Is every relation that induces a partition an equivalence relation?

Question

Let $R$ be a binary relation on a set $X$. I define a right $R$ subset of $X$ to be the set of elements that are $R$-related to some specific element $x$ in $X$. Let $P$ be the set of all right $R$ subsets of $X$. If $P$ is a partition, must $R$ be an equivalence relation?

Answer 1

As @nbritten points out, it needn't be reflexive.

Nor need it be symmetric. Consider on the integers the relation defined by the set of pairs $\{(n, n+1) \mid n \in \mathbb N \}$. Then the right-R-subsets are exactly the singletons, and hence form a partition. And come to think of it, this particular relation is neither symmetric, reflexive, nor transitive.

Thus the idea that $R$ might need to be an equivalence relation might be characterized as "extremely false." :)