Intersection (set theory)


Intersection (set theory) - Definition

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.

The intersection of A and B

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, …}.

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.

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:

<math>\left( x \in \bigcap \mathbf{M} \right) \leftrightarrow \left( \forall A \in \mathbf{M}. \ x \in A \right).<math>

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. set theorists will sometimes write "∩M", while others will instead write "∩_A∈MA". The latter notation can be generalized 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:

<math>\bigcap_{i=1}^{\infty} A_i<math>

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 σ-algebras.)

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>.)

See also