Set Identities
A, B and C are sets, and we consider them to be subsets of a universal set U. Remember that ; is
the empty set, and that Ac means\the complement" of A.
1. Commutative Laws:
A [ B = B [ A
A \ B = B \ A
2. Associative Laws:
(A [ B) [ C = A [ (B [ C)
(A \ B) \ C = A \ (B \ C)
3. Distributive Laws:
A [ (B \ C) = (A [ B) \ (A [ C)
A \ (B [ C) = (A \ B) [ (A \ C)
4. Identity Laws:
A [ ; = A
A \ U = A
5. Complement Laws:
A [ Ac = U
A \ Ac = ;
6. Double Complement Laws:
(Ac)c = A
7. Idempotent Laws:
A [ A = A
A \ A = A
8. Universal Bound Laws:
A [ U = U
A \ ; = ;
9. De Morgan's Laws:
(A [ B)c = Ac \ Bc
(A \ B)c = Ac [ Bc
10. Absorption Laws:
A [ (A \ B) = A
A \ (A [ B) = A
11. Complements of U and ;:
Uc = ;
;c = U
12. Set Di erence Law:
A ô€€€ B = A \ Bc
