Given an equivalence relation on B, construct a partition on B

Question

By definiton We know that an equivalence relation has to be: reflexive, symmetric and transitive. But, how can I construct a partition of B (set)?

Answer 1

The equivalence classes of an equivalence relation on a set partition the set.

The proof goes something like this:

Define $[x]=\{y\mid y\equiv x\}$.

Let $x\in B$. Then $x\in [x]$, by the reflexive property. Thus every $x\in B$ is in an equivalence class.

If $\exists z\in [x]\cap [y]$, then by the transitive and symmetric properties $y\equiv x$, since $z\equiv x\land z\equiv y$. Thus $[x]\cap [y]\neq\emptyset\implies [x]=[y]$. Thus if two equivalence classes intersect, they are equal.