A binary relation $R$ is symmetric if $(x,y)\in R$ implies $(y,x)\in R$ for all $x,y\in X\text{.}$
A binary relation $R$ on a set $X$ is called an equivalence relation when it is reflexive, symmetric, and transitive. Typically, symbols like, $=\text{,}$ $\cong\text{,}$ $\equiv$ and $\sim$ are used to denote equivalence relations. An equivalence relation, say $\cong\text{,}$ defines a partition on the set $X$ by setting
Note that if $x,y\in X$ and $\langle x\rangle\cap\langle y\rangle \neq\emptyset\text{,}$ then $\langle x\rangle=\langle y\rangle\text{.}$ The sets in this partition are called equivalence classes.
When using the ordered pair notation for binary relations, to indicate that a pair $(x,y)$ is not in the relation, we simply write $(x,y)\notin R\text{.}$ When using the alternate notation, this is usually denoted by using the negation symbol from logic and writing $\lnot (xRy)\text{.}$ Many of the special symbols used to denote equivalence relations come with negative versions: $x\neq y\text{,}$ $x\ncong y\text{,}$ $x\nsim y\text{,}$ etc.