Union and intersection

The union A ∪ B = {x : x ∈ A or x ∈ B} holds everything that is in A or in B or in both; common elements are written once. {2, 4, 6, 8} ∪ {6, 8, 10} = {2, 4, 6, 8, 10}.

The intersection A ∩ B = {x : x ∈ A and x ∈ B} holds only the elements common to both. {2, 4, 6, 8} ∩ {6, 8, 10} = {6, 8}.

If A ∩ B = φ, the sets are disjoint, like the even and the odd integers.

If B ⊂ A, then A ∪ B = A and A ∩ B = B: adding a part of A to A changes nothing, and the overlap of A with its part is the part.

Laws of union: A ∪ B = B ∪ A (commutative), (A ∪ B) ∪ C = A ∪ (B ∪ C) (associative), A ∪ φ = A (φ is the identity for ∪), A ∪ A = A (idempotent) and U ∪ A = U.

Laws of intersection: A ∩ B = B ∩ A, (A ∩ B) ∩ C = A ∩ (B ∩ C), φ ∩ A = φ, U ∩ A = A, A ∩ A = A, and the distributive law A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), so ∩ distributes over ∪.

If A ∪ B = A ∩ B, then A = B: any element of A lies in the union, hence in the intersection, hence in B, and the same argument runs the other way.