Converse relation

Definition

Let $R \subseteq A \times B$ be a binary relation from $A$ to $B$. The converse of $R$ is the relation from $B$ to $A$ $$R^{-1} = \{(b, a) \in B \times A \mid (a, b) \in R\}.$$ So $b \mathrel{R^{-1}} a$ if and only if $a \mathrel{R} b$.

Remarks

Other names are the inverse relation and the transpose. Another notation is $R^{T}$. The converse of $<$ on $\mathbb{R}$ is $>$. The converse of the converse of $R$ is $R$. For a bijection $f$, the converse of the graph of $f$ is the graph of the inverse function.

Explanations

L1 human

To get the converse, read each pair backwards. The converse of "is a child of" is "is a parent of".