Is a relation induced by a partition always an equivalence relation?

Question

Is a relation induced by a partition always an equivalence relation? I'm having some serious trouble understanding this concept and I was wondering if this is true.

Answer 1

Yes.

1) Reflexivity. Clearly all elements are in their own group of the partition, so we have $aRa$ for all $a$.

2) Symmetry. If $aRb$ then $a$ and $b$ are in the same group of the partition, so $b$ and $a$ are in the same group, so $bRa$.

3) Transitivity. If $aRb$ and $bRc$ then $a,b,c$ are all in the same group, so $aRc$.

Also note that the converse is true: any equivalence relation can be considered as a partition of some set. So given an equivalence relation on the set $S$ we can construct the partition $$\{[x]:x\in S\}$$ where $[x]$ denotes the equivalence class of $x$. I will leave the proof to the reader!