Chapter 1 Class 12 Maths PDF

[Class- XII , Subject- Mathematics, Chapter-1] ================================== lEcU/k ,ao Qyu (Relation and Function) ==========================...

[Class- XII , Subject- Mathematics, Chapter-1] ================================== lEcU/k ,ao Qyu (Relation and Function) ============================================== lEcU/k(Relation) leqPp; A ls leqPp; B esa dksbZ lEcU/k R = {(x, y) : x ∊ A, y ∊ B}, A X B dk mileqPp; gksrk gSA vFkZkr~ R ⊆ (A X B). tSls& A={1,8,27} , B = {1,2,3} ⇒ AXB = {(1,1), (1,2), (1,3), (8,1), (8,2), (8,3), (27,1), (27,2), (27,3)} ;fn dzfer ;qXe (x, y) ∊ R esa x = y3 vFkZkr~ (izFke vo;o) = (f}rh; vo;o) 3 rks R = {(1,1), (8,2), (27,3)}, vr% Li’V gS fd R ⊆ (A X B). i rth lEcU/k dk Mkse su ( Domain of a Relation): ;fn R ⊆ (A X B) rks R ds dzfer ;qXeksa ds lHkh izFke vo;oksa ya dk leqPp; Mksesu ;k Dom(R) dgykrk gSZA vFkZkr~ Dom(R) = {x: x ∊ A and (x, y) ∊ R}. lEcU/k dk ifjlj ( Range of a Relation): ;fn R ⊆ (A X B) rks R ds dzfer ;qXeksa ds lHkh f}rh; vo;oksa dk leqPp; ifjlj ;k Range(R) dgykrk gSZA vFkZkr~ Dom(R) = {y: y ∊ B and (x, y) ∊ R}. Ikz f rykse Qyu (Inverse Relation): ;fn leqPp; A ls leqPp; B esa dksbZ lEcU/k R gS] rks R dk izfrykse id lEcU/k og lEcU/k gS tks R ds izR;sd dzfer ;qXe ds vo;oksa ijLij cny nsus ls izkIr gksrk gSA bls R-1 ls fu#fir djrs gSA vr% R-1 = R = {(y, x) : y ∊ B, x ∊ A}. lEcU/kksa ds iz d kj (Types of Relations) eV fjDr lEcU/k (Relation): ;fn leqPp; A ds fy;s ϕ ᑕ A X A tgkW ϕ: A → A lEcU/k gS] bl lEcU/k dks fjDr lEcU/k dgrs gSA lkoZ f =d ;k le’Vh; lEcU/k (Universal Relation): ;fn leqPp; A dk izR;sd vo;o A ds lHkh vo;oksa ls lEcU/k R ds }kjk lEcfU/kr gS] rks R lEcU/k dks lkoZf=d lEcU/k dgrs gSA vFkZkr~ R = A X A ⊆ A X A vr% leqPp; A esa lEcU/k R lkoZf=d lEcU/k gS ;fn R = A X A. rRled lEcU/k (Identity Relation): ;fn leqPp; A esa lEcU/k R bl izdkj gks fd izR;sd x,y ∊ A ds fy;s] x R y ;fn x = y vFkZkr~ (x, y) ∊ R ⇒ x = y rks R lEcU/k dks rRled lEcU/k dgrs gSA leqPp; A esa bl lEcU/k dks IA ls fu#fir djrs gSA Page Nu. 01  f}vk/kkjh lEcU/k (Binary Relation): leqPp; A ds lEcU/k R dks ;qXeksa ds #i esa fu#fir djrs gSA ;qXeksa ds #i esa izkIr lEcU/k R dks f}vk/kkjh lEcU/k dgrs gSA f}vk/kkjh lEcU/k ds iz d kj (Types of Binary Relations) 1&Lorq Y ; lEcU/k (Reflexive Relation): fdlh leqPp; A esa lEcU/k R LorqY; lEcU/k dgykrk gS] ;fn vkSj dsoy ;fn (x, y) ∊ R ∀ x ∊ A, i.e, x R x, ∀ x ∊ A. tSls& x R y ;fn x ∥ y LorqY; lEcU/k gSA 2&lefer lEcU/k (Symmetric Relation): fdlh leqPp; A esa lEcU/k R Lkefer lEcU/k dgykrk gS] ;fn A ds fdUgh nks vo;oks x vkSj y ds fy;s (x, y) ∊ R ⇒ (y, x) ∊ R; i.e, xRy ⇒ yRx ; ∀ x, y ∊ A i 3&lad z e d lEcU/k (Transitive Relation): fdlh leqPp; A esa lEcU/k R Lkadzed lEcU/k dgykrk gS] ;fn vkSj rth dsoy ;fn (x, y) ∊ R rFkk (y, z) ∊ R ⇒ (z, x) ∊ R; ∀ x, y, z ∊ A. i.e, xRy and yRz ⇒ zRx ; ∀ x, y, z ∊ A 4&rq Y ;rk lEcU/k (Equivalence Relation): fdlh leqPp; A esa lEcU/k R rqY;rk lEcU/k dgykrk gS] ;fn vkSj dsoy ;fn R LorqY;] Lkefer rFkk Lkadzed rhuksa gksaA ya rq Y ;rk oxZ (Equivalence Class): ekuk fdlh vfjDr leqPp; A esa lEcU/k R rqY;rk lEcU/k gSA A dk dksbZ a vo;o gSA A dk ,slk mileqPp; ftlds lHkh vo;o x, a ds lkFk R lEcU/k cukrk gS vFkZkr~ (x, a) ∊ R, vo;o a dk rqY;rk oxZ dgykrk gSA bls [a] ls fu#fir djrs gSA rq Y ;rk oxZ ds xq. k (Properties of Equivalence Class): id (i) a ∊ [a] (ii) ;fn b∊ [a], rks [b] = [a] (iii) [a] = [b] ;fn vkSj dsoy ;fn (a, b) ∊ R eV (iv) ;k rks [a] = [b] ;k [a] Ռ [b] = ϕ leq P p; dk fcHkktu (Partition of a Set): fdlh vfjDr leqPp; A ds fcHkktu dk vFkZ gS fd A ds ,ls mileqPp; izkIr djuk] tks fd vla;qDr rFkk vfjDr gksaA vFkZkr~ A dk fcHkktu A ds vla;qDr rFkk vfjDr leqPp;ksa {Ai} dk laxzg bl izdkj gS fd & (i) izR;sd a ∊ A fdlh Ai fo|eku gSA (ii) Ai ≠ Aj rks Ai Ռ Aj = ɸ Page Nu. 02  nks lEcU/kksa dk la; kst u (Composition of Two Relatios): ;fn fdUgh vfjDr leqPp; A ls B esa lEcU/k R vkSj leqPp; B ls C esa lEcU/k S gSA nksuksa lEcU/kksa dk la;kstu SoR, A ls C esa ,d lEcU/k gS fd SoR = {(a, c) : b ∊ B ,d ,slk vo;o gS fd aRb rFkk bSc; tgkW a ∊ A rFkk c ∊ C}. i mnkgj.k (Example)% rth mnkgj.k 1& ;fn iw.kZkadks ds leqPp; Z esa lEcU/k R = {(x, y): x – y = ,d le iw.kZkad gS} ls ifjHkkf’kr gS rks fl} fdft;s fd ,d rqY;rk lEcU/k gSaA gy% (i) x – x = 0, ,d le iw.kZkad gSaA ∀ x ∊ Z : vr% R LorqY; gSA (ii) (x, y) ∊ R ⇒ x – y = ,d le iw.kZkad gSA ⇒ -(x – y) = ,d le iw.kZkad gSA ⇒ (y – x) = ,d le iw.kZkad gSA ya ⇒ y – x = ,d le iw.kZkad gSA ⇒ (y, x) ∊ R; ∀ x, y ∊ Z vr% R Lkefer gSA (iii) (x, y) ∊ R, (y, z) ∊ R, ⇒ x – y = le iw.kZkad] y – z = le iw.kZkad ⇒ x – y + y – z = le iw.kZkad ⇒ (x – z) = le iw.kZkad ⇒ (x – z) = le iw.kZkad id ⇒ (x, z) ∊ R; ∀ x, y,z ∊ Z vr% R Lkadzed gSA vr% leqPp; Z esa lEcU/k R = {(x, y): x – y = ,d le iw.kZkad gS}, ,d rqY;rk lEcU/k gSA mnkgj.k 2& ;fn R = {(4, 5), (1, 4), (4, 6), (7, 6), (3, 7)} gks] rks& (i) RoR (ii) R-1oR (ii) R-1oR-1 eV gy% (i) (1, 4) ∊ R, (4, 5) ∊ R ⇒ (1, 5) ⇒ RoR, (1, 4) ∊ R, (4, 6) ∊ R ⇒ (1, 6) ⇒ RoR, (3, 7) ∊ R, (7, 6) ∊ R ⇒ (3, 6) ⇒ RoR vr% RoR = { (1, 5), (1, 6), (3, 6) } (ii) R-1 = {(5, 4), (4, 1), (6, 4), (6, 7), (7, 3)} vc (4, 5) ∊ R , (5, 4) ∊ R-1 ⇒ (4, 4) ∊ R-1oR, (1, 4) ∊ R , (4, 1) ∊ R-1 ⇒ (1, 1) ∊ R-1oR, (4, 6) ∊ R , (6, 4) ∊ R-1 ⇒ (4, 4) ∊ R-1oR, Page Nu. 03  (7, 6) ∊ R , (6, 4) ∊ R-1 ⇒ (7, 4) ∊ R-1oR, (7, 6) ∊ R , (6, 7) ∊ R-1 ⇒ (7, 7) ∊ R-1oR, (3, 7) ∊ R , (7, 3) ∊ R-1 ⇒ (3, 3) ∊ R-1oR, rFkk (4, 6) ∊ R , (6, 7) ∊ R-1 ⇒ (4, 7) ∊ R-1oR vr% R-1oR = {(1, 1), (3, 3), (4, 4), (4, 7), (7, 4), (7, 7)} (iii) R-1 = {(5, 4), (4, 1), (6, 4), (6, 7), (7, 3)} vc (5, 4) ∊ R-1 , (4, 1) ∊ R-1 ⇒ (5, 1) ∊ R-1oR-1, (6, 4) ∊ R-1 , (4, 1) ∊ R-1 ⇒ (6, 1) ∊ R-1oR-1, rFkk (6, 7) ∊ R-1 , (7, 3) ∊ R-1 ⇒ (6, 3) ∊ R-1oR-1 i vr% R-1oR-1 = {(5, 1), (6, 1), (6, 3)}. rth Qyu (Functions) og fu;e ftlds }kjk ,d vfjDr leqPp; ds izR;sd vo;o dk nwljs vfjDr leqPp; ds vf}rh; vo;o ls lEcU/k LFkkfir fd;k tk lds] Qyu ;k izfrfp+=.k dgrs gSA fdUgh vfjDr leqPp; A ls B esa Qyu f dks izrhdkRed #i esa f: A→B fy[krs gSA x∊ A, y ∊ B leqPp; esa Qyu f ds vUrZxr y = f(x) ls iznf”kZr fd;k tkrk gSA tgkW y dks x dk izfrfcEc rFkk x dks y dk iwoZ&izfrfcEc ya dgrs gSA id Mkse su ] lgMkse su rFkk ifjlj (Domain, Co-domain and Range): Qyu f: A→B ds leqPp; A, f dk Mksesu rFkk leqPp; B, f dk lgMksesu dgykrk gSA leqPp; B ds mu vo;oksa dk leqPp; tks A ds vo;oksa dk izfrfcEc gS] f dk ifjlj f[A] dgykrk gSA Qyuks a dk iz d kj (Types of Functions) eV (i) vkPNknd Qyu (Onto Mapping): ;fn Qyu f ds lgMksesu dk izR;sd vo;o Mksesu ds fdlh u fdlh vo;o dk izfrfcEc gksa] rks ,sls Qyu dks vkPNknd Qyu dgrs gSA izrhdkRed #i esa f: A→B vkPNknd gksxk ;fn f(A) = B. Page Nu. 04  (ii)vUr%{ksi h Qyu (Into Mapping): ;fn Qyu f ds lgMksesu esa ,d ;k ,d ls vf/kd vo;o ,sls gS tks Mksesu ds fdlh Hkh vo;o ds izfrfcEc ugh gS] rks ,sls Qyu dks vUr%{ksih Qyu dgrs gSA izrhdkRed #i esa f: A→B vUr%{ksih gksxk ;fn f(A)ᑕ B. (iii) ,dS d h Qyu (One-One Mapping): ;fn Qyu esa Mksesu ds fHkUu&fHkUu vo;oksa ds izfrfcEc lnSo i fHkUu&fHkUu gS] rks ,sls Qyu dks ,dSdh Qyu dgrs gSA izrhdkRed #i esa f: A→B ,dSdh gksxk ;fn x1 ≠ x2 ⇒ f(x1) ≠ f(x2). rth ya (iv) cgq & ,d Qyu (Many-One Mapping): ;fn Qyu f esa Mksesu ds nks ;k nks ls vf/kd vo;oksa ds izfrfcEc ,d gh vo;o gksa] rks ,sls Qyu dks cgq&,d Qyu dgrs gSA izrhdkRed #i esa] f: A→B , cgq&,d gS] rks f(x1) = f(x2) ⇒ x1 ≠ x2. id eV (v) ,dS d h vUr%{ksi h Qyu (One-One Into Mapping): ;fn Qyu f vUr%{ksih Hkh gS vkSj ,dSdh Hkh gks] mls ,dSdh vUr%{ksih Qyu dgrs gSA izrhdkRed #i esa] f: A→B , ,dSdh vUr%{ksih Qyu gksxk] ;fn f(A)ᑕ B rFkk f(x1) = f(x2) ⇒ x1 = x2.. Page Nu. 05  (vi) ,dS d h vkPNknd Qyu (One-One Onto Mapping): ;fn Qyu f ,dSdh Hkh gS vkSj vkPNknd Hkh gks] mls ,dSdh vkPNknd Qyu dgrs gSA izrhdkRed #i esa] f: A→B , ,dSdh vkPNknd Qyu gksxk] ;fn f(x1) = f(x2) ⇒ x1 = x2 rFkk f(A) = B. (vii) cgq & ,d vUr%{ksi h Qyu (Many-One Into Mapping): ;fn Qyu f cgq&,d Hkh gS vkSj vUr%{ksih Hkh gks] i mls cgq&,d vUr%{ksih Qyu dgrs gSA rth izrhdkRed #i esa] f: A→B , cgq&,d vUr%{ksih Qyu gksxk] ;fn f(x1) = f(x2) ⇒ x1 ≠ x2 rFkk f(A)ᑕ B. ya (viii) cgq & ,d vkPNknd Qyu (Many-One Onto Mapping): ;fn Qyu f cgq&,d Hkh gS vkSj vkPNknd Hkh gks] mls cgq&,d vkPNknd Qyu dgrs gSA izrhdkRed #i esa] f: A→B, cgq&,d vkPNknd Qyu gksxk] ;fn f(x1) = f(x2) ⇒ x1 ≠ x2 rFkk f(A) = B. id eV (ix) rRled Qyu (Identity Function): ;fn Qyu f: R → R bl izdkj ls gS fd f(x) = x , x∊ R vFkZkr dk izR;sd vo;o Loa; dk izfrfcEc gkss] rks ,sls Qyu dks rRled Qyu dgrs gSA Page Nu. 06  (x) vpj Qyu (Constant Function): ;fn Qyu f: R → R bl izdkj ls gS fd Mksesu leLr vo;oksa dk lgMksesu esa ,d gh izfrfcEc gkss] rks ,sls Qyu dks vpj Qyu dgrs gSA (xi) ekiak d Qyu (modulus Function): ;fn Qyu f bl izdkj ls gS fd i f(x) = ∣x∣ = x, ;fn x ≥ 0 rth ;fn x ⵦ 0 -x, rc Qyu f dks ekiakd Qyu dgrs gSA (x) egRre iw. kZ ak d Qyu (Greatest Integer Function): ;fn fdlh okLrfod la[;k x ds fy, dksbZ Qyu f bl izdkj ls gS fd f(x) = [x] , rks ,sls Qyu dks egRre iw.kZakd Qyu dgrs gSA tgkW [x] ,d iw.kZakd gS tks fd ;k rks x ds cjkcj gS ;k x ls de vFkZkr~ {[x] ≤ x}. ya (xi) U;wu re iw. kZ ak d Qyu (Smallest Integer Function): ;fn fdlh okLrfod la[;k x ds fy, dksbZ Qyu f bl izdkj ls gS fd f(x) = ⌈x⌉ , rks ,sls Qyu dks U;wure iw.kZakd Qyu dgrs gSA tgkW ⌈x⌉ ,d iw.kZakd gS tks fd ;k rks x ls cM+k gS ;k x ds cjkcj vFkZkr~ {[x] ≥ x}. Qyuksa dk la; kst u (Composition of Function) id ekuk fd A, B rFkk C rhu leqPp; gSA nks Qyu f rFkk g bl izdkj ls ifjHkkf’kr gS fd f: A→B tgkWa f(x) = y, x ∊ A, y∊ B rFkk g: B→C tgkWa g(y) = z, y∊ B, z∊ C. vc ;fn Qyu h: A→C bl izdkj ls fd h(x) = z = g(y) = g{f(x)}, x∊ A rks Qyu f dks h rFkk g dk la;kstu Qyu dgrs gSA ftls gof ls fu#fir fd;k tkrk gSA eV Page Nu. 07  iz f ryksf er Qyu (Invertible Function) ;fn ,d Qyu f: A→B rFkk nwljk Qyu g: B→A Hkh lEHko gS ftlls gof = IA rFkk fog= IB Qyu f izfryksfer Qyu dgykrk gSA g dks f -1 ls fu#fir fd;k tkrk gSA vFkZkr Qyu f izfryksfer Qyu gS rks og vko”;d #i ls ,dSdh vkPNknd Qyu gksxkA rFkk foykser% Hkh lR; gksXkkA f}vk/kkjh laf dz ; k (Binary Operations) ;fn dksbZ vfjDr leqPp; gS rks Qyu f : A x A→ A leqPp; A ij f}vk/kkjh lafdz;k dgykrk gSA i mnkgj.k (Example) rth mnkgj.k&1% ;fn Qyu f rFkk okLrfod la[;kvksa ds leqPp; R ij bl izdkj ifjHkkf’kr gS fd f(x) = x+3 , ∀ x ∊ R, rks fl} dhft, fd Qyu f ,dSdh vkPNknd Qyu gSA gy&1% ;fn x1, x2 ∊ R rc f(x1) = x1 + 3, f(x2) = x2 + 3 ekuk f(x1) = f(x2) ⇒ x1 + 3 = x2 + 3 ⇒ x 1 = x2 vr% Qyu f ,dSdh gSA ya ekuk lgMksesu dk dksbZ vo;o y gSA ;fn f(x) = y, rc y = x+3 ⇒ x = y - 3 ∊ R (Mksesu) vr% f(y – 3) = (y - 3) + 3 = y vFkZkr~ ∀ y ∊ R, y - 3 ∊ R vr% f(R) = R Qyu f vkPNknd gSA id vFkZkr~ Qyu f ,dSdh vkPNknd Qyu gSA mnkgj.k&2% fl} dhft, fd Qyu f(x) = [x] nkjk iznRRk egRRke iw.kZkad Qyu f : R→ R u rks ,dSdh u vkPNknd gSA eV gy&2% fn;k gS f : R→ R rFkk f(x) = [x], rc f(1.4) = 1 rFkk f(1.7) = 1 ijUrq 1.4 ≠ 1.7 ⇒ f(1.4) = f(1.7) = 1 vr% Qyu f ,dSdh ugh gSA lgMksesu dk og vo;o tks fd iw.kZakd ugh gS] Mksesu ds fdlh Hkh vo;o dk izfrfcEc ugh gSA vFkZkr~ f (R) ⊂ R (lgMksesu), vr% f(R) = R Qyu f vkPNknd ugh gSA vFkZkr~ Qyu f u rks ,dSdh u vkPNknd gSA Page Nu. 08  mnkgj.k&3% leqPp; f{ 1, 2, 3, 4, 5} ij f}vk/kkjh lafdz;k ∗ uhps dh lafdz;k lkj.kh esa ifjHkkf’kr fd;k x;k gSA (i) (2 ∗ 3) ∗ 4 rFkk 2 ∗ (3 ∗ 4 ) Kkr djsaA (ii) D;k ∗ dze fofues; gS ? (iii) (2 ∗ 3) ∗ (4 ∗ 5 ) Kkr djsaA gy&3% (i) (2 ∗ 3) ∗ 4 = 1 ∗ 4 = 1 rFkk 2 ∗ (3 ∗ 4 ) = 2 ∗ 1 = 1 (ii) 2 ∗ 3 = 1 = 3 ∗ 4, 2 ∗ 4 = 2 = 4 ∗ 2, i 2 ∗ 5 = 1 = 5 ∗ 2, rth 3 ∗ 4 = 1 = 4 ∗ 3, ---------------------- blh izdkj vU;A (iii) (2 ∗ 3) ∗ (4 ∗ 5 ) = 1 ∗ 1 = 1 ya iz ” ukoyh 1& fl} dhft, fd leqPp; {1, 2, 3} esa R = {(1, 2), (2, 1) }kjk iznRr lEcU/k R lefer gS fdUrq u rks LorqY; gS vkSj u ladzed gSA 2& ;fn R = {a, b, c}, fl} dhft, fd A ij ifjHkkf’kr lEcU/k R rRled ugh gS] tgkW R = {(a, a), (c, c). id 3& ekuk A = {1,2,3}, B= {4,5,6,7} rFkk ekuk f = {(1, 4), (2, 5),(3, 6)} , A ls B esa ,d Qyu gSA crkb, fd f ,dSdh gS vFkok ugh ? 4& fl} dhft, fd f(x) = ∣x∣ }kjk iznRRk ekiakd Qyu f : R→ R, u rks ,dSdh u vkPNknd gSA eV 5& ;fn f}vk/kkjh lafdz;k ∗ iw.kZakdks ds leqPp; Z ij bl izdkj ifjHkkf’kr gS fd a ∗ b = a+ b +2, ∀ a, b ∊ Z, rks f}vk/kkjh lafdz;k dh dze& fofues;rk rFkk lkgp;Zrk dh tkWap dhft;sA mRRkjEkkyk% 3& gkW 5& dze& fofues;rk rFkk lkgp;Zrk vkHkkj% fo|ky;h f”k{kk ifj’kn] mRrjk[k.M }kjk fu/kZ k fjr ikB~ ; & iq L rdsa ,ao lgk;d ikB~ ; iq L rdsaA Page Nu. 09