Given a partition $\mathcal{P}$ on a set S, show how to define a relation $\sim$ on $S$ such that $\mathcal{P}$ is the corresponding partition.

Question

Given a partition $\mathcal{P}$ on a set S, show how to define a relation $\sim$ on $S$ such that $\mathcal{P}$ is the corresponding partition.

So far, all I have down is:

Let $X$ be a subset of partition $\mathcal{P}$. Then $$a \sim b \iff a \in X \ \wedge \ b \in X $$ defines an equivalence relation on S such that $\mathcal{P}_\sim = \mathcal{P}$.

I understand my next step is to show $\mathcal{P}_\sim \subseteq \mathcal{P}$ and $\mathcal{P} \subseteq \mathcal{P}_\sim$, but I'm unsure how.

Could someone nudge me in the right direction?

No complete answers please!

Answer 1

hint

Let $[a]$ be the equivalence class of the element $a \in X$ with regards to the new relation $P_{\sim}$ you have created. This consists of all elements $b$ such that both $a,b \in X$. This shows $[a] \subseteq X$. Now show the other containment.