Intersection (set theory)

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Intersection
The intersection of two sets and represented by circles. is in red.
Type	Set operation
Field	Set theory
Statement	The intersection of and is the phối of elements that lie in both phối and phối .
Symbolic statement

In phối theory, the intersection of two sets and denoted by ^[1] is the phối containing all elements of that also belong đồ sộ or equivalently, all elements of that also belong đồ sộ ^[2]

Notation and terminology[edit]

Intersection is written using the symbol "" between the terms; that is, in infix notation. For example:

The intersection of more than thở two sets (generalized intersection) can be written as:

which is similar đồ sộ capital-sigma notation.

For an explanation of the symbols used in this article, refer đồ sộ the table of mathematical symbols.

Definition[edit]

~A \cap B \cap C — Intersection of three sets:

The intersection of two sets and denoted by ,^[3] is the phối 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:

The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}.
The number 9 is not in the intersection of the phối of prime numbers {2, 3, 5, 7, 11, ...} and the phối of odd numbers {1, 3, 5, 7, 9, 11, ...}, because 9 is not prime.

Intersecting and disjoint sets[edit]

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 phối of even numbers intersects the phối of multiples of 3 at the multiples of 6.

Algebraic properties[edit]

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

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The intersection of any phối with the empty phối results in the empty set; that is, that for any phối ,

Also, the intersection operation is idempotent; that is, any phối 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 đồ sộ be the phối 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:

Arbitrary intersections[edit]

The most general notion is the intersection of an arbitrary nonempty collection of sets. If is a nonempty phối 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 đồ sộ "", which refers đồ sộ the intersection of the collection Here is a nonempty phối, and is a phối for every

In the case that the index phối is the phối of natural numbers, notation analogous đồ sộ 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.

Nullary intersection[edit]

Note that in the previous section, we excluded the case where was the empty phối (). The reason is as follows: The intersection of the collection is defined as the phối (see set-builder notation)

If is empty, there are no sets in so sánh the question becomes "which 's satisfy the stated condition?" The answer seems đồ sộ 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 phối (the identity element for the operation of intersection),^[4] but in standard (ZF) phối theory, the universal phối does not exist.

However, when restricted đồ sộ the context of subsets of a given fixed phối , 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 đồ sộ the whole underlying phối.

Also, in type theory is of a prescribed type so sánh the intersection is understood đồ sộ be of type (the type of sets whose elements are in ), and we can define đồ sộ be the universal phối of (the phối whose elements are exactly all terms of type ).