disjoint
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.
Definitions
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).
Examples
- 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:
application
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?
- 0 to 1000 euros
- 500 and more euros.
People with earnings between 500 and 1000 euros do not know which answer to choose.
properties
- 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
Individual evidence
- ↑ See the answers to the question "Is the empty family of sets pairwise disjoint?"