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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 BThe 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∈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:
- <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 ⋂, but until then, try <big>∩</big>.)
See also
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