If $\mathscr{R}$ is an [[Equivalence Relation]] on a [[[‡ Set|set]] $A$, then every $a \in A$ is in exactly one [[Equivalence Class]]. In particular, $a \mathscr{R} b$ if and only $[a]=[b]$. ### Proof Every $a\in [[Reflexive Property|a]] If there is another element $b\in [[Symmetric Property|b]] Since $\forall (a \in [a]) \exists(b\in[b]:a\mathscr{R}b)$ and $\forall (b \in [b]) \exists(a\in[a]:b\mathscr{R}a)$, $[a]$ and $[b]$ are one-and-the-same equivalence class; $[a]=[b]$