203.13.3.90: It's not actually trivial

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It's not actually trivial

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{{Short description|Concept in mathematics}}
{{More citations needed|date=February 2024}}

In [[mathematics]], '''function application''' is the act of applying a [[function (mathematics)|function]] to an argument from its [[domain of a function|domain]] so as to obtain the corresponding value from its [[range of a function|range]].<ref>{{Citation |last1=Alama |first1=Jesse |title=The Lambda Calculus |date=2023 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/win2023/entries/lambda-calculus/ |access-date=2024-02-29 |edition=Winter 2023 |publisher=Metaphysics Research Lab, Stanford University |last2=Korbmacher |first2=Johannes |editor2-last=Nodelman |editor2-first=Uri}}</ref> In this sense, function application can be thought of as the opposite of function [[Abstraction (mathematics)|abstraction]].

==Representation==
Function application is usually depicted by juxtaposing the variable representing the function with its argument encompassed in [[parentheses]]. For example, the following expression represents the application of the function ''ƒ'' to its argument ''x''.

:<math>f(x) </math>

In some instances, a different notation is used where the parentheses aren't required, and function application can be expressed just by [[juxtaposition]]. For example, the following expression can be considered the same as the previous one:

:<math>f\; x</math>

The latter notation is especially useful in combination with the [[currying]] isomorphism. Given a function <math>f : (X \times Y) \to Z</math>, its application is represented as <math>f(x, y)</math> by the former notation and <math>f\;(x,y)</math> (or <math>f \; \langle x, y \rangle</math> with the argument <math>\langle x, y \rangle \in X \times Y</math> written with the less common angle brackets) by the latter. However, functions in curried form <math>f : X \to (Y \to Z)</math> can be represented by juxtaposing their arguments: <math>f\; x \; y</math>, rather than <math>f(x)(y)</math>. This relies on function application being [[left-associative]].

{{unichar|2061|FUNCTION APPLICATION|html=}} — a contiguity operator indicating application of a function; that is an invisible zero width character intended to distinguish concatenation meaning function application from concatenation meaning multiplication.

== Set theory ==
In [[axiomatic set theory]], especially [[Zermelo–Fraenkel set theory]], a function <math>f: X \mapsto Y</math> is often defined as a [[Relation (mathematics)|relation]] (<math>f \subseteq X \times Y</math>) having the property that, for any <math>x \in X</math> [[there is a unique]] <math>y \in Y</math> such that <math>(x,y) \in f</math>.

One is usually not content to write "<math>(x,y) \in f</math>" to specify that <math>y</math>, and usually wishes for the more common function notation "<math>f(x) = y
</math>", thus function application, or more specifically, the notation "<math>f(x)</math>", is defined by an [[Axiom schema|axiom schema.]] Given any function <math>f</math> with a given [[Domain of a function|domain]] <math>X</math> and [[codomain]] <math>Y</math>:<ref>{{Cite book |last=Suppes |first=Patrick |author-link=Patrick Suppes |url=https://archive.org/details/axiomaticsettheo00supp_0/ |title=Axiomatic set theory |date=1972 |publisher=New York, Dover Publications |others=Internet Archive |isbn=978-0-486-61630-8 |pages=87}}</ref><ref>{{Cite book |last=Mendelson |first=Elliott |url=https://archive.org/details/introductiontoma0000mend/page/82/ |title=Introduction to mathematical logic |date=1964 |publisher=Princeton, N.J., Van Nostrand |others=Internet Archive |isbn=978-0-442-05300-0 |pages=82}}</ref>

<math>\forall x \in X, \forall y \in Y (f(x) = y \iff </math><math>\exists ! z\in Y((x,z) \in f) \, \and \, (x,y) \in f )</math>

Stating "For all <math>x</math> in <math>X</math> and <math>y</math> in <math>Y</math>, <math>f(x)</math> is equal to <math>y</math> [[if and only if]] there is a unique <math>z</math> in <math>Y</math> such that <math>(x,z)</math> is in <math>f</math> and <math>(x,y)</math> is in <math>f</math>". The notation <math>f(x)</math> here being defined is a new [[functional predicate]] from the underlying logic, where each y is a [[Term (logic)|term]] in x.<ref>{{Cite book |last=Lévy |first=Azriel |url=https://archive.org/details/basicsettheory00levy_0/mode/2up |title=Basic set theory |date=1979 |publisher=Springer-Verlag |isbn=978-0-387-08417-6 |location=Berlin; New York |pages=27}}</ref> Since <math>f</math>, as a functional predicate, must map every object in the language, objects not in the specified domain are chosen to map to an arbitrary object, suct as the [[empty set]].<ref>{{Cite book |last=Lévy |first=Azriel |url=https://archive.org/details/basicsettheory00levy_0/mode/2up |title=Basic set theory |date=1979 |publisher=Springer-Verlag |isbn=978-0-387-08417-6 |location=Berlin; New York |pages=15}}</ref>

==As an operator==

{{main|Apply}}

Function application can be defined as an [[Operator (mathematics)|operator]], called [[apply]] or <math>\$</math>, by the following definition:

:<math>f \mathop{\,\$\,} x = f(x)</math>

The operator may also be denoted by a [[backtick]] (`).

If the operator is understood to be of [[Order of operations|low precedence]] and [[right-associative]], the application operator can be used to cut down on the number of parentheses needed in an expression. For example;

:<math>f(g(h(j(x)))) </math>

can be rewritten as:

:<math>f \mathop{\,\$\,} g \mathop{\,\$\,} h \mathop{\,\$\,} j \mathop{\,\$\,} x</math>

However, this is perhaps more clearly expressed by using [[function composition]] instead:

:<math>(f \circ g \circ h \circ j)(x)</math>

or even:

:<math>(f \circ g \circ h \circ j \circ x)()</math>

if one considers <math>x</math> to be a [[constant function]] returning <math>x</math>.

==Other instances==
Function application in the [[lambda calculus]] is expressed by [[β-reduction]].

The [[Curry–Howard correspondence]] relates function application to the logical rule of [[modus ponens]].

==See also==
* [[Polish notation]]

==References==
{{Reflist}}

[[Category:Functions and mappings]]

{{mathanalysis-stub}}

Function application - Revision history

203.13.3.90: It's not actually trivial