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单词 Intersection
释义

intersection


The intersectionMathworldPlanetmathPlanetmath of two sets A and B is the set that contains all the elements x such thatxA and xB. The intersection of A and B is written as AB. The following Venn diagramMathworldPlanetmath illustrates the intersection of two sets A and B:

AABB..

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

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

jJAj={x:xAj for all jJ}.

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

Some elementary properties of are

  • (idempotency) AA=A,

  • (commutativity) AB=BA,

  • (associativity) A(BC)=(AB)C,

  • AA=, where A is the complement of A in some fixed universePlanetmathPlanetmath U.

Remark. What is jJAj when J=? In other words, what is the intersection of an empty family of sets? First note that if IJ, then

jJAjiIAi.

This leads the conclusionMathworldPlanetmath that the intersection of an empty family of sets should be as large as possible. How large should it be? In additionPlanetmathPlanetmath, is this intersection a set? The answer depends on what versions of set theoryMathworldPlanetmath 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 AjU, then it is sometimes a matter of convenience to define the intersection of an empty family of Aj to be U.

Titleintersection
Canonical nameIntersection
Date of creation2013-03-22 12:14:52
Last modified on2013-03-22 12:14:52
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id22
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 03E99
Synonymintersects
Synonymmeets
Related topicunion
Related topicUnion
Related topicFiniteIntersectionProperty
Related topicEmptySet
Related topicProductOfLeftAndRightIdeal
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更新时间:2025/5/4 22:54:53