intersection (original) (raw)

The intersection of two sets A and B is the set that contains all the elements x such thatx∈A and x∈B. The intersection of A and B is written as A∩B. The following Venn diagram illustrates the intersection of two sets A and B:

AA∩BB..

Example. If A={1,2,3,4,5} and B={1,3,5,7,9} then A∩B={1,3,5}.

We can also define the intersection of an arbitrary number of sets. If {Aj}j∈J is a family of sets, we define the intersection of all them, denoted ⋂j∈JAj, as the set consisting of those elements belonging to every set Aj:

A set U intersects, or meets, a set V if U∩V is non-empty.

Some elementary properties of ∩ are

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  (idempotency) A∩A=A,
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  (commutativity) A∩B=B∩A,
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  (associativity) A∩(B∩C)=(A∩B)∩C,
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  A∩A∁=∅, where A∁ is the complement of A in some fixed universe U.

Remark. What is ⋂j∈JAj when J=∅? In other words, what is the intersection of an empty family of sets? First note that if I⊆J, then

This leads the conclusion that the intersection of an empty family of sets should be as large as possible. How large should it be? In addition, is this intersection a set? The answer depends on what versions of set theory we are working in. Some theories (for example, von Neumann-Gödel-Bernays) say this is the class V of all sets, while others do not define this notion at all. However, if there is a fixed set U in advance such that each Aj⊆U, then it is sometimes a matter of convenience to define the intersection of an empty family of Aj to be U.