---
tag: GP56NBN9RY
kind: concept
name: Involution
msc: [08A02, 16W10]
sources:
  - url: https://en.wikipedia.org/wiki/Involution_(mathematics)
same_as:
  wikidata: Q846862
  wikipedia: Involution (mathematics)
---

## Summary

An involution is a function from a set to itself that undoes itself: applying it twice gives back the starting point.
Equivalently, it is a bijection that is its own [[inverse|0J2WAR83WP]].
Negation of numbers, [[complex conjugation|0FTH2E9TAS]], the [[complement|0RKF1ZT5JJ]] of a set and the inverse map of a [[group|G5S1M9]] are involutions.
In algebra the word is also used for an involution that reverses a product, $(xy)^* = y^* x^*$, as in rings with involution.

## Explanations

### L1 human

An involution is an action that, done twice, puts everything back.
Reflecting a picture in a mirror twice gives the original picture.
Changing the sign of a number twice gives the original number: $-(-5) = 5$.
Adding $1$ is not an involution, because doing it twice adds $2$.