We say that intersects (meets) if there exists some that is an element of both and in which case we also say that intersects (meets) at. Equivalently, intersects if their intersection is an inhabited set, meaning that there exists some such that
We say that and are disjoint if does not intersect In plain language, they have no elements in common. and are disjoint if their intersection is empty, denoted
For example, the sets and 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 and one has
Thus the parentheses may be omitted without ambiguity: either of the above can be written as . Intersection is also commutative. That is, for any and one has
The intersection of any set with the empty set results in the empty set; that is, that for any set ,
Also, the intersection operation is idempotent; that is, any set satisfies that . All these properties follow from analogous facts about logical conjunction.
Intersection distributes over union and union distributes over intersection. That is, for any sets and one has
Inside a universe one may define the complement of to be the set of all elements of not in Furthermore, the intersection of and 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.
If is a nonempty set whose elements are themselves sets, then is an element of the intersection of if and only if for every element of is an element of
The notation for this last concept can vary considerably. Set theorists will sometimes write "", while others will instead write "".
The latter notation can be generalized to "", which refers to the intersection of the collection
Here is a nonempty set, and is a set for every
Note that in the previous section, we excluded the case where was the empty set (). The reason is as follows: The intersection of the collection is defined as the set (see set-builder notation)
If is empty, there are no sets in so the question becomes "which 's satisfy the stated condition?" The answer seems to be every possible . When 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),
but in standard (ZF) set theory, the universal set does not exist.
In type theory however, is of a prescribed type so the intersection is understood to be of type (the type of sets whose elements are in ), and we can define to be the universal set of (the set whose elements are exactly all terms of type ).