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Saad ka saata <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/Stewards" class="extiw" title="m:Special:MyLanguage/Stewards">stjuuarditele</a> <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/VRT" class="extiw" title="m:Special:MyLanguage/VRT">VRT</a>-süsteemi aadressile <kbd>stewards@wikimedia.org</kbd> kirja, millele on lisatud kõik ülaltoodud üksikasjad.</div></div><hr /> <p>Saad vaadata ja kopeerida selle lehekülje lähteteksti. </p><textarea readonly="" accesskey="," id="wpTextbox1" cols="80" rows="25" style="" class="mw-editfont-monospace" lang="et" dir="ltr" name="wpTextbox1">{{Keeletoimeta|aasta=2022|kuu=juuli}}{{toimeta}} {{See artikkel| räägib filosoofia ja matemaatika mõistest; mineraloogia mõiste kohta vaata artiklit [[Isomorfism (mineraloogia)]]}} '''Isomorfism''' ([[kreeka keel|kreeka]]: [[wikt:ἴσος|ἴσος]] ''isos'' – ühesugune, ja [[wikt:μορφή|μορφή]] ''morphe'' – vorm) moodustavad koos [[homomorfism]]iga üldmõiste (sh ka [[filosoofiline kategooria|filosoofilise kategooria]]), mis iseloomustab vastavust objektide struktuuride vahel <ref>Schmitd, Heirich, 1991. ''Philosophisches Wörerbuch''. Stuttgart. ISBN 5250017940</ref><ref>''Новая философская энциклопедия''. 2001, Москва. ISBN 5244009613 (00962-1)</ref>. Mõned spetsiifilise suunitlusega filosoofilised koolkonnad võivad mitte tunnistada nende mõistete kuulumist kategooriate kilda. ==Selgitus== Teadaolevalt võttis isomorfismi termini kasutusele 1857. aastal A. Cayley oma keemiliste isomeeride alastes uuringutes <ref>A. Cayley, 1857. ''On the theory of the analytical forms called trees''. Phil. Mag. (4) 13 (1857), 172–176</ref>. Isomorfism tähendab vastavust, kus kaks süsteemi, vaadelduna lahus neid moodustavate elementide loomusest, vastab ühe süsteemi igale elemendile ainult üks teise süsteemi element ning ühe süsteemi igale seosele vastab ainult üks seos teises – ja vastupidi. Seega saab isomorfismist rääkida vaid niisuguste objektide puhul, millel on struktuur, st on määratletud selle elemendid (komponendid, osised) ja nendevahelised seosed (suhted). Isomorfism on määratletav kui ''struktuuri säilitav üksühene vastavus objektide vahel''. Isomorfsete objektide hulk moodustab '''isomorfismiklassi''. Kõige piltlikum näide isomorfismist on graafide isomorfism. Kaks graafi on isomorfsed, st omavad ühesugust struktuuri, vaatamata nende erinevale "välimusele". {|class="wikitable" style="margin: 1em auto 1em auto" ! Graaf G ! Graaf H ! Isomorfism<br />G ja H vahel |- |style="padding-left:2em;padding-right:2em;"|[[Pilt:Graph isomorphism a.svg|100px]] |style="padding-left:1em;padding-right:1em;"|[[Pilt:Graph isomorphism b.svg]] |align="center" style="background-color:white;"|ƒ(''a'') = 1 ƒ(''b'') = 6 ƒ(''c'') = 8 ƒ(''d'') = 3 ƒ(''g'') = 5 ƒ(''h'') = 2 ƒ(''i'') = 4 ƒ(''j'') = 7 |} ==Isomorfism matemaatikas== Matemaatikas defineeritakse isomorfismi kui süsteemi niisugust [[üksühene kujutus|üksühest kujutus]]t sama tüüpi süsteemiks, mille korral säilib süsteemide [[struktuur]]. Näiteks, kujund ja selle kujundi matemaatiline avaldis <ref>Kaasik, Ülo. 2003. ''Matemaatikaleksikon''. Tartu. ISBN 9985941772</ref>. Isomorfism on pööratav [[morfism]], millel on ''pöördmorfism'', kus nende korrutis on ''ühikmorfism''. [[topoloogia|Topoloogilist]] isomorfismi nimetatakse ''homoömorfismiks''. Isomorfismiprobleem on aktuaalne algebras, kategooria- ja [[graafiteooria]]s. [[Algebra]]s on isomorfism kujutus objektide vahel, selline mis näitab suhet kahe omaduse või operatsiooni vahel.&nbsp; Kui kahe struktuuri vahel esineb isomorfism, siis öeldakse, et vastavad objektid on ''isomorfsed''.&nbsp; Isomorfsed objektid on ''struktuurselt ekvivalentsed'', kuid võivad mõnest muus aspektis ka erineda. Teisisõnu, isomorfism on ''bijektiivne kujutus'' ''f'' niisugune, et ''f'' ja selle pöördfunktsioon ''f''<sup>&nbsp;&minus;1</sup> on struktuuri säilitavad kujutused kahe algebralise struktuuri vahel, st need mõlemad on [[homomorfism|homomorfsed]]. Isomorfism on algebras samalaadselt defineeritud ka [[Rühm (matemaatika)|rühma]], [[ring]]i ja teiste struktuuride kohta. Isomorfism [[graafiteooria]]s tähendab graafide ''G'' ja ''H'' struktuuri säilitavat tippude bijektsiooni :<math> f \colon V(G) \to V(H) \,\!</math> niisugust, et kui graafi ''G'' mingid kaks tippu ''u'' ja ''v'' on ''seotud'', siis ja ainult siis on ƒ(''u'') ja ƒ(''v'') ''seotud'' garaafis ''H''. Selle näide on selgituses esitatud. Oluline on siin nende [[substitutsioon]]ide väljatoomine: <math>\begin{pmatrix} a & b & c & d & g & h & i & j \\ 1 & 6 & 8 & 3 & 5 & 2 & 4 & 7 \end{pmatrix}</math> Kahe graafi isomorfsust tähistatakse <math>G\simeq H</math>. Juhul kui bijektsioon on graafi kujutus iseendasse, st kui ''G'' ja ''H'' on üks ja sama graaf, siis seda bijektsiooni nimetatakse graafi ''G'' [[automorfism]]iks ''AutG''. Graafide isomorfism on [[ekvivalentsus]]suhe ning isomorfsed graafid kujutavad endast ühesugust struktuuri omavaid ''graafide isomorfismiklasse''. ''Isomorfismi tuvastamine'' kujutab endast vastava [[algoritm]]i konstrueerimist, mida nimetatakse [[isomorfismiprobleem]]iks. ==Isomorfism eri valdkondades== Isomorfismi mõistet kasutatakse ka [[geoloogia]]s, [[bioloogia]]s, [[füüsika]]s jm. Korrektne on seda kasutada vaid seal, kus nende spetsiifiliste objektide struktuur ja bijektsioon on määratletav, see tähendab, kui nende geoloogiliste (bioloogiliste, füüsikaliste jt) süsteemide elemendid (komponendid, osised) ja nendevahelised seosed (suhted) on määratletud. Tegelikult ei peeta sellest nõudest alati kinni. ==Viited== {{viited}} ==Kirjandus== * McGraw-Hill dictionary of Mathematics, 1997. N. Y., ISBN 007524335. 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