# 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:

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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''.

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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.

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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.

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