Symmetric relation

A relation $R$ on a set $A$ is called symmetric, if for all elements $a, b \in A$, $(b,a) \in R$ whenever $(a, b) \in R$. In effect, all relations must be mirrored.

For example:

• Symmetric: $R_1=\{(1,1),(1,2),(2,1),(2,3),(3,2),(3,3)\}$ on $\{1,2,3\}$ All pairs of elements have the same properties in both directions.

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

• Symmetric: $R_2=\{(1,1),(2,2),(3,3)\}$ on $\{1,2,3\}$ All pairs of elements have the same properties in both directions.

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

• Symmetric: $\{(x,y)\in \mathbb{N}\times \mathbb{N}|x-y\text{ is divisible by }3\}$ on $\mathbb{N}$

• Not symmetric: $R_3=\{(1,1),(1,2),(2,1),(3,1),(3,3)\}$ on $\{1,2,3\}$ Not all pairs of elements have the same properties in both directions.

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