For an example relation $R$ on $\{1,2\}$ of $\{(1,1), (2,2)\}$.
This relation is reflexive, but is it also symmetric and transitive?
It appears to be symmetric because $(1,1)$ is present as is $(1,1)$, and $(2,2)$ is present as is $(2,2)$, in other words, if $x = 1$ and $y = 1$, then $(x,y)$ is present and $(y,x)$ is present.
Following from that, it also appears to be transitive because for $x = 1, y = 1$ and $z = 1$, $(x,y)$ is present, $(y,z)$ is present, and so is $(x,z)$.
Is there some restriction on symmetric and transitive relations in that the elements of the ordered pair cannot be equal?