Disjoint union

A disjoint union (also a disjoint union or a disjoint sum ) is a modified set union operation in set theory , which, informally speaking, consists in combining disjoint "copies" of sets. In particular, the disjoint union of two finite sets consisting of $a$ ${\ displaystyle a}$ $a$ and $b$ ${\ displaystyle b}$ $b$ elements will contain exactly $a+b$ ${\ displaystyle a + b}$ $a + b$ elements, even if the sets themselves intersect.

Content

Definition

Let be $\{A_{i}|i\in I\}$ ${\ displaystyle \ {A_ {i} | i \ in I \}}$ - family of sets listed by indices from $I$ ${\ displaystyle I}$ . Then the disjoint union of this family is the set

\bigsqcup _{i\in I}A_{i}=\bigcup _{i\in I}\{(x,i)|x\in A_{i}\}

{\ displaystyle \ bigsqcup _ {i \ in I} A_ {i} = \ bigcup _ {i \ in I} \ {(x, i) | x \ in A_ {i} \}}

The elements of a disjoint union are ordered pairs. $(x,i)$ ${\ displaystyle (x, i)}$ . In this way $i$ ${\ displaystyle i}$ there is an index indicating which set $A_{i}$ ${\ displaystyle A_ {i}}$ an item entered the union. Each of the sets $A_{i}$ ${\ displaystyle A_ {i}}$ canonically embedded in a disjoint union as a set

A_{i}^{*}=\{(x,i)|x\in A_{i}\}.

{\ displaystyle A_ {i} ^ {*} = \ {(x, i) | x \ in A_ {i} \}.}

With $\forall i,j\in I:i\neq j$ ${\ displaystyle \ forall i, j \ in i: i \ neq j}$ sets $A_{i}^{*}$ ${\ displaystyle A_ {i} ^ {*}}$ and $A_{j}^{*}$ ${\ displaystyle A_ {j} ^ {*}}$ do not have common elements, even if $A_{i}\cap A_{j}\neq \varnothing$ ${\ displaystyle A_ {i} \ cap A_ {j} \ neq \ varnothing}$ . In the degenerate case when the sets $A_{i}\forall i\in I$ ${\ displaystyle A_ {i} \ forall i \ in I}$ equal to some particular $A$ ${\ displaystyle A}$ , a disjoint union is a Cartesian product of a set $A$ ${\ displaystyle A}$ and sets $I$ ${\ displaystyle I}$ , i.e

\bigsqcup _{i\in I}A_{i}=A\times I.

{\ displaystyle \ bigsqcup _ {i \ in I} A_ {i} = A \ times I.}

Use

Sometimes you can find the designation $A+B$ ${\ displaystyle A + B}$ for a disjoint union of two sets or the following for a family of sets:

\sum _{i\in I}A_{i}.

{\ displaystyle \ sum _ {i \ in I} A_ {i}.}

Such a record implies that the cardinality of a disjoint union is equal to the sum of cardinalities of the sets of the family. For comparison, the Cartesian product has a power equal to the product of the powers.

In the category of sets, a disjoint union is a direct sum. The term disjoint union is also used to refer to the union of a family of pairwise disjoint sets. In this case, the disjoint union is denoted as the usual union of sets , coinciding with it. This designation is often found in computer science . More formally, if $C$ ${\ displaystyle C}$ Is a family of sets, then

\bigcup _{A\in C}A

{\ displaystyle \ bigcup _ {A \ in C} A}

there is a disjoint union in the above sense if and only if for all $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ of $C$ ${\ displaystyle C}$ the following condition is satisfied:

A\neq B\implies A\cap B=\varnothing .

{\ displaystyle A \ neq B \ implies A \ cap B = \ varnothing.}

Variations and generalizations

If all sets of a disjoint union are endowed with topology, then the disjoint union of topological spaces (that is, sets endowed with topology) itself has a natural topology — the strongest topology such that each inclusion is a continuous map. A disjoint union with this topology is called a disjoint union of topological spaces.

Literature

Aleksandryan R. A., Mirzakhanyan E. A. General Topology. - M .: Higher School, 1979. - p. 132.
Spanier E. Algebraic topology. - M .: Mir, 1971. - p. 9.
Melnikov, OV, and others. General algebra: В 2 т. T. 1. - M .: Nauka, 1990. - P. 13. - ISBN 5020144266

Disjoint union

Content

Definition

Use

Variations and generalizations

See also

Literature

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