Antisymmetric relation

A relation $R$ on a set $A$ is called antisymmetric, if $(a, b)$ and $(b, a)$ cannot both be in $R$ unless $a = b$.

(it also does not mean $(a,a)$ should be in $R$)

This is also not opposite to symmetric, instead it means that you can go one way but you cannot come back unless if you’re going from an element to itself.

For example:

• Antisymmetric: $<$, $>$, $leq$, $geq$ on $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ or $\mathbb{R}$.

• Antisymmetric: $R_1=\{(1,1),(1,3),(2,3),(3,3)\}$ on $\{1,2,3\}$ You can go one way but not the other.

  digraph G {
  	1->1
    1->3
    2->3
    3->3
  }

• Antisymmetric (also symmetric): $R_2=\{(1,1),(2,2),(3,3)\}$ on $\{1,2,3\}$ You can go one way but not the other.

  digraph G {
  	1->1
    2->2
    3->3
  }

• Not antisymmetric: $R_3=\{(1,1),(1,2),(2,1)\}$ on $\{1,2\}$ There are bi-directional paths here.

  digraph G {
    1->1
  	1->2
    2->1
  }