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.

For explanation of the symbols used in this article, refer to the table of mathematical symbols.

=Basic definition=

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 logical conjunction :* 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, …}.

If the intersection of two sets A and B is empty, that is they have no elements in common, then they are said to be disjoint, denoted: A ∩ B  = Ø. For example the sets {1, 2} and {3, 4} are disjoint, written {1, 2} ∩ {3, 4} = Ø.

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.

=Arbitrary intersections=

The most general notion is the intersection of an arbitrary nonempty collection of sets. If M is a empty set set whose elements are themselves sets, then x is an element of the intersection of M iff universal quantification element A of M, x is an element of A. In symbols:

:left( x in igcap mathbf{M} ight) leftrightarrow left( forall A in mathbf{M}. x in A ight).

This idea subsumes the above paragraphs, in that for example, A ∩B ∩C is the intersection of the collection {A,B,C}.

The notation for this last concept can vary considerably. Set theory will sometimes write ∩M , while others will instead write ∩A∈M A . The latter notation can be generalized to ∩i∈I 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:

:igcap_{i=1}^{infty} A_i

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 sigma algebra.)

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>.)

= Nullary intersection =

Note that in the previous section we excluded the case where M was the empty set (∅). The reason is the follows. The intersection of the collection M is defined as the set (see set-builder notation) :igcap mathbf{M} = {x : x in A; mbox{ for all } A in mathbf{M}}. If M is empty there are no sets A in M, so the question becomes which x s satisfy the stated condition The answer seems to be every possible x . When M is empty the condition given above is an example of a vacuous truth. So the intersection of the empty family should be the set of everything . The problem is, there is no such set . Assuming such a set exists leads to a famous problem in naive set theory known as Russell s paradox. For this reason the intersection of the empty set is left undefined. There is nothing that can be done about the problem, it is just a fact of life in mathematics.

A partial fix for this problem can be found if we agree to restriction our attention to subsets of a fixed set U called the universe (set theory) . In this case the intersection of a family of subsets of U can be defined as :igcap mathbf{M} = {x in U : x in A; mbox{ for all } A in mathbf{M}}. Now if M is empty there is no problem. The intersection is just the entire universe U , which is a well-defined set by assumption.

= See also =