Type | Set operation |
---|---|
Field | Set theory |
Statement | The intersection of and is the set of elements that lie in both set and set . |
Symbolic statement |
In set theory, the intersection of two sets and denoted by [1] is the set containing all elements of that also belong to or equivalently, all elements of that also belong to [2]
Intersection is written using the symbol "" between the terms; that is, in infix notation. For example:
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 , [3] 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 [3]
For example:
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
Intersection distributes over union and union distributes over intersection. That is, for any sets and one has
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.
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)
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 is well-defined. In that case, if is empty, its intersection is . Since all vacuously satisfy the required condition, the intersection of the empty collection of subsets of is all of In formulas, 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.
Also, in type theory 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 ).
Type | Set operation |
---|---|
Field | Set theory |
Statement | The intersection of and is the set of elements that lie in both set and set . |
Symbolic statement |
In set theory, the intersection of two sets and denoted by [1] is the set containing all elements of that also belong to or equivalently, all elements of that also belong to [2]
Intersection is written using the symbol "" between the terms; that is, in infix notation. For example:
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 , [3] 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 [3]
For example:
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
Intersection distributes over union and union distributes over intersection. That is, for any sets and one has
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.
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)
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 is well-defined. In that case, if is empty, its intersection is . Since all vacuously satisfy the required condition, the intersection of the empty collection of subsets of is all of In formulas, 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.
Also, in type theory 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 ).