# Domain of a Relation from A to B

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:

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.
• I have always used the terms domain and range in this way and written $\operatorname{dom}R$ and $\operatorname{ran}R$, just as I would for a function. I’m surprised that you consider this non-standard. Of course one can also write $\pi_1[R]$ and $\pi_2[R]$, after defining the obvious projection maps. – Brian M. Scott Apr 28 '13 at 9:26
The answer to your first question "What is the domain of T?" is $\{5, 6\}$ because the definition of a domain of a relation that is frequently used is quite clear: The domain of a relation is the set of the first coordinates from the ordered pairs. Since 4 has no pair, it is not part of the domain.
And the answer to your second question "Are there commonly accepted terms that apply to the set A that clarify the distinction as the terms co-domain and range do for set B ?" is simply no. There is no commonly accepted terms for that. So $\{5, 6\}$ is the domain of T. But there is no term for $\{4, 5, 6\}$. You can just call it the set of the first coordinate.