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

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