# Thread: [SOLVED] Beginner Type Proof: subsets

1. ## [SOLVED] Beginner Type Proof: subsets

Let A, B, C be sets..

Prove that (AUB)∩ C is contained in AU(B∩ C)

I know how to show this using the venn diagram, and one professor I had told us just drawing the venn diagram is proof enough, but I think this professor wants a more formal proof.

My idea is to maybe let x belong to (AUB)∩ C and then this is like (A∩ C)U(B∩ C) and then A∩ C contains AU(B∩ C), but I feel like I am just running in circles.

Thanks for any help.

2. Assume that:

$\displaystyle x \in (A \cup B)\cap C$

This means that:

$\displaystyle x \in [(A \cap C)\cup (B \cap C)]$

If we call $\displaystyle (A \cap C)$ m and $\displaystyle (B \cap C)$ n then we can rearrange them to get:

$\displaystyle x \in (n \cup m) = [(B \cap C) \cup (A \cap C)]$

Now, we expand yet again:

$\displaystyle x \in [((B \cap C)\cup A) \cap ((B \cap C) \cup C)]$

If x is in the intersection of the two, then they both have x in common, which means that:

$\displaystyle [x \in (B \cap C)\cup A] \cap [x \in (B \cap C)\cup C]$

This is sufficient to prove that:

$\displaystyle x \in (B \cap C)\cup A$