Set Operation Proofs

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# Set Operation Proofs ## Proving Distributive Law **Theorem:** $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ **Proof:** Let $x \in A \cap (B \cup C)$. Then $x \in A$ and $x \in B \cup C$.

Set Operation Proofs

Proving Distributive Law

Theorem:

A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

Proof: Let

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

. Then

x \in A

and

x \in B \cup C

. So

x \in A

and (

x \in B

x \in C

). If

x \in B

, then

x \in A \cap B

. If

x \in C

, then

x \in A \cap C

. Either way,

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

. The reverse inclusion is similar.

\square

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