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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.
The intersection operation is:
- associative : ;
- commutative : .
The intersection of any finite number of sets may be defined inductively, as
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.
- Paul Halmos (1960). Naive set theory. Van Nostrand Reinhold. Section 4.
- Keith J. Devlin (1979). Fundamentals of Contemporary Set Theory. Springer-Verlag, 6,11. ISBN 0-387-90441-7.