Definition 6.3.4.1.1 (00PN): The Domain and Range of a Relation

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Theorem Environments

Style 1

book

Alegreya Sans

tcbthm

Style 2

book

Alegreya Sans

amsthm

Style 3

book

Arno*

amsthm

Style 4

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Computer Modern

amsthm

Let $R\subset A\times B$ be a relation.^[1]^[2]

The domain of $R$ is the subset $\mathrm{dom}\webleft (R\webright )$ of $A$ defined by
\[ \mathrm{dom}\webleft (R\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ a\in A\ \middle |\ \begin{aligned} & \text{there exists some $b\in B$}\\ & \text{such that $a\sim _{R}b$}\\ \end{aligned} \webright\} . \]
The range of $R$ is the subset $\mathrm{range}\webleft (R\webright )$ of $B$ defined by
\[ \mathrm{range}\webleft (R\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ b\in B\ \middle |\ \begin{aligned} & \text{there exists some $a\in A$}\\ & \text{such that $a\sim _{R}b$}\\ \end{aligned} \webright\} . \]

Footnotes

[1] Following

, we may compute the (characteristic functions associated to the) domain and range of a relation using the following colimit formulas:

\begin{align*} \chi _{\mathrm{dom}\webleft (R\webright )}\webleft (a\webright ) & \cong \operatorname*{\text{colim}}_{b\in B}\webleft (R^{b}_{a}\webright )\qquad \webleft (a\in A\webright )\\ & \cong \bigvee _{b\in B}R^{b}_{a},\\ \chi _{\mathrm{range}\webleft (R\webright )}\webleft (b\webright ) & \cong \operatorname*{\text{colim}}_{a\in A}\webleft (R^{b}_{a}\webright )\qquad \webleft (b\in B\webright )\\ & \cong \bigvee _{a\in A}R^{b}_{a}, \end{align*}

where the join $\bigvee $ is taken in the poset $\webleft (\{ \mathsf{true},\mathsf{false}\} ,\preceq \webright )$ of Chapter 2: Constructions With Sets, Definition 1.2.2.1.3.

[2] Viewing $R$ as a function $R\colon A\to \mathcal{P}\webleft (B\webright )$, we have

\begin{align*} \mathrm{dom}\webleft (R\webright ) & \cong \operatorname*{\text{colim}}_{y\in Y}\webleft (R\webleft (y\webright )\webright )\\ & \cong \bigcup _{y\in Y}R\webleft (y\webright ),\\ \mathrm{range}\webleft (R\webright ) & \cong \operatorname*{\text{colim}}_{x\in X}\webleft (R\webleft (x\webright )\webright )\\ & \cong \bigcup _{x\in X}R\webleft (x\webright ), \end{align*}