Proving theorem on indexing sets.

Question

Let $\Lambda$ be an arbitrary indexing set and let {$B_\alpha$}$_{\alpha\in \Lambda}$ be a collection of sets indexed by $\Lambda$. Let $C$ be any set. Then the following identities hold:

1. $C\cap\bigcup_{\alpha\in\Lambda}B_\alpha = \bigcup_{\alpha\in\Lambda}(C\cap B_{\alpha})$

I haven't got much into it at all, but would the notion of DeMorgan's Laws apply to this theorem at all? Just poking ideas at it.

My attempt at reading this, which is probably incorrect: "The intersection of a set $C$ and the union over $\alpha\in\Lambda$ of the $B_{\alpha}$'s is equal to the union over $\alpha\in\Lambda$ on the set $C$'s intersection over $B_{\alpha}$'s.

Apologies if I completely butchered it - this is the first time seeing these notations presented in this form. Also, we are supposed to prove why that theorem (1) holds true.

Answer 1

The general case is probably a bit distracting. Try a simple example first. Prove that

$C \cap (B_1 \cup B_2) = (C \cap B_1) \cup (C \cap B_2)$

That is, show that each set is a subset of the other. If you can get this, then the ideas involved in the proof to your general case is essentially identical.

Answer 2

Tough it out point by point.
x in C $\cap$ $\cup${ B$_a$ : a in A }
iff x in C and exists a in A with x in B$_a$
iff ....