involution
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Involutory function is a function that is its own inverse.
An involution, or an involutory function, is a function that is its own inverse,
for all in the domain of
Equivalently, applying twice produces the original value.
The term anti-involution refers to involutions based on antihomomorphisms (see Quaternion algebra, groups, semigroups)
such that
)
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