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.