What is involution in mathematics

Involutions

Under one involution one understands a self-inverse mapping.

definition

A mapping f: A → A is called Involution, if for all x∈A the following applies: f (f (x)) = x, so if f∘f = idA applies for f (idA is the identical mapping to A).

Examples

x↦ − x for x∈R
is an involution because - (- x) = x.
x↦x1 for x∈R ∖ {0}
an involution because of 1 / x1 = x for all x = / 0

Theorem C86B (Involutions as Bijections)

proof

Injectivity: f (a) = f (b) ⟹f (f (a)) = f (f (b)) ⟹a = b. Surjectivity: Let a∈A, then f (a) is an archetype of a, because of f (f (a)) = a. □

More examples of involutions

Group inverse

If G is a group, the mapping g↦ − g (with additive notation) or g↦g − 1 (with multiplicative notation) is an involution.

Complex conjugation

Transposing matrices

⋅t: Mat (n × n, K) → Mat (n × n, K)

There is no silver bullet to mathematics.

Euclid

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