Main Page | See live article | Alphabetical index

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.

AB 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 "AM A". The latter notation can be generalised to "iI Ai", which refers to the intersection of the collection {Ai : i ∈ I}. Here I is a nonempty set, and Ai 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>&cap;</big>.)

See also: Basic set theory