From my own understanding, a relation is anti-symmetric if it has $(a, b)$ but does not have $(b, a)$ while $(a, a)$ and $(b, b)$ are allowed.
But in my college textbook, the relation $R = \{(1,1), (2,2), (3,3), (2,1), (1,3)\}$ is given as only reflexive and neither transitive nor anti-symmetric. I'm confused now.
Best Answer
A relation $R$ on a set $X$ is antisymmetric, if from $(x,y) \in R$ and $(y,x) \in R$ with $x,y \in X$, follows that $x = y$, which is the case for your set $X$ and relation $R$. It is not transitive, since $(2,1) \in R$ and $(1,3) \in R$ but not $(2,3) \in R$.