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.