Converse Relations

Why

If $x$ is related to $y$, then $y$ is related to $x$, but how?

Definition

If $R$ is a relation between $X$ and $Y$, then the converse or inverse relation of $R$ is a relation on $Y$ and $X$ relating $y \in Y$ to $x \in X$ if and only if $x\,R\,y$. If $R = R^{-1}$ then $R$ is symmetric.

Notation

We denote the converse relation of $R$ by $R^{-1}$.

Example

Let $X$ be the set of people and let $R$ be a relation in $X$. If $R$ is “is a father of”, then $R^{-1}$ is “is a son of”. If $R$ is “is a mother of”, then $R^{-1}$ is “is a daughter of”. If $R$ is “is a brother of”, then $R^{-1}$ is “is a brother of”. The relation “is a brother of” is symmetric.