Let $R=\{ (a,b) : \ \mid a-b\mid \ \leq1\ \}$ on $\Bbb Z$
Well I know it's reflexive and symmetric and not anti-symmetric, although I don't see why it's not transitive. if $\mid a-b\mid \leq1 \ \land \mid b-c\mid \leq1 \rightarrow \mid a-c \mid\leq1$. I can also define a set $\{ (1,0),(0,1),(1,1),(2,1),(1,2),(2,2) \}$ which would imply that it is.
You have $(0, 1), (1, 2) \in R$ but do you have $(0, 2) \in R$?