DEF-EQV. Equivalence Relation.
An equivalence relation is a binary relation that is reflexive, symmetric and transitive.
DEF-EQV-CLS. Equivalence Class.
Let $\sim$ be an equivalence relation on $X$ and $a\in X$. The equivalence class of $a$ with respect to $\sim$ is
\[[a]_\sim = \{x\in X:a\sim x\}.\]If there is no ambiguity, $[a]_\sim$ can be simply written as $[a]$.
Let $\sim$ be an equivalence relation on $X$. The quotient set of $X$ by $\sim$, or $X$ modulo $\sim$, is the set
\[X/{\sim} = \{[x]:x\in X\}.\]
Let $\sim$ be an equivalence relation on $X$ and $x$, $y\in X$. The following statements are equivalent:
- \[x \sim y.\]
- \[[x] = [y].\]
- \[x \in [y].\]
- \[[x]\cap[y] \neq \varnothing.\]