Laws of Set Algebra

Set operations obey algebraic laws closely parallel to ordinary arithmetic, and every one of them can be checked directly against the definitions above (or read off a Venn diagram).

Commutative laws — order does not matter for union or intersection:

Associative laws — grouping does not matter:

Distributive laws — union and intersection distribute over each other, just as multiplication distributes over addition in ordinary algebra:

De Morgan's laws — complementing a union or intersection flips the operation:

In words: "not (A or B)" means "not A, and not B" — and, symmetrically, "not (A and B)" means "not A, or not B." This is the same logic used in everyday reasoning: saying a customer bought "neither product A nor product B" is identical to saying they bought "not A, and also not B."

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