I am reviewing some elementary abstract algebra and set theory, and I am just wondering the following:
If $R$ is an equivalence relation on a set $X$, then is $X$ considered the field of $R$? Field is defined in the textbook as: the union of the domain and the range of $R$.
It's not just "considered" the field of $R$, you can prove it!
To say that $R$ is a relation on $X$ is to say that $R\subseteq X\times X$. So if $(x,y)\in R$, then $x\in X$ and $y\in X$. This shows that $\text{field}(R)\subseteq X$.
Conversely, if $x\in X$, then $(x,x)\in R$, so $x$ is in the field of $R$. This shows that $X\subseteq \text{field}(R)$.
The only thing we used about equivalence relations here is reflexivity.