An involution is a function which, when applied twice, brings one back to the starting point.

In mathematics, an involution, or an involutary function, is a function that is its own inverse, so that

f(f(x)) = x for all x in the domain of f. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined

General properties

Any involution is a bijection. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

The identity map is a trivial example of an involution. In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that Common examples in mathematics of more interesting involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part.

Other examples include circle inversion, the ROT13 transformation, and the Beaufort polyalphabetic cipher. In Geometry, inversive geometry is the study of a type of transformations of the Euclidean plane, called inversions. ROT13 (" rotate by 13 places " sometimes hyphenated ROT-13) is a simple Substitution cipher used in Online forums as a means of The Beaufort cipher, created by Sir Francis Beaufort, is a Substitution cipher that is similar to the Vigenère cipher but uses a slightly modified A polyalphabetic cipher is any cipher based on substitution, using multiple substitution alphabets

Involutions in Euclidean geometry

A simple example of an involution of the three-dimensional Euclidean space is reflection against a plane. In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. Doing a reflection twice, brings us back where we started.

This transformation is a particular case of an affine involution. In Euclidean geometry, of special interest are Involutions which are linear or Affine transformations over the Euclidean space R

Involutions in linear algebra

In linear algebra, an involution is a linear operator T such that T² = I. Except for in characteristic 2, such operators are diagonalizable with 1's and -1's on the diagonal. If the operator is orthogonal (an orthogonal involution), it is orthonormally diagonalizable.

Involutions are related to idempotents; if 2 is invertible, (in a field of characteristic other than 2), then they are equivalent. Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation

Involutions in ring theory

In ring theory, the word involution is customarily taken to mean an antihomomorphism that is its own inverse function. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In Mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication Examples include complex conjugation and the transpose of a matrix. In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a

See also star-algebra. -ring In Mathematics, a *-ring is an Associative ring with a map *: A &rarr A which is an Antiautomorphism

Involutions in group theory

In group theory, an element of a group is an involution if it has order 2; i. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is e. an involution is an element a such that a ≠ e and a² = e, where e is the identity element. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that Originally, this definition differed not at all from the first definition above, since members of groups were always bijections from a set into itself, i. e. , group was taken to mean permutation group. In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation By the end of the 19th century, group was defined more broadly, and accordingly so was involution. The group of bijections generated by an involution through composition, is isomorphic with cyclic group C₂. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an

A permutation is an involution precisely if it can be written as a product of one or more non-overlapping transpositions. In several fields of Mathematics the term permutation is used with different but closely related meanings In informal language a transposition is a function that swaps two elements of a set

The involutions of a group have a large impact on the group's structure. The study of involutions was instrumental in the classification of finite simple groups. The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups.

Coxeter groups are groups generated by involutions with the relations determined only by relations given for pairs of the generating involutions. In Mathematics, a Coxeter group, named after HSM Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries Coxeter groups can be used, among other things, to describe the possible regular polyhedra and their generalizations to higher dimensions. In Geometry, a Platonic solid is a convex Regular polyhedron. In Mathematics, a regular polytope is a Polytope whose Symmetry is transitive on its flags, thus giving it the highest degree of symmetry

Involutions in mathematical logic

The operation of complement in Boolean algebras is an involution. In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. Accordingly, negation in classical logic satisfies the law of double negation: ¬¬A is equivalent to A. In Logic and Mathematics, negation or not is an operation on Logical values for example the logical value of a Proposition

Generally in non-classical logics, negation which satisfies the law of double negation is called involutive. In algebraic semantics, such a negation is realized as an involution on the algebra of truth values. In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true Examples of logics which have involutive negation are, e. g. , Kleene and Bochvar three-valued logics, Łukasiewicz many-valued logic, fuzzy logic IMTL, etc. A ternary, three-valued or trivalent logic (sometimes abbreviated 3VL) is a term to describe any of several Multi-valued logic systems in which In Mathematics, Łukasiewicz logic is a non-classical, many valued logic Fuzzy logic is a form of Multi-valued logic derived from Fuzzy set theory to deal with Reasoning that is approximate rather than precise Involutive negation is sometimes added as an additional connective to logics with non-involutive negation; this is usual e. g. in t-norm fuzzy logics. T-norm fuzzy logics are a family of Non-classical logics informally delimited by having a semantics which takes the real unit interval for the system of truth values and functions

The involutiveness of negation is an important characterization property for logics and the corresponding varieties of algebras. In Universal algebra, a variety of algebras is the class of all Algebraic structures of a given signature satisfying a given set of identities For instance, involutive negation characterizes Boolean algebras among Heyting algebras. In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. In Mathematics, Heyting algebras are special Partially ordered sets that constitute a generalization of Boolean algebras named after Arend Heyting Correspondingly, classical Boolean logic arises by adding the law of double negation to intuitionistic logic. Classical logic identifies a class of Formal logics that have been most intensively studied and most widely used Intuitionistic logic, or constructivist logic, is the Symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwer The same relationship holds also between MV-algebras and BL-algebras (and so correspondingly between Łukasiewicz logic and fuzzy logic BL), IMTL and MTL, and other pairs of important varieties of algebras (resp. In Abstract algebra, a branch of pure Mathematics, an MV-algebra is an Algebraic structure with a Binary operation \oplus a In Mathematics, Łukasiewicz logic is a non-classical, many valued logic Basic fuzzy Logic (or shortly BL) the logic of continuous T-norms is one of T-norm fuzzy logics It belongs to the broader class of Substructural Monoidal t-norm based logic (or shortly MTL) the logic of left-continuous T-norms is one of T-norm fuzzy logics It belongs to the broader class of corresponding logics).

Count of involutions

The number of involutions on a set with n = 0, 1, 2, . . . elements is given by the recurrence relation:

a(0) = a(1) = 1;

a(n) = a(n − 1) + (n − 1) × a(n − 2), for n > 1. "Difference equation" redirects here It should not be confused with a Differential equation.

The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 (sequence A000085 in OEIS). Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In mathematics Two has many properties in Mathematics. An Integer is called Even if it is divisible by 2 In mathematics Four is the smallest Composite number, its proper Divisors being and. 26 ( twenty-six) is the Natural number following 25 and preceding 27. 76 ( seventy-six) is the Natural number following 75 and preceding 77. 230 ( two hundred Thirty) is the Natural number following 229 and preceding 231 The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

See also

• Automorphism