# Function

(Redirected from Involution)

A **function** is a correspondence between two sets , where every element of corresponds to exactly one element of . Such a function can be denoted as , and the element that corresponds to is denoted by .

Simple examples of functions are constant functions satisfying for any , and the **identity** function which satisfies for all .

## Glossary[edit | edit source]

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

- If and are two functions, we may define their
**composition**through the equation .

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

- If all values in 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**such that .

- A function where is an
**involution**.

- For a function and a set the
**image**of a function on A , written , is the set of all for a in A .

- For a function and a set the
**preimage**of a function on A , written , is the set of all a such that 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, , is any x such that f (x )=x .

## External links[edit | edit source]

- Wikipedia contributors. "Function".
- Weisstein, Eric W. "Function" at MathWorld.
- nLab contributors. "Function" on nLab.