Disjoint union

In the mathematical branch of set theory there are two slightly different uses of the term disjoint union .

definition

The following distinction corresponds exactly to the difference between inner and outer direct sum . The two definitions represent the different states of affairs, but both are referred to as disjoint union. Therefore, the term must be understood depending on its context. The notations in the article are not only used in this way in the literature, mostly the latter for the former.

Union of disjoint sets

A set is the disjoint union of a system of subsets , written ${\displaystyle X}$ ${\displaystyle (X_{i})_{i\in I}}$ ${\displaystyle X_{i}\subseteq X}$

${\displaystyle X={\dot {\bigcup _{i\in I}}}X_{i},}$

if the following two conditions are met:

• ${\displaystyle X_{i}\cap X_{j}=\varnothing ,}$if , that is to say, they are pairwise disjoint ; ${\displaystyle i\neq j}$ ${\displaystyle X_{i}}$
• ${\displaystyle \textstyle X=\bigcup \limits _{i\in I}X_{i}}$, that is, is the union of all sets . ${\displaystyle X}$ ${\displaystyle X_{i}}$

Disjoint union of arbitrary sets

Are sets of given, it means the amount ${\displaystyle X_{i}}$ ${\displaystyle i\in I}$

${\displaystyle \bigsqcup _{i\in I}X_{i}=\bigcup _{i\in I}\{(i,x)\mid x\in X_{i}\}}$

the disjoint union of the sets . It is roughly a union in which the sets are artificially disjointed beforehand. ${\displaystyle X_{i}}$

properties

• For the widths applies: . In cardinal number arithmetic , the sum is precisely defined by this relationship. ${\displaystyle \left|\bigsqcup \limits _{i\in I}X_{i}\right|=\sum _{i\in I}|X_{i}|}$
• The disjoint union is the categorical coproduct in the category of sets. This means: images correspond to one-to-one systems of images with . ${\displaystyle \bigsqcup \limits _{i\in I}X_{i}}$ ${\displaystyle f\colon \bigsqcup \limits _{i\in I}X_{i}\to Y}$ ${\displaystyle (f_{i})_{i\in I}}$ ${\displaystyle f_{i}\colon X_{i}\to Y}$
• If the sets are disjoint, the canonical mapping is bijective. ${\displaystyle X_{i}}$ ${\displaystyle \bigsqcup \limits _{i\in I}X_{i}\to \bigcup \limits _{i\in I}X_{i}}$

Examples

Example of the union of disjoint sets

Disjoint union of and . ${\displaystyle A=\{1,2,3\}}$ ${\displaystyle B=\{4,5,6\}}$

• ${\displaystyle A\cap B=\varnothing }$ Both sets are disjoint
• ${\displaystyle A\;{\dot {\cup }}\;B=C}$
• ${\displaystyle C}$is the disjoint union of the sets and ⇒ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C=\{1,2,3,4,5,6\}}$
• The sets and form a partition of the set ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$
• The disjoint union in the second sense yields the pair set . The projection maps bijectively on . ${\displaystyle A\sqcup B}$ ${\displaystyle D=\{(1,1),(1,2),(1,3),(2,4),(2,5),(2,6)\}}$ ${\displaystyle \pi _{2}}$ ${\displaystyle D}$ ${\displaystyle C}$

Example of a disjoint union of arbitrary sets

Disjoint union of and . ${\displaystyle X_{1}=\{1,2,3\}}$ ${\displaystyle X_{2}=\{1,2,3,4\}}$

• ${\displaystyle I=\{1,2\}}$
• ${\displaystyle \textstyle \bigsqcup \limits _{i\in I}X_{i}=\bigcup \limits _{i\in I}\{(i,x)\mid x\in X_{i}\}=\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(2,4)\}}$