Domain of a Relation from A to B

Question

The latest versions of Susanna S. Epp's Discrete Mathematics book (both the First Brief Edition, and the full Applications 4th edition) define a relation from $\mathcal{A}$ to $\mathcal{B}$ as a subset of $\mathcal{A} \times \mathcal{B}$ and state that the set $\mathcal{A}$ is the domain, and $\mathcal{B}$ is the co-domain.

This to me indicates that there is a different meaning of domain in "Domain of a relation" and "Domain of a function". For example, let $\mathcal{A} = \{4, 5, 6\}$, $\mathcal{B} = \{5, 6, 7\}$ and define relation $T$ from $\mathcal{A}$ to $\mathcal{B}$ is defined as

$$ (x,y) \in T \text{ means } x \ge y.$$

In this case, there is no ordered pair in $T$ which has $4$ as the first member.

For the set $\mathcal{B}$ there are two specific terms: co-domain, and range which define the entire set $\mathcal{B}$ and the subset of $\mathcal{B}$ for which $T$ has elements with $b \in \mathcal{B}$ as the second member of the ordered pair.

So, my questions are:

1. What is the domain of $T$?
2. Are there commonly accepted terms that apply to the set $\mathcal{A}$ that clarify the distinction as the terms co-domain and range do for set $\mathcal{B}$ ?

To summarize, I am looking for a definition of "Domain of a Relation"?

Notes:

• It seems that Partial functions expands this issue further for functions, but not relations.
• I am looking at this from a first (or perhaps second) year undergraduate math level.

References:

• The only other reference to this issue that I can find is Proof Wiki in the "Also Defined As" section.
• Should the domain of a function be inferred?

Answer 1

As far as I know, the range of a relation is not a concept that is frequently considered, and hence the dual concept within the domain is equally not frequently considered.

In the context of functions, we assume that each element of the domain is in the relation exactly once, but we make no such assumption on the codomain. Hence in this context it is useful to distinguish those codomain elements used by the function, ergo "range''.

Answer 2

A function is just a very special type of relation. You could define the range of a relation, and it is sometimes awkward that there isn't a commonly accepted notation for the set $\{y \colon x \mathrel{R} y \text{ for an } x \in \mathcal{A}\}$ (like it is said $f(\mathcal{A})$ for a function $f$) or the other other way around (like $f^{-1}(\mathcal{A})$). But it is rare enough that it isn't a real problem.