Item 2 (02AW) — The Clowder Project

Let $A$ and $B$ be sets.

Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation.^1,2

1.
The domain of $R$ is the subset $\operatorname {Dom}(R)$ of $A$ defined by

\[ \operatorname {Dom}(R)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ a\in A\ \middle |\ \begin{aligned} & \text{there exists some $b\in B$}\\ & \text{such that $a\sim _{R}b$}\\ \end{aligned} \right\} . \]

2.
The image of $R$ is the subset $\mathrm{Im}(R)$ of $B$ defined by

\[ \mathrm{Im}(R)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ b\in B\ \middle |\ \begin{aligned} & \text{there exists some $a\in A$}\\ & \text{such that $a\sim _{R}b$}\\ \end{aligned} \right\} . \]

¹Following , , we may compute the (characteristic functions associated to the) domain and image of a relation using the following colimit formulas:
\begin{align*} \chi _{\operatorname {Dom}(R)}(a) & \cong \operatorname*{\operatorname {\mathrm{colim}}}_{b\in B}(R^{b}_{a})\qquad (a\in A)\\ & \cong \bigvee _{b\in B}R^{b}_{a},\\ \chi _{\mathrm{Im}(R)}(b) & \cong \operatorname*{\operatorname {\mathrm{colim}}}_{a\in A}(R^{b}_{a})\qquad (b\in B)\\ & \cong \bigvee _{a\in A}R^{b}_{a}, \end{align*}
where the join $\bigvee $ is taken in the poset $(\{ \mathsf{true},\mathsf{false}\} ,\preceq )$ of Chapter 4: Constructions With Sets, Definition 3.2.2.1.3.
²Viewing $R$ as a function $R\colon A\to \mathcal{P}(B)$, we have
\begin{align*} \operatorname {Dom}(R) & \cong \operatorname*{\operatorname {\mathrm{colim}}}_{y\in Y}(R(y))\\ & \cong \bigcup _{y\in Y}R(y),\\ \mathrm{Im}(R) & \cong \operatorname*{\operatorname {\mathrm{colim}}}_{x\in X}(R(x))\\ & \cong \bigcup _{x\in X}R(x), \end{align*}

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