---
tag: 0JCSQNNP2M
kind: definition
name: Converse relation
msc: ["03E20"]
defines: 054K3HNN5Z
uses: [08EJC6Q25H]
sources:
  - url: https://en.wikipedia.org/wiki/Converse_relation
    locator: lead section
---

## Definition

Let $R \subseteq A \times B$ be a [[binary relation|08EJC6Q25H]] 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|0SNQEF7Z53]] $f$, the converse of the graph of $f$ is the graph of the [[inverse function|0W766JAPSN]].

## Explanations

### L1 human

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