which is similar to capital-sigma notation.

is an inhabited set, meaning that there exists some

In plain language, they have no elements in common.

are disjoint if their intersection is empty, denoted

are disjoint, while the set of even numbers intersects the set of multiples of 3 at the multiples of 6.

Binary intersection is an associative operation; that is, for any sets

Thus the parentheses may be omitted without ambiguity: either of the above can be written as

Also, the intersection operation is idempotent; that is, any set

All these properties follow from analogous facts about logical conjunction.

Intersection distributes over union and union distributes over intersection.

may be written as the complement of the union of their complements, derived easily from De Morgan's laws:

The most general notion is the intersection of an arbitrary nonempty collection of sets.

The notation for this last concept can vary considerably.

Set theorists will sometimes write "

", which refers to the intersection of the collection

In the case that the index set

is the set of natural numbers, notation analogous to that of an infinite product may be seen:

This last example, an intersection of countably many sets, is actually very common; for an example, see the article on σ-algebras.

In the previous section, we excluded the case where

The reason is as follows: The intersection of the collection

is defined as the set (see set-builder notation)

's satisfy the stated condition?"

is empty, the condition given above is an example of a vacuous truth.

So the intersection of the empty family should be the universal set (the identity element for the operation of intersection),[4] but in standard (ZF) set theory, the universal set does not exist.

However, when restricted to the context of subsets of a given fixed set

, the notion of the intersection of an empty collection of subsets of

vacuously satisfy the required condition, the intersection of the empty collection of subsets of

This matches the intuition that as collections of subsets become smaller, their respective intersections become larger; in the extreme case, the empty collection has an intersection equal to the whole underlying set.

so the intersection is understood to be of type

(the set whose elements are exactly all terms of type