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:
- What is the domain of $T$?
- 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?