Let $\mathcal{P}$ be a partition on a set $A$, and define a relation $\sim$ on $A$ by:
$x \sim y \Leftrightarrow$ $x$ and $y$ belong to the same set $P \in \mathcal{P}$.
Check that $\sim$ really is an equivalence relation.
Is this just as straightforward as this?
(i) Reflexivity: $x \sim x \Leftrightarrow$ $x$ and $x$ belong to the same subset $P \in \mathcal{P}$
(ii) Symmetry: if $x$ and $y$ belong to the same subset $P \in \mathcal{P}$, then $y$ and $x$ belong to the same subset $P \in \mathcal{P}$, so if $x \sim y$ then $y \sim x$
(iii) Transitivity: if $x$ and $y$ belong to the same subset $P \in \mathcal{P}$ and $y$ and $z$ belong to the same subset $P \in \mathcal{P}$, then $x$ and $z$ belong to the same subset $P \in \mathcal{P}$. So if $x \sim y$ and $y \sim z$ then $x \sim z$.
It seems very straightforward, but perhaps I have misunderstood the complexity. Also, the definition of "partition" seems a bit more "oral" than written with symbols...
I really hope I have not made a duplicate now! The question is if this "checks out", i.e. a solution verification...