Converse Relations

Let get back to general relations from some set $S$ to some set $T$ and define a sort of a 'backwards' relation.

Definition. [Converse Relations] Let $R$ be a relation from set $S$ to set $T$. The converse relation of $R$, denoted by $R^{\leftarrow}$ (

$R^{\leftarrow}$

), contains pairs of the form $(t, s)$ only if $R$ contains the pairs $(s, t)$ (where $t \in T$ and $s \in S$).

Example #1: The converse of the relation $\text{LessThanOrEqualTo}$ ($\le$) is $\text{GreaterThanOrEqualTo}$ ($\ge$) because, for every pair $x \le y$, which exists in $\text{LessThanOrEqualTo}$, the pair $y \ge x$ exists in $\text{GreaterThanOrEqualTo}$.

Example #2: The converse of the relation $\text{EvenToOdd}$ is $\text{OddToEven}$ because, for every pair $(x, y)$ with $x$ being even and $y$ being odd, which exists in $\text{EvenToOdd}$, the pair $(y, x)$ exists in $\text{OddToEven}$.

Example #3: The converse of the relation $\text{Smiles} = \;$$\{(😀, 😄), \,$$(😀, 😆), \,$$(😁, 😄), \,$$(😁, 😆)\}$ is:
$\{(😄, 😀), \,$$(😆, 😀), \,$$(😄, 😁), \,$$(😆, 😁)\}$. (Simply exchange the 1st coordinate with the 2nd coordinate.)