Set theory defines an intersection operation. For any nonempty set A, the set
B=∩A = { x : ∀y.y∈A⇒x∈y }.
B is the set of "all
elements of all elements" of A: those elements that are in
each member of A.
No axiom is required to guarantee the existence of ∩A: A is non-empty, so there is some a∈A. And ∩A can be generated as a subset of a. Define the predicate P(x) = ∀y.y∈A⇒x∈y, and then define
∩A = { x∈a : P(x) }
In essence (and unlike the case for the
union ∪A), the
cardinality of
∩A is (easily) bounded, so no new axiom is required to generate it.
When A={a,b} is a pair (derived e.g. by the axiom of pairing), we use the more common notation a∩b = ∩A. This notation is also used when a=b.
When A is an indexed set A = {ai: i∈I} for some index set I, notations based on ∩i∈Iai = ∩A are common. Thus, we may see
∅ = ∩n=1∞ {k∈N : k≥n}.
These less formal notations are more common in most of mathematics.
Given a "universal" set in which to take complements, intersections are related to unions by means of DeMorgan's laws. But as we've seen, in the general case (when there is no a priori universal set), intersections are much simpler than unions: They require no axioms. They also give less, as they only ever decrease cardinality.