Intersection

The intersection of two sets A and B, written A ∩ B, is the set of elements belonging to both A and B: A ∩ B = {x : x ∈ A and x ∈ B}. More generally, the intersection of a family {Aᵢ}ᵢ∈I is ⋂ᵢ Aᵢ = {x : x ∈ Aᵢ for all i ∈ I}.

Intersection is commutative, associative, and idempotent. It distributes over union: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). Together with union and complement, intersection gives the power set of any set the structure of a Boolean algebra.

In a lattice, intersection corresponds to meet (greatest lower bound). In the open sets of a topological space, only finite intersections are guaranteed to be open — this asymmetry between union and intersection is what makes the open-set lattice a frame rather than a Boolean algebra. In the Heyting algebra H, meet (a ∧ b) is the semantic analogue of intersection: the strongest value entailed by both a and b. The Heyting implication is defined through meet by the residuation law: c ≤ (a → b) if and only if c ∧ a ≤ b.