Disjoint union: Difference between revisions
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In [[mathematics]], the '''disjoint union''' of two | {{subpages}} | ||
In [[mathematics]], the '''disjoint union''' of two [[set (mathematics)|set]]s ''X'' and ''Y'' is a set which contains "copies" of each of ''X'' and ''Y'': it is denoted <math>X \amalg Y</math> or, less often, <math>X \uplus Y</math>. | |||
There are ''injection maps'' in<sub>1</sub> and in<sub>2</sub> from ''X'' and ''Y'' to the disjoint union, which are [[injective function]]s with disjoint images. | There are ''injection maps'' in<sub>1</sub> and in<sub>2</sub> from ''X'' and ''Y'' to the disjoint union, which are [[injective function]]s with disjoint images. | ||
If ''X'' and ''Y'' are disjoint, then the usual union is also a disjoint union. In general, the disjoint union can be realised in a number of ways, for example as | If ''X'' and ''Y'' are disjoint, then the usual [[union]] is also a disjoint union. In general, the disjoint union can be realised in a number of ways, for example as | ||
:<math>X \amalg Y = \{0\} \times X \cup \{1\} \times Y . \, </math> | :<math>X \amalg Y = \{0\} \times X \cup \{1\} \times Y . \, </math> | ||
The disjoint union has a [[universal property]]: if there is a set ''Z'' with maps <math>f:X \rightarrow Z</math> and <math>g:Y \rightarrow Z</math>, then there is a map <math>h : X \amalg Y \rightarrow Z</math> such that the compositions <math>\mathrm{in}_1 \cdot h = f</math> and <math>\mathrm{in}_2 \cdot h = g</math>. | The disjoint union has a [[universal property]]: if there is a set ''Z'' with maps <math>f:X \rightarrow Z</math> and <math>g:Y \rightarrow Z</math>, then there is a map <math>h : X \amalg Y \rightarrow Z</math> such that the compositions <math>\mathrm{in}_1 \cdot h = f</math> and <math>\mathrm{in}_2 \cdot h = g</math>. | ||
The disjoint union is [[commutative]], in the sense that there is a natural [[bijection]] between <math>X \amalg Y</math> and <math>Y \amalg X</math>; it is [[associative]] again in the sense that there is a natural bijection between <math>X \amalg (Y \amalg Z)</math> and <math>(X \amalg Y) \amalg Z</math>. | |||
==General unions== | ==General unions== | ||
The disjoint of any finite number of sets may be defined inductively, as | The disjoint union of any finite number of sets may be defined inductively, as | ||
:<math>\coprod_{i=1}^n X_i = X_1 \ | :<math>\coprod_{i=1}^n X_i = X_1 \amalg (X_2 \amalg (X_3 \amalg (\cdots X_n)\cdots))) . \, </math> | ||
The disjoint union of a general family of sets ''X''<sub>λ</sub> as λ ranges over a general index set Λ may be defined as | The disjoint union of a general family of sets ''X''<sub>λ</sub> as λ ranges over a general index set Λ may be defined as | ||
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==References== | ==References== | ||
* {{cite book | author= | * {{cite book | author=Michael D. Potter | title=Sets: An Introduction | publisher=[[Oxford University Press]] | year=1990 | isbn=0-19-853399-3 | pages=36-37 }}[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 17:00, 7 August 2024
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In mathematics, the disjoint union of two sets X and Y is a set which contains "copies" of each of X and Y: it is denoted or, less often, .
There are injection maps in1 and in2 from X and Y to the disjoint union, which are injective functions with disjoint images.
If X and Y are disjoint, then the usual union is also a disjoint union. In general, the disjoint union can be realised in a number of ways, for example as
The disjoint union has a universal property: if there is a set Z with maps and , then there is a map such that the compositions and .
The disjoint union is commutative, in the sense that there is a natural bijection between and ; it is associative again in the sense that there is a natural bijection between and .
General unions
The disjoint union of any finite number of sets may be defined inductively, as
The disjoint union of a general family of sets Xλ as λ ranges over a general index set Λ may be defined as
References
- Michael D. Potter (1990). Sets: An Introduction. Oxford University Press, 36-37. ISBN 0-19-853399-3.