Proof: Suppose A, B, and C are any sets. [To show that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), we must show that A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ (A ∩ C) and that (A ∩ B) ∪ (A ∩ C) ⊆ A ∩ (B ∪ C).] g

I suppose you're supposed to prove that set intersection is distributive across a union,

Two sets are equal if they are subsets of one another. To prove a set is a subset of another set , you have to show that any element also belongs to .

Let . By definition of intersection, both and . By definition of union, either or . If , then clearly ; if , then . Either way, . Hence .

The proof in the other direction uses the same sort of reasoning. Let . Then either or . If , then both and ; if , then both and . So certainly , and either or so that . Hence .

Both sets are subsets of one another, so they are equal.