How to show that symmetry holds for equivalence classes of an equivalence relation $R$ on a set $A$?

Question

Given that equivalence classes of an equivalence relation form a partition of a set $A$, how does one show that symmetric holds? If two equivalence class $[a]$ and $[b]$ of an equivalence relation are disjoint and since equivalence relations are reflexive, $a\in[a]$ and $b\in[b]$, which means that $a$ cannot be in $[b]$ and vice versa. However, since equivalence relations are also symmetric, $aRb$ implies $bRa$, which to me implies that $b$ should be in $[a]$ and vice versa? Is there a way (by example) to show that symmetry is satisfied despite the equivalence classes being pairwise disjoint?

Answer 1

If $C_i, i \in I$ form a partition of a set $X$ (so $\cup_{i \in I} C_i=X$ and $\forall_{i,j \in I}: i \neq j \implies C_i \cap C_j= \emptyset$), defining $xRy$ iff $\exists_{i \in I}: (x \in R_i \land y \in R_i)$ then $R$ is symmetric because "and" is symmetric, essentially: $x \in R_i \land y \in R_i \iff y \in R_i \land x \in R_i$. It turns out that $R$ is indeed an equivalence relation and the class of $x$ is the unique $C_i$ it is in. So if you start with a set of classes the induced relation $R$ is very easily symmetric.