# disjoint

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. ${\ displaystyle A}$${\ displaystyle B}$

## Definitions

A disjoint set system

Two sets and are disjoint if their intersection is empty , so if: ${\ displaystyle A}$${\ displaystyle B}$

${\ displaystyle A \ cap B = \ emptyset}$.

A family of sets is a disjoint family of sets if its elements are pairwise disjoint , so if: ${\ displaystyle (M_ {i}) _ {i \ in I}}$

${\ displaystyle M_ {i} \ cap M_ {j} = \ emptyset}$for and .${\ displaystyle i \ neq j}$${\ displaystyle i, j \ in I}$

The union of a disjoint set family is called a disjoint union and it is written as ${\ displaystyle M}$

${\ displaystyle M = {\ dot {\ bigcup _ {i \ in I}}} M_ {i}}$.

In addition, if all sets of the family are not empty, there is a partition of . ${\ displaystyle M}$

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.${\ displaystyle A = \ {1,2,3 \}}$${\ displaystyle B = \ {7,8,11 \}}$
• The sets and are not disjoint because they have the element in common.${\ displaystyle A = \ {1,2,7 \}}$${\ displaystyle B = \ {6,7,8,11 \}}$${\ displaystyle 7}$
• The three sets , and are not pairwise disjoint, since at least one of the three possible intersections (namely ) is not empty.${\ displaystyle A = \ {1,2,3 \}}$${\ displaystyle B = \ {4,5 \}}$${\ displaystyle C = \ {5,6,7 \}}$${\ displaystyle B \ cap C}$
• The following list defined an (infinite) disjoint family that a partition of the integers is: .${\ displaystyle \ {0 \}, \ {1, -1 \}, \ {2, -2 \}, \ {3, -3 \}, \ {4, -4 \}, \ ldots}$
• 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.${\ displaystyle g}$${\ displaystyle h}$${\ displaystyle g}$

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?

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.

## properties

• The empty set is disjoint to any set.${\ displaystyle \ emptyset}$
• ${\ displaystyle \ {a \}}$and are disjoint if and only if .${\ displaystyle B}$${\ displaystyle a \ notin B}$
• 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