Disjoint sets

In set theory, the names of two quantities and disjoint (Latin disiunctum, isolated '), disjoint or average foreign, if they have no common element. Several sets are called pairwise disjoint if any two of them are disjoint.

Definitions

Two sets and are disjoint if their intersection is empty, ie, if the following applies:

A family of sets is a disjoint family of sets, if its elements are pairwise disjoint, ie, if the following applies:

The union of a disjoint family of sets is called a disjoint union and writes it as

Also are all amounts the family is not empty, then there is a partition of.

The terms are also used for analogous systems of sets (instead of families of sets ).

Examples

• The quantities and are disjoint, because they have no common element.

• The quantities and are not disjoint, since they have the element together.

• The three quantities, 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 set family, which represents a partition of the integers: .

• Two different lines and in the Euclidean plane are disjoint if and only if they are parallel. The totality of all parallels to a given line forms a partition of the plane.

Application

In the questionnaire design questions need to be formulated so that the answer choices ( term relationships ) are disjoint and exhaustive.

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

People with an income between 500 and 1000 euros not know which answer choice they should choose.

Properties

• The empty set is disjoint from any quantity.

• 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 widths. For non- disjoint unions the Siebformel applies.