Function (mathematics): Difference between revisions

Revision as of 13:52, 14 March 2024

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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.

AGiven ''function'' with ___domainsets {{mathmvar|''X''}} and codomain {{mathmvar|''Y''}}, a ''function'' can be defined to isbe a binary relation {{mvar|R}} between {{math|''X''}} and {{math|''Y''}} that satisfies the two following conditions:

* 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>

* <math>(x,y)\in R</math> and <math>(x,z)\in R</math> imply <math>y=z.</math>

Viewing the relation {{mvar|R}} together with the inclusion into <math>X\times Y</math> allows to speak of the function with ___domain {{math|''X''}} and codomain {{math|''Y''}} or say that it is a function from {{math|''X''}} to {{math|''Y''}}. Some authors do this implicitly, some explicitly, and some view functions as only the relation itself.

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

AWithout referring explicitly to the concept of a relation, but using more notation (including [[set-builder notation]]), a function with ___domain {{math|''X''}} and codomain {{math|''Y''}} 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\subset \{(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>