Complement and De Morgan's laws

The complement of A with respect to U is A′ = {x : x ∈ U and x ∉ A} = U − A: everything in the universal set that is outside A. A′ is itself a subset of U, and it changes if U changes.

Complement laws: A ∪ A′ = U and A ∩ A′ = φ. Double complement: (A′)′ = A. Also φ′ = U and U′ = φ.

De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′. Taking the complement swaps ∪ and ∩ and puts a dash on each set.

Check with U = {1, …, 6}, A = {2, 3}, B = {3, 4, 5}: A ∪ B = {2, 3, 4, 5}, so (A ∪ B)′ = {1, 6}; A′ = {1, 4, 5, 6} and B′ = {1, 2, 6}, so A′ ∩ B′ = {1, 6} as well.

In a Venn diagram A′ is the part of the rectangle outside A's circle, and (A ∪ B)′ is the region outside both circles.

A useful rewrite that follows from the definitions: A − B = A ∩ B′, the part of A that is also outside B.