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Two disjoint sets

In set theory called two sets and disjoint ( Latin disjunctus (-a, -um) , separated), disjoint or average foreign if they have no common element. Several sets are pairwise disjoint if any two of them are disjoint.


A disjoint set system

Two sets and are disjoint if their intersection is empty , so if:


A family of sets is a disjoint family of sets if its elements are pairwise disjoint , so if:

for and .

The union of a disjoint set family is called a disjoint union and it is written as


In addition, if all sets of the family are not empty, there is a partition of .

The terms are also used analogously for set systems (instead of set families).


  • The sets and are disjoint because they have no common element.
  • The sets and are not disjoint because they have the element in common.
  • The three sets , and are not pairwise disjoint, since at least one of the three possible intersections (namely ) is not empty.
  • The following list defined an (infinite) disjoint family that a partition of the integers is: .
  • Two different straight lines and in the Euclidean plane are disjoint if and only if they are parallel . The totality of all parallels to a given straight line forms a partition of the plane.

Further examples:


When constructing the questionnaire , questions must be formulated in such a way that the answer options ( conceptual relationships ) are disjoint and exhaustive .

Example of non-disjoint answer options: How much do you earn?

  1. 0 to 1000 euros
  2. 500 and more euros.

People with earnings between 500 and 1000 euros do not know which answer to choose.


  • The empty set is disjoint to any set.
  • and are disjoint if and only if .
  • The cardinality of a finite disjoint union of finite sets is equal to the sum of the individual cardinalities. The sieve formula applies to non-disjoint associations .
  • One-element set systems are always pairwise disjoint.
  • The empty set system is pairwise disjoint

See also

  • Linear disjunct , a concept of abstract algebra in connection with field extensions , which has only in common with the disjunct considered here that the intersection of linear disjunct fields is the smallest possible .

Web links

Wiktionary: disjunkt  - explanations of meanings, word origins, synonyms, translations

Individual evidence

  1. See the answers to the question "Is the empty family of sets pairwise disjoint?"