Is there a similar notion to the domain dual to codomain and range

Question

Given a function $f\colon X \to Y$, $X$ is called the "domain", $Y$ is called the "codomain" and the range of $f$ is defined to be the set, $\{ y \in Y \mid \exists x \in X \colon y=f(x) \}$

We can extend this concept to binary relations. We call a set "graph(of a binary relation)" $R$ a set whose only elements are ordered pairs, i.e. there exist sets $A$ and $B$ such that $R \subseteq A \times B$. Notice that the sets $A$ and $B$ are not unique, since $\emptyset \subseteq \emptyset \times \emptyset$ and $\emptyset \subseteq \{ \emptyset \} \times \{ \emptyset \}$.

For this reason if $R \subseteq A \times B$, we define a binary relation as the triplet $(A, B, R)$. Now, if we were to analogously define the "codomain" and the "range", the codomain is defined as $B$ and the range is defined as $\{ y \mid (\exists x)((x, y) \in R) \}$. But how would we define the domain? We could define is as $A$, but also as the set $\{ x \mid (\exists y)((x, y) \in R) \}$. With functions the two sets are equal since functions are total relations. I was wondering if there is a standard notion for these two sets, since they would pop up in algebra with sets.

Answer 1

I think it would be standard to say that, for a relation $(A,B,R)$,

• $A$ is the domain,
• $B$ is the codomain,
• $\{y \mid \exists x ((x,y) \in R)\}$ is the range, and
• $\{x \mid \exists y ((x,y) \in R)\}$ is the corange.

As you note, the range and corange are determined entirely by the set $R$, while the domain and codomain cannot be determined uniquely by $R$ and instead must be specified as part of the data of the relation.