# Function

(Redirected from Involution)

A function is a correspondence between two sets ${\displaystyle X,Y}$, where every element of ${\displaystyle X}$ corresponds to exactly one element of ${\displaystyle Y}$. Such a function can be denoted as ${\displaystyle f:X\to Y}$, and the element that corresponds to ${\displaystyle x\in X}$ is denoted by ${\displaystyle f(x)}$.

Simple examples of functions are constant functions ${\displaystyle f:X\to Y}$ satisfying ${\displaystyle f(x_{1})=f(x_{2})}$ for any ${\displaystyle x_{1},x_{2}\in X}$, and the identity function ${\displaystyle \iota :X\to X}$ which satisfies ${\displaystyle \iota (x)=x}$ for all ${\displaystyle x\in X}$.

## Glossary

The following are common terms used with or to describe functions:

• If ${\displaystyle f:X\to Y}$ and ${\displaystyle g:Y\to Z}$ are two functions, we may define their composition ${\displaystyle g\circ f:X\to Z}$ through the equation ${\displaystyle (g\circ f)(x)=g(f(x))}$.

• If different inputs give different outputs, the function is said to be injective.

• If all values in ${\displaystyle Y}$ are attained as outputs, the function is said to be surjective.

• A function that is both injective and surjective is called bijective.

• If a function is bijective, then there exists an inverse function ${\displaystyle f^{-1}}$ such that ${\displaystyle f\circ f^{-1}=f^{-1}\circ f=\iota }$.

• A function ${\displaystyle f:X\to X}$ where ${\displaystyle f\circ f=\iota }$ is an involution.

• For a function ${\displaystyle f:X\to Y}$ and a set ${\displaystyle A\subseteq X}$ the image of a function on A , written ${\displaystyle f[A]}$, is the set of all ${\displaystyle f(a)}$ for a  in A .

• For a function ${\displaystyle f:X\to Y}$ and a set ${\displaystyle A\subseteq Y}$ the preimage of a function on A , written ${\displaystyle f^{-1}[A]}$, is the set of all a  such that ${\displaystyle f(a)}$ is in A . Despite what the notation may imply, it is not necessary that f  have an inverse in order to take the preimage of a set.

• A fixed point of a function, ${\displaystyle f:X\to X}$, is any x  such that f (x )=x .