# Function

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− | Mathematically speaking, a [http://mathworld.wolfram.com/Function.html function] relates all values in a set <math>A</math> to values in a set <math>B</math>. The function <math>square\ x = x^2</math>, given that <math>x</math> is an integer, will map all elements of the set of integers into another set -- in this case the set of square integers. In Haskell functions can be specified as below in the examples, with an optional [[Type specification|type specification] that gives the compiler (and other programmers) a hint as to the use of the function. |
+ | Mathematically speaking, a [http://mathworld.wolfram.com/Function.html function] relates all values in a set <math>A</math> to values in a set <math>B</math>. The function <math>square\ x = x^2</math>, given that <math>x</math> is an integer, will map all elements of the set of integers into another set -- in this case the set of square integers. In Haskell functions can be specified as below in the examples, with an optional [[type specification]] that gives the compiler (and other programmers) a hint as to the use of the function. |

==Examples== |
==Examples== |

## Revision as of 00:16, 19 October 2006

Mathematically speaking, a function relates all values in a set *A* to values in a set *B*. The function , given that *x* is an integer, will map all elements of the set of integers into another set -- in this case the set of square integers. In Haskell functions can be specified as below in the examples, with an optional type specification that gives the compiler (and other programmers) a hint as to the use of the function.

## Contents |

## 1 Examples

square :: Int -> Int square x = x * x

In other words, a function has *input* and *output*, and it describes how to produce the output from its input. Functions can be *applied*, which just means that you give an input value as argument to the function and can then expect to receive the corresponding output value.

Haskell functions are *first class* entities, which means that they

- can be given names
- can be the value of some expression
- can be members of a list
- can be elements of a tuple
- can be passed as parameters to a function
- can be returned from a function as a result

(quoted from Davie's *Introduction to Functional Programming Systems using Haskell.*)

### 1.1 map example

As an example of the power of first-class functions, consider the function *map*:

map :: (a -> b) -> [a] -> [b] map f xs = [f x | x <- xs]

(Note this is a Higher order function.)

This function takes two arguments: a function*f*which maps

*a*s to

*b*s, and a list

*xs*of

*a*s. It returns a list of

*b*s which are the results of applying

*f*to every member of

*xs*. So

*b*s that

*map*returns can itself be a list of functions, things start to get interesting. Suppose you have some data structure (e.g.

*Set*) that has the function

*mySet*and

*myList,*a set and a list of values to be added to the set, respectively. One could write a function to recurse over the list of integers, each time inserting a single member of

*myList,*but with first-class functions this is not necessary. Look at the expression

*insert*takes an

*Int*and a

*Set*, but only

*Int*s were given, the resulting list will be of functions that take a set and return a set. Conceptually, the code

### 1.2 Composition / folding example

Haskell supports a Function composition operator:

(.) :: (b -> c) -> (a ->b) -> (a->c) (f . g) x = f (g x)

*folding.*Several variants of the

*fold*function are defined, but the basic concept is the same: given a function and a list, "collapse" the list by applying the function "between" its elements. This is easiest to see with simple binary operators, but it is not limited to them. For example,

*myList.*What is the type of this expression? It is

*myList*inserted. To complete the example,

newSet = (foldr1 (.) (map insert myList)) mySet

will define *newSet* as *mySet* with the elements of *myList* inserted.

## 2 See also

- The Functions section in the Gentle Introduction.