# The purpose of the domain and range of a relation

This is just a general question regarding the way relations are defined. A relation is just a subset fn the cartesian product of two sets. More formally:

Definiton Suppose $A$ and $B$ are sets. Then a set $R \subseteq A \times B$ is called a relation from A to B. Its domain is given by $\text{Dom}(R) = \{ a \in A |\, \exists b \in B((a,b) \in R) \}$ and its range is given by $\text{Ran}(R) = \{ b \in B |\, \exists a \in A((a,b) \in R) \}$.

So in essence the Domain and the Range are just subsets of $A$ and $B$ respectively, or intuitively the values of $A$ and $B$ that "pop up" in the relation. Why not just define relations on these Dom($R$) and Ran($R$) instead of the supersets $A$ and $B$? We could theoretically also take arbitrary supersets of the domain and range. What was the motivation for defining them like this?

• I think there might be a problem with circularity. $\mathrm{Dom}(R)$ has a definition that depends on $R$, so you can't define $R$ in terms of $\mathrm{Dom}(R)$. – fred Oct 8 '15 at 13:50

Let $X$ and $Y$ be such-and-such sets, and let $R$ be any relation from $X$ to $Y$. Then blah blah blah.
and we then want the theorem to be useful whenever $R$ is a subset of $X\times Y$. If you insisted of defining relations only in terms of their narrow domains and ranges, our theorem would need to be
Let $X$ and $Y$ be such-and-such sets, and let $R$ be a relation whose domain is a subset of $X$ and whose range is a subset of $Y$. Then blah blah blah.