Intersection: Difference between revisions

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In [[set theory]], the '''intersection''' of two sets is the set of elements that they have in common:
In [[set theory]], the '''intersection''' of two sets is the set of elements that they have in common:


:<math> A \cap B = \{ x : x \in A \wedge x \in B \} , \, </math>
:<math> A \cap B = \{ x : x \in A \wedge x \in B \} , \, </math>


where <math>\wedge</math> denotes logical and.
where <math>\wedge</math> denotes [[logical and]].  Two sets are '''disjoint''' if their intersection is the [[empty set]].


==Properties==
==Properties==
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==References==
==References==
* {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[Van Nostrand Reinhold]] | year=1960 }}  Section 4.
* {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[Van Nostrand Reinhold]] | year=1960 }}  Section 4.
* {{cite book | author=Keith J. Devlin | authorlink=Keith Devlin | title=Fundamentals of Contemporary Set Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90441-7 | pages=6,11 }}
* {{cite book | author=Keith J. Devlin | authorlink=Keith Devlin | title=Fundamentals of Contemporary Set Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90441-7 | pages=6,11 }}[[Category:Suggestion Bot Tag]]

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In set theory, the intersection of two sets is the set of elements that they have in common:

where denotes logical and. Two sets are disjoint if their intersection is the empty set.

Properties

The intersection operation is:

  • associative : ;
  • commutative : .

General intersections

Finite intersections

The intersection of any finite number of sets may be defined inductively, as

Infinite intersections

The intersection of a general family of sets Xλ as λ ranges over a general index set Λ may be written in similar notation as

We may drop the indexing notation and define the intersection of a set to be the set of elements contained in all the elements of that set:

In this notation the intersection of two sets A and B may be expressed as

The correct definition of the intersection of the empty set needs careful consideration.

See also

References