Question

Consider the statement. For all sets A, B, and C, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). Fill in the blanks in the following proof for the statement. (In the proof, let ∩ and ∪ stand for the words "intersection" and "union," respectively.) 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).]

Proof that A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ (A ∩ C): Let x ∈ A ∩ (B ∪ C).

Answer 1

By definition it follows that

`x \in (A \cap B) \cup(A \cap C)\\`

Explanation:

`\text{if} \,\,\,\,\, x \in A \cap (B \cup C)`

by definition

`x\in A \,\,\,\,\,\,\,\, \text{and} \,\,\,\,\,\,\,\, x \in (B \cup C). \\ \text{therefore} \,\,\,\,\,x\in A \,\,\,\,\, \text{and} \,\,\,\,\, x\in B \,\,\,\,\, \text{or} \,\,\,\,\,\, x\in \text{C}`

Then it follows that

`x \in (A \cap B) \cup(A \cap C)\\`

The other side is pretty much the same.