Intersection

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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