Relation, $\mathrm{dom}R$, $\mathrm{ran}R$, $\mathrm{fld}R$, and the specialisations "relation from $A$ to $B$" and "relation on $A$"

A relation is a set $R$ every element of which is an ordered pair (The Kuratowski ordered pair $(a,b):=\{\{a\},\{a,b\}\}$). We write $aRb$ for $(a,b)\in R$.

By If every element of $R$ is an ordered pair, then $\{a:\mathrm{\exists }b(a,b)\in R\}$ and $\{b:\mathrm{\exists }a(a,b)\in R\}$ are sets, both included in $\bigcup \bigcup R$ the following two classes are sets, so the notation is legitimate:

$\mathrm{dom}R:=\{a:\mathrm{\exists }b(a,b)\in R\},\mathrm{ran}R:=\{b:\mathrm{\exists }a(a,b)\in R\},$

the domain and the range of $R$. The field of $R$ is $\mathrm{fld}R:=\mathrm{dom}R\cup \mathrm{ran}R$ (The union $\bigcup x$ of a set, and the binary union $a\cup b:=\bigcup \{a,b\}$).

$R$ is a relation from $A$ to $B$ when $R\subseteq A\times B$ (The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$, Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$), and a relation on $A$ when $R\subseteq A\times A$.

• Domain and range are intrinsic; the sets $A$ and $B$ are not. A relation determines its own domain and range, but it is a relation from $A$ to $B$ for many different $A$ and $B$: any $A$ containing $\mathrm{dom}R$ and any $B$ containing $\mathrm{ran}R$ will do. That is the content of Every relation $R$ satisfies $R\subseteq \mathrm{dom}R\times \mathrm{ran}R$, and $R$ is a relation from $A$ to $B$ if and only if $\mathrm{dom}R\subseteq A$ and $\mathrm{ran}R\subseteq B$, and it is the same distinction that makes a codomain part of the declaration $f:A\to B$ rather than of the set $f$ in A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain.

• The empty set is a relation. It has no elements, so the requirement that every element be an ordered pair holds vacuously, and $\mathrm{dom}\mathrm{\varnothing }=\mathrm{ran}\mathrm{\varnothing }=\mathrm{\varnothing }$. It is a relation on every set.