Let X,Y,Z be sets. Prove that X union (Y intersecting Z) = (X union Y) intersecting (X union Z).

The proof would start off by showing that each side is a subset of the other, I believe... But there would be several cases? I'm not really sure where to go with this.. Any help would be wonderful.

2. $\displaystyle X\cup(Y \cap Z)=(X\cup Y)\cap(X \cup Z)$

That's the deMorgan law proof for sets or whatever.

You take the left side or the right side, write out what it means to be a union and stuff just a base definition.

So say the left side:

$\displaystyle Y \cap Z) = \left \{ x: x \in Y and \text{ } x \in Z \right \}$

Let's let $\displaystyle E = Y \cap Z$

$\displaystyle X \cup E = \left \{ x: x \in X or \text{ } x \in E \right \}$

So combine it: (I don't really remember what I did when I did this the first time but it gets kinda messy)

$\displaystyle =\left \{ x: x \in X or \text{ } (x \in Y and \text{ } x \in Z )\right \}$

Then you do the same to the right side and you can see they're equal.