Given the relation $$\mathcal{R}=\{(1,1), (1,2), (2,2), (2,3), (3,3), (3,1)\}$$ the problem is to determine whether this relation is reflexive/symmetric/antisymmetric/transitive or not.
I understand why this relation is reflexive, and also why it is not symmetric and transitive.
But the textbook says it is NOT antisymmetric whereas I think it is antisymmetric.
My idea:
Def. of antisymmetry: $x\mathcal{R}y$ and $y\mathcal{R}x \implies x=y$
For $(a,b)$ in $\mathcal{R}$, there are only 3 elements in $\mathcal{R}$ that satisfy the condition "$x\mathcal{R}y$ and $y\mathcal{R}x$"; $(1,1), (2,2), (3,3)$. So the condition "$x=y$" trivially holds for those elements. Also, there is NO counterexample: that is, there is no $(2,1), (3,2)$ and $(1,3)$ in $\mathcal{R}$. So I think it is antisymmetric, but the book says it is NOT antisymmetric and there is no explanation.
Am I wrong? Please give your help!
Your argument is perfectly fine, there must be a mistake in the book. You can also use the equivalent form of antysimmetry: If $ x \mathcal{R} y $ and $ x \neq y $ then $(y,x) \not \in \mathcal{R} $ which is the second part of your argument