbott's Understanding Analysis and I'm trying to show the following,

For an arbitrary function g :\mathbb{R}\longrightarrow \mathbb{R} it is always true that g(A\bigcap B) \subseteq g(A) \bigcap g(B) for all sets A, B \subseteq \mathbb{R}.

I'm confused on how to get going with this--any help or hints would be greatly appreciated. Thanks.

 

[statdad]

If X, Y are sets for which you know the definitions or other properties, the classical way to show X \subset Y is this:

1: Pick an arbitrary a \in X

2: Use the definitions of the sets to show that a \in Y

As a start, if you know that a \in g(A \cap B), then you know that there is a value x_0 such that a = g(x_0) and that x_0 \in A \cap B. What else do you know about x_0, and how can you use that information?

 

[jeffreydk]

Thanks, that really helps, I think I've got it now.