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function (Definition)

A function is a triple $ (f,A,B)$ where:

  1. $ A$ is a set (called the domain of the function).
  2. $ B$ is a set (called the codomain of the function).
  3. $ f$ is a relation from $ A$ to $ B$.
  4. For every $ a \in A$, there exists $ b \in B$ such that $ (a,b) \in f$.
  5. If $ a \in A$, $ b_1,b_2 \in B$, and $ (a,b_1) \in f$ and $ (a,b_2) \in f$, then $ b_1 = b_2$.
The triple $ (f,A,B)$ is usually written with the specialized notation $ f\colon A \to B$. This notation visually conveys the fact that $ f$ maps elements of $ A$ into elements of $ B$.

Other standard notations for functions are as follows:

  • For $ a \in A$, one denotes by $ f(a)$ the unique element $ b \in B$ such that $ (a,b) \in f$.
  • The image of $ f$, denoted $ f(A)$, is the set
    $\displaystyle \{b \in B \mid f(a) = b$    for some $\displaystyle a \in A\} $
    consisting of all elements of $ B$ which equal $ f(a)$ for some element $ a \in A$.
  • In cases where the function $ f$ is clear from context, the notation $ a \mapsto b$ is equivalent to the statement $ f(a) = b$.
  • Given two functions $ f\colon A \to B$ and $ g\colon B \to C$, there exists a unique function $ g \circ f\colon A \to C$ satisfying the equation $ g \circ f(a) = g(f(a))$. The function $ g \circ f$ is called the composition of $ f$ and $ g$.
  • When a function $ f\colon A \to A$ has its domain equal to its codomain, one often writes $ f^n$ for the $ n$-fold composition
    $\displaystyle \underbrace{f \circ f \circ \cdots \circ f}_{n\text{ times}} $
    where $ n$ is any natural number. Occasionally this can be confused with ordinary exponentiation (for example the function $ x\mapsto (\sin x)(\sin x)$ is conventionally written as $ \sin^2$); in such cases one usually writes $ f^{[n]}$ to denote the $ n$-fold composition.
There is no universal agreement as to the definition of the range of a function. Some authors define the range of a function to be equal to the codomain, and others define the range of a function to be equal to the image.



"function" is owned by djao. [ full author list (2) ]
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See Also: mapping, injective function, surjective, bijection, relation

Other names:  map
Also defines:  domain, codomain, composition, image, range

Attachments:
mapping (Definition) by rmilson
real function (Definition) by pahio
complex function (Definition) by pahio
argument (Definition) by Wkbj79

Cross-references: universal, natural number, equation, equivalent, clear, relation from
There are 1420 references to this entry.

This is version 11 of function, born on 2001-10-19, modified 2005-06-01.
Object id is 360, canonical name is Function.
Accessed 52632 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
Pending Errata and Addenda
None.
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