Counter-symmetric binary relations

Question

In the lecture pdf our teacher sent us, some of binary relation properties are described. I understood all of properties except counter-symmetric. Here is the definition of it from the pdf: $$\forall x, y \in X: xRy \Rightarrow \neg(yRx)$$ I understand the antisymmetric relations - if there is $(a,b)$ then $(b,a)$ can't be in the relation set, unless $a=b$. That given, I have no idea what the logic behind counter-symmetry can be.

Could someone please explain it with some examples?

Answer 1

In my experience this property is usually called asymmetry of $R$.

Given a relation $R$ on a set $X$ and distinct elements $x,y\in X$, four things are possible:

• $x\mathrel{R}y$ and $y\mathrel{R}x$;
• $x\mathrel{R}y$ and $y\not\mathrel{R}x$;
• $x\not\mathrel{R}y$ and $y\mathrel{R}x$; or
• $x\not\mathrel{R}y$ and $y\not\mathrel{R}x$.

$R$ is antisymmetric the first is true only when $x=y$. $R$ is asymmetric (or counter-symmetric) if the first is never true. (Note that in particular this implies that $R$ is irreflexive: there is no $x\in X$ such that $x\mathrel{R}x$.)