Lesson 1 PDF

vè;k; 1 leqPp; (Sets) vIn these days of conflict between ancient and modern studies; there must purely be something to be said for a study which did not begin with Pythagoras and will not end with Einstein; but...

vè;k; 1 leqPp; (Sets) vIn these days of conflict between ancient and modern studies; there must purely be something to be said for a study which did not begin with Pythagoras and will not end with Einstein; but is the oldest and the youngest — G.H.HARDY v 1.1 Hkwfedk (Introduction) orZeku le; esa xf.kr osQ vè;;u esa leqPp; dh ifjdYiuk vk/kjHkwr gSA vktdy bl ifjdYiuk dk iz;ksx xf.kr dh izk;% lHkh 'kk[kkvksa esa gksrk gSA leqPp; dk iz;ksx lac/ a ,oa iQyu dks ifjHkkf"kr djus osQ fy, fd;k tkrk gSA T;kferh;] vuqØe] izkf;drk vkfn osQ vè;;u esa leqPp; osQ Kku dh vko';drk iM+rh gSA leqPp; fl¼kar dk fodkl teZu xf.krK Georg Cantor (1845&1918) }kjk fd;k x;k FkkA f=kdks.kferh; Js.kh osQ iz'uksa dks ljy djrs le; mudk leqPp; ls igyh ckj ifjp; gqvk FkkA bl vè;k; esa ge leqPp; ls lacaf/r oqQN ewyHkwr ifjHkk"kkvksa vkSj Georg Cantor lafØ;kvksa ij fopkj djsaxsA (1845-1918 A.D.) 1.2 leqPp; vkSj mudk fu:i.k (Sets and their Representations) nSfud thou esa ge cgq/k oLrqvksa osQ laxzg dh ppkZ djrs gSa] tSls rk'k dh xM~Mh] O;fDr;ksa dh HkhM+] fØosQV Vhe vkfnA xf.kr esa Hkh ge fofHkUu laxzgksa] dh ppkZ djrs gSa] mnkgj.kkFkZ] izko`Qr la[;kvksa dk laxzg fcanqvksa dk laxzg] vHkkT; la[;kvksa dk laxzg vkfnA fo'ks"kr%] ge fuEufyf[kr laxzg ij fopkj djsaxs% (i) 10 ls de fo"ke izko`Qr la[;k,¡] vFkkZr~ 1] 3] 5] 7] 9 (ii) Hkkjr dh ufn;k¡] (iii) vaxzs”kh o.kZekyk osQ Loj] ;kuh] a, e, i, o, u, (iv) fofHkUu izdkj osQ f=kHkqt] 2024-25 2 xf.kr (v) la[;k 210 osQ vHkkT; xq.ku[kaM] vFkkZr~] 2] 3] 5 rFkk 7] (vi) lehdj.k x2 – 5x + 6 = 0, osQ ewy vFkkZr~] 2 rFkk 3 ;gk¡ ge ;g ns[krs gSa fd mi;qZDr izR;sd mnkgj.kksa esa ls oLrqvksa dk ,d lqifjHkkf"kr laxzg bl vFkZ esa gS fd fdlh oLrq osQ laca/ esa ge ;g fu.kZ; fuf'pr :i ls ys ldrs gSa fd og oLrq ,d iznÙk laxzg esa gS vFkok ugha gSA mnkgj.kr% ge ;g fuf'pr :i ls dg ldrs gSa fd ^uhy unh*] Hkkjr dh ufn;ksa osQ laxzg esa ugha gSA blosQ foijhr xaxk unh bl laxzg esa fuf'pr:i ls gSA ge uhps ,sls leqPp; osQ dqN vkSj mnkgj.k ns jgs gSa] ftudk iz;ksx xf.kr esa fo'ks"k:i ls fd;k tkrk gS_ N : izko`Qr la[;kvksa dk leqPp; Z : iw.kk±dksa dk leqPp; Q : ifjes; la[;kvksa dk leqPp; R : okLrfod la[;kvksa dk leqPp; Z+ : /u iw.kk±dksa dk leqPp; Q + : /u ifjes; la[;kvksa dk leqPp; R + : /u okLrfod la[;kvksa dk leqPp; bu fo'ks"k leqPp;ksa osQ fy, fu/kZfjr mi;qZDr izrhdksa dk iz;ksx ge bl iqLrd esa fujarj djrs jgsaxsA blosQ vfrfjDr fo'o osQ ik¡p lokZf/d fo[;kr xf.krKksa dk laxzg ,d lqifjHkkf"kr leqPp; ugha gS] D;ksafd lokZf/d fo[;kr xf.krKksa osQ fu.kZ; djus dk ekinaM ,d O;fDr ls nwljs O;fDr osQ fy, fHkUu&fHkUu gks ldrk gSA vr% ;g ,d lqifjHkkf"kr laxzg ugha gSA vr% ^oLrqvksa osQ lqifjHkkf"kr laxzg* dks ge ,d leqPp; dgrs gSaA ;gk¡ ij gesa fuEufyf[kr fcanqvksa ij è;ku nsuk gS% (i) leqP;; osQ fy, oLrq,¡] vo;o rFkk lnL; i;kZ;okph in gSaA (ii) leq P ;; dks iz k ;% va x z s ” kh o.kZ e kyk os Q cM+ s v{kjks a ls fu:fir djrs gS a ] tSls A, B, C, X, Y, Z vkfn (iii) leqPp; osQ vo;oksa dks vaxzs”kh o.kZekyk osQ NksVs v{kjksa }kjk iznf'kZr djrs gSa] tSls a, b, c, x, y, z vkfn ;fn a, leqPp; A dk ,d vo;o gS] rks ge dgrs gaS fd ^a leqPp; A esa gS*A okD;ka'k ^vo;o gS* ^lnL; gS* ;k ^esa gS* dks lwfpr djus osQ fy, ;wukuh izrhd ^^∈ (epsilon)** dk iz;ksx fd;k tkrk gSA vr% ge ‘a ∈ A’ fy[krs gSaA ;fn b, leqPp; A dk vo;o ugha gS] rks ge ‘b ∉ A’ fy[krs gSa vkSj bls ^^b leqPp; A esa ugha gS** iys dks"Bd osQ Hkhrj fy[krs gSaA mnkgj.kkFkZ] 7 ls de lHkh le /u iw.kk±dksa osQ leqPp; dk o.kZu jksLVj :i esa {2, 4, 6} }kjk fd;k tkrk gSA fdlh leqPp; dks jksLVj :i esa iznf'kZr djus osQ dqN vkSj mnkgj.k uhps fn, gSa% (a) la [ ;k 42 dks foHkkftr djus okyh lHkh iz k o` Q r la [ ;kvks a dk leq P p; {1, 2, 3, 6, 7, 14, 21, 42} gSA (b) vaxzs”kh o.kZekyk osQ lHkh Lojksa dk leqPp; {a, e, i, o, u} gSA (c) fo"ke izko`Qr la[;kvksa dk leqPp; {1, 3, 5,...} gSA var osQ fcanq] ftudh la[;k rhu gksrh gS] ;g crykrs gSa fd bu fo"ke la[;kvksa dh lwph varghu gSA uksV dhft, fd jksLVj :i esa vo;oksa dks lwphc¼ djus esa muosQ Øe dk egRo ugha gksrk gSA bl izdkj mi;ZqDr leqPp; dks {1, 3, 7, 21, 2, 6, 14, 42}izdkj Hkh iznf'kZr dj ldrs gSaA AfVIi.kh ;g è;ku j[kuk pkfg, fd leqPp; dks jksLVj :i esa fy[krs le; fdlh vo;o dks lkekU;r% nksckjk ugha fy[krs gSa] vFkkZr~] izR;sd vo;o nwljs ls fHkUu gksrk gSA mnkgj.k osQ fy, 'kCn ‘SCHOOL’ esa iz;qDr v{kjksa dk leqPp; { S, C, H, O, L} gSA (ii) leqPp; fuekZ.k :i esa] fdlh leqPp; osQ lHkh vo;oksa esa ,d loZfu"B xq.k/eZ gksrk gS tks leqPp; ls ckgj osQ fdlh vo;o esa ugha gksrk gSA mnkgj.kkZFk leqPp; {a, e, i, o, u} osQ lHkh vo;oksa esa ,d loZfu"B xq.k/eZ gS fd buesa ls izR;sd vo;o vaxzs”kh o.kZekyk dk ,d Loj gS vkSj bl xq.k/eZ okyk dksbZ vU; v{kj ugha gSA bl leqPp; dks V ls fu:fir djrs gq, ge fy[krs gSa fd] V = {x : x vaxzs”kh o.kZekyk dk ,d Loj gS}A ;gk¡ è;ku nsuk pkfg, fd fdlh leqPp; osQ vo;oksa dk o.kZu djus osQ fy, ge izrhd ‘x’ dk iz;ksx djrs gS]a (x osQ LFkku ij fdlh vU; izrhd dk Hkh iz;ksx fd;k tk ldrk gS] tSl]s v{kj y, z vkfnA) ftlosQ mijkar dksyu dk fpÉ ¶%¸ fy[krs gSAa dksyu osQ fpÉ osQ ckn leqPp; osQ vo;oksa osQ fof'k"V xq.k/eZ dks fy[krs gSa vkSj fiQj lai.w kZ dFku dks e>ys dks"Bd { } osQ Hkhrj fy[krs gSaA leqPp; V osQ mi;qDZ r o.kZu dks fuEufyf[kr izdkj ls i