For broader coverage of this topic, see Intersection.
In mathematics, the intersection of two sets and denoted by [1][2] is the set containing all elements of that also belong to or equivalently, all elements of that also belong to [3]
Intersection is written using the symbol "" between the terms; that is, in infix notation. For example:
The intersection of more than two sets (generalized intersection) can be written as:[1]
which is similar to capital-sigma notation.
For an explanation of the symbols used in this article, refer to the table of mathematical symbols.
The intersection of two sets and denoted by [1][4] is the set of all objects that are members of both the sets and In symbols:
That is, is an element of the intersection if and only if is both an element of and an element of [4]
For example:
Intersection is an associative operation; that is, for any sets and one has Intersection is also commutative; for any and one has It thus makes sense to talk about intersections of multiple sets. The intersection of for example, is unambiguously written disjoint sets also called Unoin sets
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:
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.
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 In symbols:
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
In the case that the index set is the set of natural numbers, notation analogous to that of an infinite product may be seen:
When formatting is difficult, this can also be written "". This last example, an intersection of countably many sets, is actually very common; for an example, see the article on σ-algebras.
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),[5] 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 ).
