# Whether the relation $R = \{(1,1), (2,2), (3,3), (2,1), (1,3)\}$ is anti-symmetric or not?

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.

• Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please edit the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. Commented Jun 29, 2023 at 4:09
• You are right: your book is wrong indeed. Commented Jun 29, 2023 at 5:26

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 $$(2,3) \in R$$.