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In mathematics, the intersection A &cap; B 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 two sets A and B, denoted by $A &cap; B$, is the set of all objects that are members of both the sets $A$ and $B$. In symbols,


 * A \cap B = \{ x: x \in A \text{ and } x \in B\}.

That is, x is an element of the intersection A &cap; B if and only if x is both an element of A and 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 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, ...}, because 9 is not prime.

Intersection is an associative operation; that is, for any sets A, B, and C, one has A &cap; (B &cap; C) = (A &cap; B) &cap; C. Intersection is also commutative; for any A and B, one has A &cap; B = B &cap; A. It thus makes sense to talk about in