El I-Theorem ye-Bolzano Ingenye yezinsika eziyisisekelo ekuhlaziyeni izibalo, ikakhulukazi mayelana nocwaningo lwemisebenzi yangempela neqhubekayo. Le theorem ayibalulekile nje kuphela ekuqondeni umqondo wokuqhubeka, kodwa futhi ikhombisa ukuba khona kwezixazululo zezibalo ngaphakathi kwezikhawu ezithile. Uma ubuzibuza ukuthi ungaqinisekisa kanjani ukuthi umsebenzi uwela i-eksisi engu-X, le theorem iphethe ukhiye. Kulesi sihloko sizohlola ukwakheka kwayo, ukusetshenziswa okusebenzayo, izibonelo nomlando wayo.
Ubuwazi ukuthi le theorem inezimpande emalini yezibalo kaBernhard Bolzano, isazi sefilosofi saseCzech nesazi sezibalo sekhulu le-19? Nakuba ekuqaleni umsebenzi wakhe wawungenawo umthelela ongatheni ngenxa yezindaba zezombangazwe nezenkolo yangaleso sikhathi, namuhla ubhekwa njengobalulekile emkhakheni wezibalo. Sizohlukanisa zonke izici zayo ezibalulekile nokuthi yini eyenza ibe ukutholwa okufanelekile.
Yini eyasungulwa yi-Bolzano Theorem?
Isitatimende se I-Theorem ye-Bolzano Kuyacaca: uma umsebenzi uqhubeka f (x) esikhathini esivaliwe [a, b] ithatha amanani ophawu oluphambene ezindaweni ezeqisayo zesikhawu, okungukuthi, uma f(a)·f(b) <0, bese kuba khona okungenani iphuzu elilodwa c phakathi nesikhathi (a, b) kuphi f(c) = 0.
Lesi sitatimende sisekelwe ku- ukuqhubeka kwempahla yemisebenzi, eqinisekisa ukuthi akukho "ukweqa" kugrafu yomsebenzi. Cabanga ngejika elixhuma amaphuzu amabili ngamavelu aphikisanayo; I-Logic isitshela ukuthi endaweni ethile phakathi kwalawo maphuzu, ijika kufanele liwele i-eksisi engu-X.
Isibonelo esihle se-Bolzano Theorem
Thatha njengesibonelo umsebenzi f(x) = x³ + x − 1. Siyazi ukuthi kuwumsebenzi oqhubekayo ngoba i-polynomial. Uma sihlola umsebenzi ekupheleni kwesikhathi [0,1], sine:
- f(0) = -1 (negative)
- f(1) = 1 (enhle)
Njengoba i-theorem idinga ukuthi izimpawu zihluke, singasebenzisa i-Bolzano ukuphetha ngokuthi kukhona inani c phakathi nesikhathi (0,1) kuphi f(c) = 0. Lo mphumela awusitsheli kahle ukuthi liyini lelo nani, kodwa uqinisekisa ubukhona balo.
Izicelo zeTheorem kaBolzano
Le theory inokuningi Izicelo ezingokoqobo kanye nethiyori, okwenza kube a ithuluzi esisemqoka emikhakheni eminingi:
- Thola izimpande: Iwusizo ikakhulukazi ezindleleni zezinombolo ezifana nokuhlukanisa kabili, okuhlukanisa izikhathi izikhathi ukuze kufinyelelwe impande ngokunembe kakhudlwana.
- Ukuhlaziywa kwemisebenzi eqhubekayo: Kuyasiza ukuqonda ukuziphatha kwemisebenzi ngezikhathi ezithile, ukuhlonza amaphuzu abalulekile njengezimpande noma amaphuzu abalulekile.
- Ukuxazulula izinkinga zobunjiniyela: Kusukela ekwakhiweni kwesakhiwo kuya ekuhlaziyweni okuphoqayo, ithiyori isetshenziselwa ukukhomba amaphuzu lapho kuhlangatshezwana khona nezimo ezithile ezibucayi.
- Ama-algorithms kukhompyutha: Isetshenziswa ezinhlelweni zokuhlaziya izinombolo ukuze kuxazululwe izibalo ezingezona umugqa ezingenaso isixazululo sokuhlaziya esiqondile.
Umlando we-Bolzano Theorem
I-theorem yasungulwa isazi sezibalo u-Bernhard Bolzano (1781-1848) njengengxenye yokuhlaziywa kwakhe kokuqhubeka nokungapheli. Imibono yakhe yayinjalo oguquguqukayo, nakuba bengazange bakuthole ukuqashelwa okubafanele ngesikhathi sabo ngenxa yokuhlolwa kwabo kwezombusazwe. Umsebenzi wakhe othi "Rein analytischer Beweis" owanyatheliswa ngo-1817 wawuqukethe le theory okokuqala ngqa, emaka ngaphambi nangemva kokuhlaziywa kwezibalo.
UBolzano waphinde watholwa eminyakeni embalwa ngemva kokufa kwakhe, futhi imibono yakhe yamukelwa izazi zezibalo ezinjengoKarl Weierstrass, owasungula imiqondo eminingi uBolzano ayeyihlongoze.
Ubufakazi beTheorem kaBolzano
Enye yezindlela ezinembile nezisetshenziswa kakhulu zokufakazela ukuthi i-theoem idlula indlela yokuhlukanisa kabili. Le nqubo iqukethe:
- Hlukanisa isikhawu sokuqala sibe izingxenye ezimbili ezilinganayo bese uhlola umsebenzi endaweni emaphakathi.
- Nquma kuziphi izikhawu lapho inani lomsebenzi lishintsha uphawu.
- Phinda inqubo esikhathini esingaphansi esikhethiwe kuze kube yilapho kufinyelelwa ukunemba okufunayo, ngokuqhubekayo ukuqinisekisa ukuthi sisondela kumpande.
Le ndlela iwusizo ikakhulukazi kuma-algorithms wokubala kanye nokuhlaziywa kwezibalo ezisetshenziswayo.
El I-Theorem ye-Bolzano Yethulwa njengethuluzi elibalulekile lokubonisa ubukhona bezimpande nokuhlaziya ukuziphatha kwemisebenzi eqhubekayo ngaphakathi kwezikhawu ezithile. Izicelo zayo zisukela ekuxazululeni izinkinga zezibalo kuye ekuthuthukisweni kwama-algorithms kubunjiniyela nesayensi yekhompiyutha. Amagugu ayo omlando nethiyori alibeka njengensika eyisisekelo ekuqeqeshweni kwezibalo nasekuxazululeni izinkinga zomhlaba wangempela.