Does a relation on a set that partitions it an equivalence relation?

Question

I know that if a relation is an equivalence relation on a set, it partitions the set. However, I do not know whether the inverse is also true. Namely, if a relation partitions a set, is it an equivalence relation on the set?

Answer 1

Yes, that's true. If $A$ is a set and $\mathcal A=\{A_i~:~i\in I\}$ is a partition of $A$ with an arbitrary index set $I$, then you can define $\sim$ by $$ x\sim y :\Leftrightarrow \exists i\in I : x,y\in A_i. $$ You can check that this defines an equivalence relation.