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

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.

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.