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Charta functionis exemplaris,

Functio in mathematica est congruentia inter duas copias, quae determinat unum secundae copiae elementum ad elementum quemque primae copiae.[1] Prima copia dicitur dominium ; altera, codominium. Si nominat quoddam primae copiae elementum, est variabilis independens. Si est functio, possumus scribere , quod significat " est elementum codominii ad elementum dominii respondens." Deinde est variabilis dependens.

Si sunt pluria elementa possibilia ad elementum respondentia, congruentia non est functio. Exempli gratia: sit , et sint dominium et codominium copia numerorum realium . Haec congruentia non est functio, quod ad elementum (sicut 4) respondent duae elementa (sicut 2, -2). Sed si codominium est copia numerorum realium non-negativorum, vel si functio est , 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 ; analysis analysis numerorum complexorum, earum quarum dominium est . G. H. Hardy dicit, "Haec notio, ut quantitas variabilis dependet ex alia, est fortasse notio maximi momenti per totam rem mathematicam."[2]

Si dominium est copia quantitatum binarum, sicut , functio habet duas variabiles independentes. Exempli gratia, . Hac functione par ad unum elementum codominii (quod est ) congruit, sicut par cum congruit. Possumus habere functiones trium, quattuor, vel plurimorum variabilum independentium.

Altera notatio functionum est notatio lambda, quae nominat variabiles independentis post lambda litteram. Scribimus: vel ad eandem functionem describendam. Forma sicut est combinator.

Si ad elementum quendam codominii respondat aut nullum aut unum modo elementum dominii, functio est functio iniectiva, aut functio unum elementum ad unum elementum attribuens. Si omne elementum codominii habet elementum (aut plura elementa ...) dominii quod ad correspondet, functio est functio superiectiva. Functio et iniectiva et superiectiva est functio biiectiva.

Si functio est biiectiva, habet functionem inversam , cuius dominium est codominium functionis , et codominium est dominium functionis . Si , est ergo . Exempli gratia, sit ; deinde functio inversa . 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 sunt functiones, et si dominium functionis est (aut continet) codominium functionis , possumus scribere . Exempli gratia, sint . Deinde , et . Non sunt eaedem functiones: si , sed .

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

Notae[recensere | fontem recensere]

  1. Behnke et al, p. 64.
  2. Hardy, p. 40.

Bibliographia[recensere | fontem recensere]

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