When defining a binary relation $R \subseteq X^2$, does there have to be a definite "true" or "false" value for a pair $(x,y) \in R$, or does it only have to be "true" to be included, and excluded otherwise?
For example, can I define the relation on $\mathbb{Z}$, $(x,y)\in R$ if and only if $x|y$? Even though $(0,y)$ is undefined for any value $y \in \mathbb{Z}$ (except $0$, depending on which text book you read..)
EDIT:
changed a typo from "$(x,0)$ is undefined..." to "$(0,y)$ is undefined..."
Either $x$ divides $y$, or it does not; whether $x\mid y$ (and hence whether $\langle x,y\rangle\in R$) is not undefined for any $x,y\in\Bbb Z$, though whether $\langle 0,0\rangle\in R$ may depend on your conventions. In particular, $\langle n,0\rangle\in R$ for every $n$ except possibly $0$, and $\langle 0,n\rangle\notin R$ for any $n$.
More generally, a binary relation $R$ on $X$ is simply a well-defined subset of $X\times X$. We may not always be able to tell whether some particular pair $\langle x,y\rangle$ is in $R$, but it either is or is not; there is no fuzzy in-between state. For instance, let
$$R=\{\langle n,d\rangle:\text{the }n\text{-th digit of the decimal expansion of }\pi\text{ is }d\}\;.$$
For sufficiently large values of $n$ we don’t know whether $\langle n,1\rangle\in R$, but we do know that this question has a definite answer.
(I suppose that I should mention that there actually is a notion of fuzzy sets with a corresponding notion of degree of membership, but I assume that we are talking here about the usual notion of binary relation.)