In mathematics, there are two ways to present relation between two objects. One is through the use of predicate symbol. For instance, in set theory, we introduce predicate =:
\begin{equation}
\forall x\ \forall y\ x = y \leftrightarrow \left(\forall z\ z \in x \leftrightarrow z \in y\right).
\end{equation}
On the other hand, we have equivalence relation as sets: a set $R$ is an equivalence relation on $X$ if (1) for any $x \in X$ $\left(x,x\right) \in R$; (2) for any $x, y \in X$, $\left(x,y\right) \in R$ leads to $\left(y,x\right) \in R$; (3) for any $x, y, z \in X$, $\left(x,y\right) \in R$ and $\left(y,z\right) \in R$ leads to $\left(x,z\right) \in R$.
We may define the set-based relation from the predicate =. For instance, suppose $X$ is a set, then the set $=_{X}$ defined by $=_{X} = \left\{\left(x,y\right)\vert x \in X, y \in X, x = y\right\}$ is an equivalence relation. Of course, it doesn't hurt to mean either when we write $x = y$ in math when $x, y \in X$. It could mean $=(x,y)$, or $\left(x,y\right) \in =_{X}$. Honestly, non-math major people may choose the first understanding. But I am still wondering, which one is mathematician's way of thinking when writing relations, set or predicate?
