Functio

[thumb|[graphum (mathematica)|Charta functionis exemplaris, \begin{align}&\scriptstyle  \\ &\textstyle f(x) = \frac{(4x^3-6x^2+1)\sqrt{x+1}}{3-x}\end{align}  ]]
**Functio** in arte mathematica est congruentia inter duas copias, quae ad quodque elementum primae copiae unum elementum secundae copiae destinat. Prima copia *dominium* dicitur,  altera *codominium.* Si x quoddam elementum primae copiae designat, x variabilis independens est. Si f functio est, quae mathematice per y = f(x) scribitur et significat "y esse elementum codominii ad elementum x dominii destinatum", variabilis y dependens appellatur.

Si plura elementa sunt, quae y ad elementum x destinari possint, congruentia *non* est
functio. Exempli gratia: sit f(x) = \pm \sqrt{x} sintque dominium et codominium copiae
numerorum realium \mathbb{R}. Haec congruentia non est functio, quod elemento x (velut 4) duo elementa (i. e. 2, -2) attribuuntur. Sin autem codominium est copia numerorum realium non-negativorum vel functio est velut f(x) = + \sqrt{x}, haec congruentia functio appellatur.

Functionem definire licet per formulam aut regulam aut tabulam, dum modo unum elementum codominii sit quod ad quodque elementum dominii destinetur.  Functiones sunt species relationis:  functio f est copia parium ordinatarum (a, b) ut f(a) = b.

Analysis est theoria functionum. Analysis numerorum realium est theoria functionum quarum dominium (et codominium) est \mathbb{R}; analysis numerorum complexorum est analysis earum, quarum dominium est \mathbb{C}. G. H. Hardy dicit, "Illa notio, quantitatem variabilis ex quadam alia dependere, fortasse notio potissima in tota arte mathematica est."

Si dominium est copia quantitatum binarum, sicut \mathbb{R}^2, functio duas variabiles independentes habet. Exempli gratia, f(x, y) = x^2 + y^2. Quae functio f par elementorum (x, y) ad unum elementum codominii (quod est \mathbb{R}) destinat, velut par (2, 3) ad 2^2 + 3^2 = 4 + 9 = 13. Functiones autem tres, quattuor pluresve variabiles independentes habere possunt.

Altera notatio functionum est *notatio lambda,* quae variabiles independentes post lambda litteram enumerat. Scribitur: f = \lambda(x).x^2 quod eandem functionem atque f(x) = x^2 describit. Forma sicut \lambda(x).x^2 appellatur combinatoria.

Si ad quoddam elementum y codominii aut nullum aut unum elementum x dominii destinatur, functio appellatur functio iniectiva, aut functio unum elementum uni elemento attribuens. Si omne elementum y codominii habet elementum x (aut plura elementa x_1, x_2, x_3, ...) dominii quod ad y destinatur, functio appellatur functio superiectiva. Functio quae simul iniectiva et superiectiva est, functio biiectiva appellatur.

Quaedam functio biiectiva f habet functionem inversam f^{-1}, cuius dominium est codominium functionis f cuiusque codominium dominium functionis f. Si f(x) = y, tum est f^{-1}(y) = x. Exempli gratia: functio f(x) = x/2 habet functionem inversam f^{-1}(x) = 2x. Formulam functionis inversae describere saepenumero haud facile est.

*Compositio* functionum est nova functio per quam elementum dominii primae functionis cum elemento codominii secundae functionis congruit. Si y = f(x), y = g(x) sunt functiones, et si dominium functionis f est (aut continet) codominium functionis g, scribi potest f \circ g = f(g(x)). Exempli gratia, sint f(x) = x^2, g(x) = \sin(x). Nunc f \circ g = f(g(x)) = (\sin(x))^2, et g \circ f = g(f(x)) = \sin(x^2). Non sunt eaedem functiones: si x = \pi, f(g(x)) = (\sin(\pi))^2 = 0, sed g(f(x)) = \sin(\pi^2) \approx -0.43.

Copia omnium functionum invertibilium quarum dominium et codominium eadem copia est caterva appellatur. Idem factor catervae est functio quae quodque elementum cum eodum elemento coniungit: f(x) = x; operatio catervae est compositio.

## Notae 
*Functio biiectiva
*Functio differentiabilis
*Functio exponentialis
*Functio linearis
*Graphum (mathematica)
*Polynomium
*Programmatura functionalis
*Signum (functio)

## Bibliographia 
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* Behnke, H., F. Bachmann, K. Fladt, et W. Süss, eds. 1974. *Fundamentals of Mathematics,* vol 1: *Foundations of Mathematics: The Real Number System and Algebra.* Convertit S. H. Gould.  Cantabrigiae Massachusettae: MIT Press.
* Hardy, G. H. 1952. *A Course of Pure Mathematics.* Editio 10a. Cantabrigiae: Cambridge University Press.
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* Halmos, Paul R. 1970. *Naive Set Theory*, Springer-Verlag, New York, ISBN 0-387-90092-6.
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* Reichenbach, Hans. 1947. *Elements of Symbolic Logic*, Dover Publishing Inc., New York NY, ISBN 0-486-24004-5.
* Russell, Bertrand. 1903. *The Principles of Mathematics: Vol. 1*, Cambridge at the University Press, Cambridge, UK, republished as a googlebook.
* Russell, Bertrand. 1920. *Introduction to Mathematical Philosophy* (second edition), Dover Publishing Inc., New York NY, ISBN 0-486-27724-0 (pbk).
* Suppes, Patrick. 1960. *Axiomatic Set Theory*, Dover Publications, Inc, New York NY, ISBN 0-486-61630-4. cf his *Chapter 1 Introduction*.
* Tarski, Alfred. 1946/ *Introduction to Logic and to the Methodolgy of Deductive Sciences*, republished 1195 by Dover Publications, Inc., New York, NY ISBN 0-486-28462-X
* Venn, John. 1881. *Symbolic Logic*, Macmillian and Co., London UK. Republished as a googlebook.
* van Heijenoort, Jean (1967, 3rd printing 1976), *From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931*, Harvard University Press, Cambridge, MA, ISBN 0-674-32449-8 (pbk)
** Gottlob Frege (1879) *Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought* with commentary by van Heijenoort, pages 1–82
** Giuseppe Peano (1889) *The principles of arithmetic, presented by a new method* with commentary by van Heijenoort, pages 83–97
** Bertrand Russell (1902) *Letter to Frege'' with commentary by van Heijenoort, pages 124–125. Wherein Russell announces his discovery of a "paradox" in Frege's work.
** Gottlob Frege (1902) *Letter to Russell* with commentary by van Heijenoort, pages 126–128.
** David Hilbert (1904) *On the foundations of logic and arithmetic*, with commentary by van Heijenoort, pages 129–138.
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** Ernst Zermelo (1908) *A new proof of the possibility of a well-ordering'', with commentary by van Heijenoort, pages 183–198. Wherein Zermelo rails against Poincaré's (and therefore Russell's) notion of impredicative definition.
** Ernst Zermelo (1908a) *Investigations in the foundations of set theory I*, with commentary by van Heijenoort, pages 199–215. Wherein Zermelo attempts to solve Russell's paradox by structuring his axioms to restrict the universal domain B (from which objects and sets are pulled by *definite properties*) so that it itself cannot be a set, i.e., his axioms disallow a universal set.
** Norbert Wiener (1914) *A simplification of the logic of relations*, with commentary by van Heijenoort, pages 224–227
** Thoralf Skolem. 1922. *Some remarks on axiomatized set theory*, with commentary by van Heijenoort, pages 290–301. Wherein Skolem defines Zermelo's vague "definite property".
** Moses Schönfinkel. 1924. *On the building blocks of mathematical logic*, with commentary by Willard Quine, pages 355–366. The start of *combinatory logic*.
** John von Neumann. 1925. *An axiomatization of set theory*, with commentary by van Heijenoort , pages 393–413. Wherein von Neumann creates "classes" as distinct from "sets" (the "classes" are Zermelo's "definite properties"), and now there is a universal set, etc.
** Hilbert, David. 1927. *The foundations of mathematics* by van Heijenoort, with commentary, pages 464–479.
* Whitehead, Alfred North, et Bertrand Russell. 1913, 1962. *Principia Mathematica to *56*. Cantabrigiae: at the University Press, Londinii.

Categoria:Mathematica elementaria