Intersection

From Citizendium, the Citizens' Compendium

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.

Contents 1 Properties 2 General intersections 2.1 Finite intersections 2.2 Infinite intersections 3 See also 4 References

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 X_λ 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

• Union

References

• 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.