Editing Function (mathematics) (section)

=== Formal definition ===

[[file:Injection keine Injektion 2a.svg|thumb|Diagram of a function]]
[[file:Injection keine Injektion 1.svg|thumb|Diagram of a relation that is not a function. One reason is that 2 is the first element in more than one ordered pair.  Another reason is that neither 3 nor 4 are the first element (input) of any ordered pair.]]

The above definition of a function is essentially that of the founders of [[calculus]], [[Leibniz]], [[Isaac Newton|Newton]] and [[Euler]]. However, it cannot be [[formal proof|formalized]], since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of [[set theory]]. This set-theoretic definition is based on the fact that a function establishes a ''relation'' between the elements of the domain and some (possibly all) elements of the codomain. Mathematically, a [[binary relation]] between two sets {{math|''X''}} and {{math|''Y''}} is a [[subset]] of the set of all [[ordered pair]]s <math>(x, y)</math> such that <math>x\in X</math> and <math>y\in Y.</math> The set of all these pairs is called the [[Cartesian product]] of {{math|''X''}} and {{math|''Y''}} and denoted <math>X\times Y.</math> Thus, the above definition may be formalized as follows.

A ''function'' with domain {{math|''X''}} and codomain {{math|''Y''}} is a binary relation {{mvar|R}} between {{math|''X''}} and {{math|''Y''}} that satisfies the two following conditions:<ref>{{cite book | last=Halmos | first=Paul R. | title=Naive Set Theory | publisher=Springer | year=1974 | pages=30–33}}</ref>
* For every <math>x</math> in <math>X</math> there exists <math>y</math> in <math>Y</math> such that <math>(x,y)\in R.</math>
* If <math>(x,y)\in R</math> and <math>(x,z)\in R,</math> then <math>y=z.</math>

This definition may be rewritten more formally, without referring explicitly to the concept of a relation, but using more notation (including [[set-builder notation]]):

A function is formed by three sets, the ''domain'' <math>X,</math> the ''codomain'' <math>Y,</math> and the ''graph'' <math>R</math> that satisfy the three following conditions. 
*<math>R \subseteq \{(x,y) \mid x\in X, y\in Y\}</math>
*<math>\forall x\in X, \exists y\in Y, \left(x, y\right) \in R \qquad</math>
*<math>(x,y)\in R \land (x,z)\in R \implies y=z\qquad</math>