I came across an exercise in a book which introduces the circular relation as:
$C$ is a relation from $R \to R$ such that $(x,y) \in C$ means $x^2+y^2 = 1$.
It then says that the domain of $C$ is $R$.
However, as I understand things, the domain can only be the set of all $x$ that satisfy the relation. For example, the number 5 will never be in the domain of $C$, no matter what $y$ we choose. What is going on here? Is the book in error?
This is a more general context than that of functions - a relation need not be "defined" everywhere on the domain. A relation $R$ between sets $X$ and $Y$ (also sometimes phrased as "from $X$ to $Y$") is just a subset $R\subseteq X\times Y$. The set $X$ is, by definition, the domain of $R$. Here's the Wikipedia article.