Intersection (set theory)
In mathematics, the intersection A ∩ 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.
The intersection of A and B is written "A ∩ B". Formally:
that is
For example:
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. Additionally, intersection is commutative; thus A ∩ B = B ∩ A.
Inside a universe U one may define the complement Ac of A to be the set of all elements of U not in A. Now the intersection of A and B may be written as the complement of the union of their complements, derived easily from De Morgan's laws:
A ∩ B = (Ac ∪ Bc)c
We say that A intersects (meets) B at an element x if x belongs to A and B. We say that A intersects (meets) B if A intersects B at some element. A intersects B if their intersection is inhabited.
We say that A and B are disjoint if A does not intersect B. In plain language, they have no elements in common. A and B are disjoint if their intersection is empty, denoted .
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