Example of a relation that is symmetric and transitive, but not reflexive [duplicate]
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.
Solution 1:
Take $X=\{0,1,2\}$ and let the relation be $\{(0,0),(1,1),(0,1),(1,0)\}$
This is not reflexive because $(2,2)$ isn't in the relation.
Addendum: More generally, if we regard the relation $R$ as a subset of $X\times X$, then $R$ can't be reflexive if the projections $\pi_1(R)$ and $\pi_2(R)$ onto the two factors of $X\times X$ aren't both equal to $X$.
Solution 2:
Take a set where no element is in relation with the other ones.
P.S. If $xRy$, then $yRx$ by symmetry, hence $xRx$ by transitivity. In particular, reflexivity holds in all points in relation with something other.