involution

Involution

https://en.wikipedia.org/wiki/Involution_(mathematics)

Involutory function is a function that is its own inverse.

An involution, or an involutory function, is a function $f$ that is its own inverse, $f(f(x)) = x$

for all $x$ in the domain of $f$

Equivalently, applying $f$ twice produces the original value.

The term anti-involution refers to involutions based on antihomomorphisms (see Quaternion algebra, groups, semigroups)

$f(xy) = f(y) f(x)$ such that

$xy = f(f(xy)) = f( f(y) f(x) ) = f(f(x)) f(f(y)) = xy$

Involution

https://en.wikipedia.org/wiki/Involution_(mathematics)

An involution, or an involutory function, is a function f that is its own inverse, f(f(x)) = f(x) (for all x in the domain of f). Equivalently, applying f twice produces the original value.

Any involution is a bijection.

Example of an involuntary function: f(x) = -1 * x

In set theory, involution or double complement law states: (Aᶜ)ᶜ = A

The term *anti-involution refers to involutions based on antihomomorphisms (see Quaternion algebra, groups, semigroups) f(xy) = f(x)f(y) such that xy = f(f(xy)) = f( f(x) f(y) ) = f(f(x)) f(f(y)) = xy

Examples of involution:

• identity map is a trivial example of an involution

• multiplication by −1 in arithmetic

• taking of reciprocals

• complementation in set theory

• complex conjugation

• circle inversion

• rotation by a half-turn

• reciprocal ciphers such as the ROT13 transformation