Exercise 575Logic, sets and proofs › Sets and mapsHigh schoolAdded 3 Nov 2026

Distributive laws for sets

#distributivity#union#intersection

Statement

Prove that $A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$ .

Hint

Show both inclusions element by element.

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Solution

$\subseteq$ . If $x\in A\cap(B\cup C)$ , then $x\in A$ and ( $x\in B$ or $x\in C$ ). So either ( $x\in A$ and $x\in B$ ) or ( $x\in A$ and $x\in C$ ), i.e. $x\in(A\cap B)\cup(A\cap C)$ .

$\supseteq$ . If $x\in(A\cap B)\cup(A\cap C)$ , then $x\in A\cap B$ or $x\in A\cap C$ . In both cases $x\in A$ , and $x\in B$ or $x\in C$ , i.e. $x\in B\cup C$ . So $x\in A\cap(B\cup C)$ .

Conclusion. The two sets are equal.

Distributive laws for sets

Statement

Solution

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