I was reading one of my algebra courses and one particular exercise required us, amongst other things, to prove that a binary relation is well-defined. I don't know any special requirements for a binary relation to be well-defined, beyond mapping to elements that actually exist in the given set, I guess. Is there something I'm missing?
EDIT: I will add the relation itself to the question. There is given a non-empty set $A$ and a preorder relation $ρ$. An equivalence relation $σ$ is defined on the set $A$ as follows: $aσb\Leftrightarrow a\rho b\wedge b\rho a$. On the set $A/ \sigma$ the relation $\tau $ is defined such that $\bar{a} \tau \bar{b}\Leftrightarrow a \rho b$. The relation $\tau$ is the one in question(whether it is well-defined or not).
A binary relation on a set $A$ can also be viewed as a function $A\times A\to\{0,1\}$.
Now we have defined $\tau$ on $A/\sigma$ using elements of $A$, that is, a priori, the given definition of $\tau$ is only a relation on $A$ (namely, it's exactly $\varrho$ in this example), which can be viewed as a function $A\times A\to\{0,1\}$, and the question is: is this function factoring through (the square of) the quotient $A/\sigma\times A/\sigma$?
That is, you have to prove that ($a\,\sigma\,a',\ b\,\sigma\,b'$ and $a\,\varrho\,b$) implies $a'\,\varrho\,b'$.