Intersection (set theory)

In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.

A∩B in Venn diagram

The intersection of A and B is written "A ∩ B". Formally:

x is an element of A ∩ B if and only if

x is an element of A and

x is an element of B.

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

More generally, one can take the intersection of several sets at once. The intersection of A, B, C, and D, for example, is A ∩ B ∩ C ∩ D = A ∩ (B ∩ (C ∩ D)). Intersection is an associative operation; thus, A ∩ (B ∩ C) = (A ∩ B) ∩ C.

The most general notion is the intersection of an arbitrary nonempty collection of sets. If M is a nonempty set whose elements are themselves sets, then x is an element of the intersection of M if and only if for every element A of M, x is an element of A. In symbols:

This idea subsumes the above paragraphs, in that for example, A ∩ B ∩ C is the intersection of the collection {A,B,C}. (The case where M is empty can sometimes be made sense of; see nullary intersection.)

The notation for this last concept can vary considerably. Hardcore set theorists will simply write "∩M", while most people will instead write "∩_A∈MA". The latter notation can be generalised to "∩_i∈I A_i", which refers to the intersection of the collection {A_i : i ∈ I}. Here I is a nonempty set, and A_i is a set for every i in I.

In the case that the index set I is the set of natural numbers, you might see notation analogous to that of an infinite series:

When formatting is difficult, this can also be written "A₁ ∩ A₂ ∩ A₃ ∩ ...", even though strictly speaking, A₁ ∩ (A₂ ∩ (A₃ ∩ ... makes no sense. (This last example, an intersection of countably many sets, is actually very common; for an example see the article on σalgebras.)

Finally, let us note that whenever the symbol "∩" is placed before other symbols instead of between them, it should be of a larger size. (Eventually this will be available in HTML as the character entity &bigcap;, but until then, try <big>∩</big>.)