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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?

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    $\begingroup$ 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 $\mathrm{Dom}(R)$. $\endgroup$ – fred Oct 8 '15 at 13:50
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The definitions are there to make it easy to say things we want to say. In this particular case we're often interested in stating and using theorems of the general shape

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.

This would mean more words to write, and more words to digest and understand each time such a theorem is stated -- and for very little benefit.

It is simply more useful to have a short term for a relation between two already given sets, even if it may not use all of the sets we have at hand to speak about.

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