Functio

Charta functionis exemplaris,
${\begin{aligned}&\scriptstyle \\&\textstyle f(x)={\frac {(4x^{3}-6x^{2}+1){\sqrt {x+1}}}{3-x}}\end{aligned}}$

Functio est congruentia inter duas copias, quae determinat unum secundae copiae elementum ad elementum quemque primae copiae. Prima copia dicitur dominium ; altera, codominium. Si $x$ nominat quoddam primae copiae elementum, $x$ est variabilis dependens. Si $f$ est functio, possumus scribere $y=f(x)$, quod significat "'y' est elementum codominii ad x elementum dominii respondens." Deinde $y$ est variabilis independens.

Si sunt plures quantitates $y$ ad x elementum respondentes, congruentia non est functio. Exempli gratia: sit $f(x)=\pm {\sqrt {x}}$, et sint dominium et codominium copia numerorum realium $\mathbb {R}$. Haec congruentia non est functio, quod ad elementum $x$ (sicut 4) respondent duae elementa (sicut 2, -2). Sed si codominium est copia numerorum realium non-negativorum, vel si functio est $f(x)=+{\sqrt {x}}$, haec congruentia functio est.

Licet functio definire per formulam aut regulam aut tabulam, dum sit modo unum elementum codominii quod ad elementum quemque dominii respondat.

Analysis est theoria functionum. Analysis numerorum realium est theoria functionum quarum dominium (et codominium) est $\mathbb {R}$; analysis numerorum complexorum, earum quarum dominium est $\mathbb {C}$. G. H. Hardy dicit, "Haec notio, ut quantitas variabilis dependet ex alia, est fortasse notio maximi momenti per totam rem mathematicam."

Si dominium est copia quantitatum binarum, sicut $\mathbb {R} ^{2}$, functio habet duas variabiles dependentes. Exempli gratia, $f(x,y)=x^{2}+y^{2}$. Hac functione $f$ par $(x,y)$ ad unum elementum codominii (quod est $\mathbb {R}$) congruit, sicut par $(2,3)$ cum $2^{2}+3^{2}=4+9=13$ congruit. Possumus habere functiones trium, quattuor, vel plurimorum variabilum dependentium.

Altera notatio functionum est notatio lambda, quae nominat variabiles dependentis post lambda litteram. Scribimus: $f=\lambda (x).x^{2}$ vel $f(x)=x^{2}$ ad eandem functionem describendam. Forma sicut $\lambda (x).x^{2}$ est combinator.

Si ad elementum quendam $y$ codominii respondat aut nullum aut unum modo elementum $x$ dominii, functio est functio iniectiva, aut functio unum elementum ad unum elementum attribuens. Si omne elementum $y$ codominii habet elementum $x$ (aut plura elementa $x_{1},x_{2},x_{3},$ ...) dominii quod ad $y$ correspondet, functio est functio suriectiva. Functio et iniectiva et suriectiva est functio biiectiva.

Si functio $f$ est biiectiva, habet functionem inversam $f^{-1}$, cuius dominium est codominium functionis $f$, et codominium est dominium functionis $f$. Si $f(x)=y$, est ergo $f^{-1}y=x$. Exempli gratia, sit $f(x)=x/2$; deinde functio inversa $f^{-1}(x)=2x$. Saepius difficile est scribere formulae functionis inversae.

Compositio functionum est nova functio per quam elementum dominii primae functionis correspondit cum elemento codominii secundae functionis. Si $y=f(x),y=g(x)$ sunt functiones, et si dominum functionis $f$ est (aut continet) codominium functionis $g$, possumus scribere $f\circ g=f(g(x))$. Exempli gratia, sint $f(x)=x^{2},g(x)=\sin(x)$. Deinde $f\circ g=f(g(x))=(\sin(x))^{2}$, et $g\circ f=g(f(x))=\sin(x^{2})$. Non sunt eaedem functiones: si $x=\pi ,f(g(x))=(\sin(\pi ))^{2}=1$, sed $g(f(x))=\sin(\pi ^{2})\approx -0.43$.

Copia omnium functionum invertibilium quarum dominium et codominium est eadem copia est caterva. Idemfactor catervae est functio quae ad omne elementum idem elementum coniungit, $f(x)=x$; operatio catervae est compositio.

Bibliographia

• Anton, Howard (1980), Calculus with Analytical Geometry, Wiley, ISBN 978-0-471-03248-9 
• Bartle, Robert G. (1976), The Elements of Real Analysis (2nd ed.), Wiley, ISBN 978-0-471-05464-1 
• 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.
• Husch, Lawrence S. (2001), Visual Calculus, University of Tennessee 
• Katz, Robert (1964), Axiomatic Analysis, D. C. Heath and Company .
• Ponte, João Pedro (1992), "The history of the concept of function and some educational implications", The Mathematics Educator 3 (2): 3–8 
• Thomas, George B.; Finney, Ross L. (1995), Calculus and Analytic Geometry (9th ed.), Addison-Wesley, ISBN 978-0-201-53174-9 
• Youschkevitch, A. P. (1976), "The concept of function up to the middle of the 19th century", Archive for History of Exact Sciences 16 (1): 37–85 .
• Monna, A. F. (1972), "The concept of function in the 19th and 20th centuries, in particular with regard to the discussions between Baire, Borel and Lebesgue", Archive for History of Exact Sciences 9 (1): 57–84 .
• Kleiner, Israel (1989), "Evolution of the Function Concept: A Brief Survey", The College Mathematics Journal (Mathematical Association of America) 20 (4): 282–300 .
• Ruthing, D. (1984), "Some definitions of the concept of function from Bernoulli, Joh. to Bourbaki, N.", Mathematical Intelligencer 6 (4): 72–77 .
• Dubinsky, Ed; Harel, Guershon (1992), The Concept of Function: Aspects of Epistemology and Pedagogy, Mathematical Association of America, ISBN 0883850818 .
• Malik, M. A. (1980), "Historical and pedagogical aspects of the definition of function", International Journal of Mathematical Education in Science and Technology 11 (4): 489–492 .
• Boole, George (1854), An Investigation into the Laws of Thought on which are founded the Laws of Thought and Probabilities", Walton and Marberly, London UK; Macmillian and Company, Cambridge UK. Republished as a googlebook.
• Eves, Howard. (1990), Foundations and Fundamental Concepts of Mathematics: Third Edition, Dover Publications, Inc. Mineola, NY, ISBN 0-486-69609-X (pbk) 
• Frege, Gottlob. (1879), Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle 
• Grattan-Guinness, Ivor and Bornet, Gérard (1997), George Boole: Selected Manuscripts on Logic and its Philosophy, Springer-Verlag, Berlin, ISBN 3-7643-5456-9 (Berlin...) 
• Halmos, Paul R. 1970. Naive Set Theory, Springer-Verlag, New York, ISBN 0-387-90092-6.
• Hardy, Godfrey Harold (1908), A Course of Pure Mathematics, Cambridge University Press (published 1993), ISBN 978-0-521-09227-2 
• 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.
  • Jules Richard (1905) The principles of mathematics and the problem of sets, with commentary by van Heijenoort, pages 142–144. The Richard paradox.
  • Russell, Bertrand. 1908a. Mathematical logic as based on the theory of types, with commentary by Willard Quine, pages 150–182.
  • 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.