TITLE: Example of a relation that is symmetric and transitive, but not reflexive
QUESTION [16 upvotes]: Can you give an example of a relation that is symmetric and transitive, but not reflexive? 
By definition,

$R$, a relation in a set $X$, is reflexive if and only if $\forall x\in X$, $x\,R\,x$. 
$R$ is symmetric if and only if $\forall x, y\in X$,  $x\,R\,y\implies y\,R\,x$.
$R$ is transitive if and only if $\forall x, y, z\in X$, $x\,R\,y\land y\,R\,z\implies x\,R\,z$. 

I can give a relation $\leqslant$, in a set of real numbers, as an example of reflexive and transitive, but not symmetric. But I can't think of a relation that is symmetric and transitive, but not reflexive.

REPLY [2 votes]: This is called a “partial equivalence relation (PER)”. PERs can be used to simultaneously quotient a set and imbue the quotiented set with a notion of equivalence.
A genuinely useful example (copied straight from the linked page) is functions that respect equivalence relations of the domain and codomain. Suppose we have sets $X$ and $Y$, with (partial or otherwise) equivalence relations $\approx_X$ and $\approx_Y$, respectively. We define the partial equivalence relation $\approx_{X \to Y}$ by
$$
f \approx_{X \to Y} f' := \forall x,x' \in X. x \approx_X x' \implies f(x) \approx_Y f'(x')
$$
Then, $\{~f~|~f \approx_{X \to Y} f~\}$ are exactly the functions that preserve equivalence. Furthermore, $\approx_{X \to Y}$ is an equivalence relation on that set, equating $f$ and $f'$ exactly when they map the same inputs (up to $\approx_X$) to the same outputs (up to $\approx_Y$). This is true even if equality of functions in the underlying set theory is more intensional – hence the use of this technique in type theory.