Nazariyai majmuho

Nazariyai majmuho talimotest dar borai khosiyathoi umumii majmuho, alalkhusus majmuhoi beintiho. Mafhumi majmu az jumlai soddatarin mafhumhoi  matematikist, on tarif doda nameshavad, ammo bo misolho sharh dodan mumkin ast. Chunonchi, dar borai majmui kitobhoi yagon. kitobkhona, majmui nuqtahoi khati rost, majmui halhoi yagon muodila sukhan ronda metavonem. Kitobhoi kitobkhonai mazkur, nuqtahoi khati rost, halhoi muodilai intikhobshuda muvofiqan ele­menthoi majmu meboshand. Bo maqsadi muayyan kardani majmu khosiyati kharakteristikii elementho, yane on khosiyatero nishon dodan kofist, ki hamai elementhoi in majmu sohiband. Chunin ham meshavad, ki hej yak element in khosiyatro doro nest, on goh in guna majmuro majmui kholi menomand. Element majmui M budani kh in tavr navishta meshavad: khϵ M (khondanash kh mutaalliqi majmui M).

Agar har yak elementi majmui A dar ayni hol elementi majmui Vboshad, on vaqt A-ro zermajmu va yo qismi majmui V menomand va chunin menavisand: A⊇Vyo V⊆A. Majmui kholiro nazar ba tarif, zermajmui hama guna majmu mehisoband.

Avvalin masalae, ki dar sohai majmuhoi beintiho ba miyon omad, in masalai bayni yakdigar muqoisa kardani miqdori elementhoi maj­muho bud. In masala va masalahoi ba on nazdikro solhoi 70 asri 19 G. Kantor hal namud va Nazariyai marifatro chun ilmi matematiki asosnok kard. Mu- qoisai miqdori elementhoi majmuho ba mafhumi bayni yakdigar yakqimata muvofiq guzoshtani ele­menthoi du majmu istinod mekunad. Bigzor ba har yak elementi maj­mui A, az rui yagon qonunu qoida, elementi malumi majmui V muvofiq guzoshta shuda boshad. Agar dar in mavrid ba har yak elementi machjmui A yak va faqat yak elementi majmui V muvofiq guzoshta shavad, on gokhh meguyand, ki bayni majmuhoi A va V muvofiqati baynan yakqimata muqarrar karda shudaast [ishorai mukhtasarash (1—1)]. Zohiran, bayni du majmui okhirnok tanho hamon vaqt muvofiqatii (1— 1)-ro muqarrar namudan mumkin ast, ki agar har du majmu az hamon yak miqdori elementho tashkil yofta boshand. Baroi tamimi in omil iq tidori majmuro muayyan mekunand. Mohiyati mafhumi iqtidori majmu bo mavjudiyati majmuhoi beintihoi iqtidorashon gunogun muayyan megardad. Masalan, iqtidori hama i zer- majmuhoi majmui M az iqtidori khudi majmui M kalon ast. Majmue, ki iqtidorash ba iqtidori majmui hamai adadhoi naturali barobar ast, majmui h i s o b i nom dorad. Iqtidori majmui hisobi khurdtarin iqtidorest, ki onro majmui beintiho dorad. G. Kantor isbot namud, ki majmui hamai adadhoi ratsionali va hatto majmui hamai adadhoi algebravi hisobiand, vale majmui hamai adadhoi haqiqi gayrihisobi. Iqtidori majmui hamai adadhoi haqiqiro iqtidori kontinuum menomand (nig. Kontinuum). Majmui ha­mai zermajmuhoi majmui hisobi, majmui hamai adadhoi kompleksi, majmui hamai nuqtaho dar hamvori, inchunin majmui hamai nuqtahoi fazoi sechena na yo umuman fazoi n-chena (p — adadi ikhtiyori) ba majmui hamai adadhoi haqiqi barobariqtidor meboshand. Kantor far- ziyaero (ba nom kontinuumfarziya) pesh guzosht, ki bino ba on har guna majmui iborat az adadhoi haqiqi yo bointiho, yo hisobi va yo bo majmui hamai adadhoi haqiqi barobariqtidor khohad bud (nig. Problemai kontinuum).

Dar Nazariyai marifat mafhumhoi tamimi funksiya, mafhumi geometrii inikos yo tabdiloti shakl va monandi inho tahti yak mafhumi umumii inikosi yak majmu ba digarash mujtame meyobad. Bigzor majmuhoi X va Y dodashuda boshad. Faraz kunem, ki ba har kadom elementn muayyani kh ϵ X yagon elementi muayyani u = f (x) az Y muvofiqat karda bo­shad; dar chunin hol meguyand, ini­kosi majmui X dar majmui Y vujud dorad. Ba iborai digar, bayni maj­mui X va Y funksiyae vujud dorad ki argumenti on kh az X qimat me- giradu qimati funksiya az majui Y. Elementn u = f (kh), ki baroi shar kh-i malum az X dar Y mavjud ast, o b r a z i elementi kh ϵ X dar Y (hangomi in inikos) va yo qimati funksiyai mazkur nom dorad. Masalan, kh majmui hamai adadhoi haqiqi bo­shad; agar ba har yak kh ϵ X funksiya u = f(x) = arc tg kh boshad, pas mu­qarrar khohad shud, ki majmui X dar

intervali (-  , ) inikos gardidaast.

Amalho bo majmuho niz ba monandi amalho bo adadho (gayr az amali taqsim) vujud dorand, chunonchi summay du, se va gayra majmuhoi dilkhohi okhirnok va beokhir gufta majmuero menomand, ki elementhoi on loaqal yakto elementi majmuhoi jamshavandaro dar khud doshta boshad. B u r i sh i majmuho gufta majmuero guyand, ki elementhoi on baroi hamai majmuhoi intikhobshuda umumi meboshad. Burishi du majmui gayriholi majmui holi shuda metavolad. F a r q i majmui A va majmui V majmuest, ki elementhoyash az elemelthoi majmui V iborat buda, mutaalliqi majmui A nestand. Amalhoi jam va burishi majmuho qonunhoi assosiativi va kommutativiro qone megardonand. Agar ξ va η muvofiqan iqtidori majmuhoi X va U boshad, on goh ξη  va ηξ  muvofiqan chun iqtidori zarbi berunai in majmuho X∙ U (majmuhoi jufti imkonpazir (kh, u), kh ϵ X, u ϵ Y] va YX darajai onho (majmui hamin inikoshoi imkolpaziri majmui X ba majmui Y) boyad muoina kard. Ba monandi hamin summai iqtidorhoro chun iqtidori summai majmuhoi juft- juft hamdigarro naburanda muayyan kardan mumkin ast.

Dar majmui X muayyan namudanl mafhumi tartibi elementho az mu­ayyan kardani paydarpaii jufti elementhoi kh’ va kh” iborat ast. Bigzor kh‘ < kh ” ast. Agar kh < kh ‘ va kh'< kh “ boshad, on goh kh < kh” (yane qonuni tranzitivi) khohad shud. Maj­mue, ki az rui yagon tartibi muayyan muoina megardad, majmui murattab- shuda nomida meshavad. Vale majmui murattabshuda majmui qisman mu- rattabshudaest, ki boyad sharthoi zerinro qone gardonand. 1) hej yak element pesh az khud omada nametavonad; 2) az du elementi gunoguni kh, kh’, yake az digare pesh omada metavonad, yane yo kh < kh’, yo kh’ < kh. Masalan, har guna majmui adadhoi haqiqi khatti murattabshuda meboshad; hamon adadero peshoyandi adadi soni menomand, ki on nisbat ba adadi soni khurdtar boshad.

Tasiri Nazariyai marifat dar inkishofi matematikai muosir baso kalon ast. Pesh az hama Nazariyai marifat asosi yak qator fanhoi navi matematiki ba monan­di nazariyai funksiyahoi tagyiryobandahoyash haqiqi topologiya i umumi, algebrai umumi, tahlili funksionali va gayra gardid. Metodhoi nazariyaviyu majmui dar sohahoi klassikii matematika niz tadrijan tatbiq yofta istodaast. Masalan, yak metodro dar sohai tahlili matematiki, dar nazariyai sifatii muodilahoi differensiali, hisobi variatsioni, nazariyai ehtimoliyat va gayra istifoda mebarand. Nihoyat Nazariyai marifat baroi fahmishi khudi predmeti matematika yo khud chunin fasli kaloni on — ge­ometriya tasiri kalon rasond.