The reason we define quotient sets for equivalence relations only is that each of the properties of being an equivalence relation—reflexivity, symmetry, and transitivity—ensures that the equivalences classes $\webleft [a\webright ]$ of $X$ under $R$ are well-behaved:
- Reflexivity. If $R$ is reflexive, then, for each $a\in X$, we have $a\in \webleft [a\webright ]$.
- Symmetry. The equivalence class $\webleft [a\webright ]$ of an element $a$ of $X$ is defined by
\[ \webleft [a\webright ]\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in X\ \middle |\ x\sim _{R}a\webright\} , \]
but we could equally well define
\[ \webleft [a\webright ]'\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in X\ \middle |\ a\sim _{R}x\webright\} \]instead. This is not a problem when $R$ is symmetric, as we then have $\webleft [a\webright ]=\webleft [a\webright ]'$.[1]
- Transitivity. If $R$ is transitive, then $\webleft [a\webright ]$ and $\webleft [b\webright ]$ are disjoint iff $a\nsim _{R}b$, and equal otherwise.
