Relations and Functions PDF

vè;k; 1 laca/ ,oa iQyu (Relations and Functions) vThere is no permanent place in the world for ugly mathematics.... It may be very hard to define mathematical beauty but that is just as true of beauty of any kind, we may not know quite what we mean b...

vè;k; 1 laca/ ,oa iQyu (Relations and Functions) vThere is no permanent place in the world for ugly mathematics.... It may be very hard to define mathematical beauty but that is just as true of beauty of any kind, we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognising one when we read it. — G. H. Hardy v 1.1 Hkwfedk (Introduction) Lej.k dhft, fd d{kk XI esa] laca/ ,oa iQyu] izkar] lgizkar rFkk ifjlj vkfn dh vo/kj.kkvksa dk] fofHkUu izdkj osQ okLrfod ekuh; iQyuksa vkSj muosQ vkys[kksa lfgr ifjp; djk;k tk pqdk gSA xf.kr esa 'kCn ^laca/ (Relation)* dh loaQYiuk dks vaxzs”kh Hkk"kk esa bl 'kCn osQ vFkZ ls fy;k x;k gS] ftlosQ vuqlkj nks oLrq,¡ ijLij lacaf/r gksrh gS] ;fn muosQ chp ,d vfHkKs; (Recognisable) dM+h gksA eku yhft, fd A, fdlh LowQy dh d{kk XII osQ fo|kfFkZ;ksa dk leqPp; gS rFkk B mlh LowQy dh d{kk XI osQ fo|kfFkZ;ksa dk leqPp; gSaA vc leqPp; A ls leqPp; B rd osQ laca/ksa osQ oqQN mnkgj.k bl izdkj gSa (i) {(a, b) ∈ A × B: a, b dk HkkbZ gS}, Lejeune Dirichlet (1805-1859) (ii) {(a, b) ∈ A × B: a, b dh cgu gS}, (iii) {(a, b) ∈ A × B: a dh vk;q b dh vk;q ls vf/d gS}, (iv) {(a, b) ∈ A × B: fiNyh vafre ijh{kk esa a }kjk izkIr iw.kk±d b }kjk izkIr iw.kk±d ls de gS }, (v) {(a, b) ∈ A × B: a mlh txg jgrk gS tgk¡ b jgrk gS}. rFkkfi A ls B rd osQ fdlh laca/ R dks vewrZ:i (Abstracting) ls ge xf.kr esa A × B osQ ,d LosPN (Arbitrary) mileqPp; dh rjg ifjHkkf"kr djrs gSaA Reprint 2024-25 2 xf.kr ;fn (a, b) ∈ R, rks ge dgrs gSa fd laca/ R osQ varxZr a, b ls lacaf/r gS vkSj ge bls a R b fy[krs gSaA lkekU;r%] ;fn (a, b) ∈ R, rks ge bl ckr dh fpark ugha djrs gSa fd a rFkk b osQ chp dksbZ vfHkKs; dM+h gS vFkok ugha gSA tSlk fd d{kk XI esa ns[k pqosQ gSa] iQyu ,d fo'ks"k izdkj osQ laca/ gksrk gSaA bl vè;k; esa] ge fofHkUu izdkj osQ laca/ksa ,oa iQyuksa] iQyuksa osQ la;kstu (composition)] O;qRØe.kh; (Invertible) iQyuksa vkSj f}vk/kjh lafØ;kvksa dk vè;;u djsaxsA 1.2 laca/ksa osQ izdkj (Types of Relations) bl vuqPNsn esa ge fofHkUu izdkj osQ laca/ksa dk vè;;u djsaxsA gesa Kkr gS fd fdlh leqPp; A esa laca/] A × A dk ,d mileqPp; gksrk gSA vr% fjDr leqPp; φ ⊂ A × A rFkk A × A Lo;a] nks vUR; laca/ gSaA Li"Vhdj.k gsrq] R = {(a, b): a – b = 10} }kjk iznÙk leqPp; A = {1, 2, 3, 4} ij ifjHkkf"kr ,d laca/ R ij fopkj dhft,A ;g ,d fjDr leqPp; gS] D;ksafd ,slk dksbZ Hkh ;qXe (pair) ugha gS tks izfrca/ a – b = 10 dks larq"V djrk gSA blh izdkj R′ = {(a, b) : | a – b | ≥ 0}] laiw.kZ leqPp; A × A osQ rqY; gS] D;ksafd A × A osQ lHkh ;qXe (a, b), | a – b | ≥ 0 dks larq"V djrs gSaA ;g nksuksa vUR; osQ mnkgj.k gesa fuEufyf[kr ifjHkk"kkvksa osQ fy, izsfjr djrs gSaA ifjHkk"kk 1 leqPp; A ij ifjHkkf"kr laca/ R ,d fjDr laca/ dgykrk gS] ;fn A dk dksbZ Hkh vo;o A osQ fdlh Hkh vo;o ls lacaf/r ugha gS] vFkkZr~ R = φ ⊂ A × A. ifjHkk"kk 2 leqPp; A ij ifjHkkf"kr laca/ R, ,d lkoZf=kd (universal) laca/ dgykrk gS] ;fn A dk izR;sd vo;o A osQ lHkh vo;oksa ls lacaf/r gS] vFkkZr~ R = A × A. fjDr laca/ rFkk lkoZf=kd laca/ dks dHkh&dHkh rqPN (trivial) laca/ Hkh dgrs gSaA mnkgj.k 1 eku yhft, fd A fdlh ckydksa osQ LowQy osQ lHkh fo|kfFkZ;ksa dk leqPp; gSA n'kkZb, fd R = {(a, b) : a, b dh cgu gS } }kjk iznÙk laca/ ,d fjDr laca/ gS rFkk R′ = {(a, b) : a rFkk b dh Å¡pkbZ;ksa dk varj 3 ehVj ls de gS } }kjk iznÙk laca/ ,d lkoZf=kd laca/ gSA gy iz'ukuqlkj] D;ksafd LowQy ckydksa dk gS] vr,o LowQy dk dksbZ Hkh fo|kFkhZ] LowQy osQ fdlh Hkh fo|kFkhZ dh cgu ugha gks ldrk gSA vr% R = φ] ftlls iznf'kZr gksrk gS fd R fjDr laca/ gSA ;g Hkh Li"V gS fd fdUgha Hkh nks fo|kfFkZ;ksa dh Å¡pkb;ksa dk varj 3 ehVj ls de gksuk gh pkfg,A blls izdV gksrk gS fd R′ = A × A lkoZf=kd laca/ gSA fVIi.kh d{kk XI esa fo|kFkhZx.k lh[k pqosQ gSa fd fdlh laca/ dks nks izdkj ls fu:fir fd;k tk ldrk gS] uker% jksLVj fof/ rFkk leqPp; fuekZ.k fof/A rFkkfi cgqr ls ys[kdksa }kjk leqPp; {1, 2, 3, 4} ij ifjHkkf"kr laca/ R = {(a, b) : b = a + 1} dks a R b }kjk Hkh fu:fir fd;k tkrk gS] ;fn vkSj osQoy ;fn b = a + 1 gksA tc dHkh lqfo/ktud gksxk] ge Hkh bl laosQru (notation) dk iz;ksx djsaxsA Reprint 2024-25  laca/ ,oa iQyu 3 ;fn (a, b) ∈ R, rks ge dgrs gSa fd a,b ls lacaf/r gS* vkSj bl ckr dks ge a R b }kjk izdV djrs gSaA ,d vR;Ur egRoiw.kZ laca/] ftldh xf.kr esa ,d lkFkZd (significant) Hkwfedk gS] rqY;rk laca/ (Equivalence Relation) dgykrk gSA rqY;rk laca/ dk vè;;u djus osQ fy, ge igys rhu izdkj osQ laca/ksa] uker% LorqY; (Reflexive)] lefer (Symmetric) rFkk laØked (Transitive) laca/ksa ij fopkj djrs gSaA ifjHkk"kk 3 leqPp; A ij ifjHkkf"kr laca/ R; (i) LorqY; (reflexive) dgykrk gS] ;fn izR;sd a ∈ A osQ fy, (a, a) ∈ R, (ii) lefer (symmetric) dgykrk gS] ;fn leLr a1, a2 ∈ A osQ fy, (a1, a2) ∈ R ls (a2, a1) ∈ R izkIr gksA (iii) laØked (transitive) dgykrk gS] ;fn leLr, a1, a2, a3 ∈ A osQ fy, (a1, a2) ∈ R rFkk (a2, a3) ∈ R ls (a1, a3) ∈ R izkIr gksA ifjHkk"kk 4 A ij ifjHkkf"kr laca/ R ,d rqY;rk laca/ dgykrk gS] ;fn R LorqY;] lefer rFkk laØked gSA mnkgj.k 2 eku yhft, fd T fdlh lery esa fLFkr leLr f=kHkqtksa dk ,d leqPp; gSA leqPp; T esa R = {(T1, T2) : T1, T2osQ lokZxale gS} ,d laca/ gSA fl¼ dhft, fd R ,d rqY;rk laca/ gSA gy la c a / R Lorq Y ; gS ] D;ks a f d iz R ;s d f=kHkq t Lo;a os Q lokx± l e gks r k gS A iq u % (T1, T2) ∈ R ⇒T1 , T2 osQ lokZxale gS ⇒T2 , T1 osQ lokZxale gS ⇒(T2, T1) ∈ R. vr% laca/ R lefer gSA blosQ vfrfjDr (T1, T2), (T2, T3) ∈ R ⇒T1 , T2 osQ lokZxale gS rFkk T2, T3 osQ lokZxale gS ⇒T1, T3 osQ lokZxale gS ⇒(T1, T3) ∈ R. vr% laca/ R laØked gSA bl izdkj R ,d rqY;rk laca/ gSA mnkgj.k 3 eku yhft, fd L fdlh lery esa fLFkr leLr js[kkvksa dk ,d leqPp; gS rFkk R = {(L1, L2) : L1, L2 ij yac gS} leqPp; L esa ifjHkkf"kr ,d laca/ gSA fl¼ dhft, fd R lefer gS ¯drq ;g u rks LorqY; gS vkSj u laØked gSA gy R LorqY; ugha gS] D;ksafd dksbZ js[kk L1 vius vki ij yac ugha gks ldrh gS] vFkkZr~ (L1, L1) ∉ R- R lefer gS] D;ksafd (L1, L2) ∈ R ⇒ L1, L2 ij yac gS ⇒ L2 , L1 ij yac gS ⇒ (L2, L1) ∈ R Reprint 2024-25 4 xf.kr R laØked ugha gSA fu'p; gh] ;fn L1, L2 ij yac gS rFkk L2 , L3 ij yac gS] rks L1 , L3 ij dHkh Hkh yac ugha gks ldrh gSA okLro esa ,slh n'kk esa L1 , L3 osQ lekUrj gksxhA vFkkZr~] (L1, L2) ∈ R, (L2, L3) ∈ R ijarq (L1, L3) ∉ R mnkgj.k 4 fl¼ dhft, fd leqPp; {1, 2, 3} esa R = {(1, 1), (2, 2), vko`Qfr 1-1 (3, 3), (1, 2), (2, 3)} }kjk iznÙk laca/ LorqY; gS] ijarq u rks lefer gS vkSj u laØked gSA gy R LorqY; gS D;ksafd (1, 1), (2, 2) vkSj (3, 3), R osQ vo;o gSaA R lefer ugha gS] D;ksafd (1, 2) ∈ R ¯drq (2, 1) ∉ R. blh izdkj R laØked ugha gS] D;ksafd (1, 2) ∈ R rFkk (2, 3)∈R ijarq (1, 3) ∉ R mnkgj.k 5 fl¼ dhft, fd iw.kk±dksa osQ leqPp; Z esa R = {(a, b) : la[;k 2, (a – b) dks foHkkftr djrh gS} }kjk iznÙk laca/ ,d rqY;rk laca/ gSA gy R LorqY; gS] D;ksafd leLr a ∈ Z osQ fy, 2] (a – a) dks foHkkftr djrk gSA vr% (a, a) ∈ R. iqu%] ;fn (a, b) ∈ R, rks 2] a – b dks foHkkftr djrk gS A vr,o b – a dks Hkh 2 foHkkftr djrk gSA vr% (b, a) ∈ R, ftlls fl¼ gksrk gS fd R lefer gSA blh izdkj] ;fn (a, b) ∈ R rFkk (b, c) ∈ R, rks a – b rFkk b – c la [ ;k 2 ls HkkT; gS A vc] a – c = (a – b) + (b – c) le (even) gS (D;ksa\)A vr% (a – c) Hkh 2 ls HkkT; gSA blls fl¼ gksrk gS fd R laØked gSA vr% leqPp; Z esa R ,d rqY;rk laca/ gSA mnkgj.k 5 esa] uksV dhft, fd lHkh le iw.kk±d 'kwU; ls lacaf/r gSa] D;ksafd (0, ± 2), (0, ± 4), ---vkfn R esa gSa vkSj dksbZ Hkh fo"ke iw.kk±d 0 ls lacaf/r ugha gS] D;ksafd (0, ± 1), (0, ± 3), ---vkfn R esa ugha gSAa blh izdkj lHkh fo"ke iw.kk±d 1 ls lacfa /r gSa vkSj dksbZ Hkh le iw.kk±d 1 ls lacaf/r ugha gSA vr,o] leLr le iw.kk±dksa dk leqPp; E rFkk leLr fo"ke iw.kk±dksa dk leqPp; O leqPp; Z osQ mi leqPp; gSa] tks fuEufyf[kr izfrca/ksa dks larq"V djrs gSaA (i) E osQ leLr vo;o ,d nwljs ls lacaf/r gSa rFkk O osQ leLr vo;o ,d nwljs ls lacaf/r gSaA (ii) E dk dksbZ Hkh vo;o O osQ fdlh Hkh vo;o ls lacaf/r ugha gS vkSj foykser% O dk dksbZ Hkh vo;o E osQ fdlh Hkh vo;o ls lacaf/r ugha gSA (iii) E rFkk O vla;qDr gS vkSj Z = E ∪ O gSA mileqPp; E, 'kwU; dks varfoZ"V (contain) djus okyk rqY;rk&oxZ (Equivalence Class) dgykrk gS vkSj ftls izrhd ls fu:fir djrs gSaA blh izdkj O, 1 dks varfoZ"V djus okyk rqY;rk&oxZ gS] ftls }kjk fu:fir djrs gSaA uksV dhft, fd ≠ , = [2r] vkSj Reprint 2024-25  laca/ ,oa iQyu 5 = [2r + 1], r ∈ Z. okLro esa] tks oqQN geus Åij ns[kk gS] og fdlh Hkh leqPp; X esa ,d LosPN rqY;rk laca/ R osQ fy, lR; gksrk gSA fdlh iznÙk LosPN leqPp; X esa iznÙk ,d LosPN (arbitrary) rqY;rk laca/ R, X dks ijLij vla;qDr mileqPp;ksa Ai esa foHkkftr dj nsrk gS] ftUgsa X dk foHkktu (Partition) dgrs gSa vksj tks fuEufyf[kr izfrca/ksa dks larq"V djrs gSa% (i) leLr i osQ fy, Ai osQ lHkh vo;o ,d nwljs ls lacaf/r gksrs gSaA (ii) Ai dk dksbZ Hkh vo;o] Aj osQ fdlh Hkh vo;o ls lacaf/r ugha gksrk gS] tgk¡ i ≠ j (iii) ∪ Aj = X rFkk Ai ∩ Aj = φ, i ≠ j mileqPp; Ai rqY;rk&oxZ dgykrs gSaA bl fLFkfr dk jkspd i{k ;g gS fd ge foijhr fØ;k Hkh dj ldrs gSaA mnkgj.k osQ fy, Z osQ mu mifoHkktuksa ij fopkj dhft,] tks Z osQ ,sls rhu ijLij vla;qDr mileqPp;ksa A1, A2 rFkk A3 }kjk iznÙk gSa] ftudk lfEeyu (Union) Z gS] A1 = {x ∈ Z : x la[;k 3 dk xq.kt gS } = {..., – 6, – 3, 0, 3, 6,...} A2 = {x ∈ Z : x – 1 la[;k 3 dk xq.kt gS } = {..., – 5, – 2, 1, 4, 7,...} A3 = {x ∈ Z : x – 2 la[;k 3 dk xq.kt gS } = {..., – 4, – 1, 2, 5, 8,...} Z esa ,d laca/ R = {(a, b) : 3, a – b dks foHkkftr djrk gS} ifjHkkf"kr dhft,A mnkgj.k 5 esa iz;qDr roZQ osQ vuqlkj ge fl¼ dj ldrs gSa fd R ,d rqY;rk laca/ gSaA blosQ vfrfjDr A1, Z osQ mu lHkh iw.kk±dksa osQ leqPp; osQ cjkcj gS] tks 'kwU; ls lacaf/r gSa] A2, Z osQ mu lHkh iw.kk±dksa osQ leqPp; osQ cjkcj gS] tks 1 ls lacaf/r gSa vkSj A3 , Z osQ mu lHkh iw.kk±dksa osQ leqPp; cjkcj gS] tks 2 ls lacaf/r gSaA vr% A1 = , A2 = vkSj A3 =. okLro esa A1 = [3r], A2 = [3r + 1] vkSj A3 = [3r + 2], tgk¡ r ∈ Z. mnkgj.k 6 eku yhft, fd leqPp; A = {1, 2, 3, 4, 5, 6, 7} esa R = {(a, b) : a rFkk b nksuksa gh ;k rks fo"ke gSa ;k le gSa} }kjk ifjHkkf"kr ,d laca/ gSA fl¼ dhft, fd R ,d rqY;rk laca/ gSA lkFk gh fl¼ dhft, fd mileqPp; {1, 3, 5, 7} osQ lHkh vo;o ,d nwljs ls lacaf/r gS] vkSj mileqPp; {2, 4, 6} osQ lHkh vo;o ,d nwljs ls lacaf/r gS] ijarq mileqPp; {1, 3, 5,7} dk dksbZ Hkh vo;o mileqPp; {2, 4, 6} osQ fdlh Hkh vo;o ls lacaf/r ugha gSA gy A dk iznÙk dksbZ vo;o a ;k rks fo"ke gS ;k le gS] vr,o (a, a) ∈ R- blosQ vfrfjDr (a, b) ∈ R ⇒a rFkk b nksuksa gh] ;k rks fo"ke gSa ;k le gSa ⇒(b, a) ∈ R- blh izdkj (a, b) ∈ R rFkk (b, c) ∈ R ⇒vo;o a, b, c, lHkh ;k rks fo"ke gSa ;k le gSa ⇒(a, c) ∈ R. vr% R ,d rqY;rk laca/ gSA iqu%] {1, 3, 5, 7} osQ lHkh vo;o ,d nwljs ls lacaf/r gSa] D;ksafd bl mileqPp; osQ lHkh vo;o fo"ke gSaA blh izdkj {2, 4, 6,} osQ lHkh vo;o ,d nwljs ls lacaf/r gSa] D;ksafd ;s lHkh le gSaA lkFk gh mileqPp; {1, 3, 5, 7} dk dksbZ Hkh vo;o {2, 4, 6} osQ fdlh Hkh vo;o ls lacaf/r ugha gks ldrk gS] D;ksafd {1, 3, 5, 7} osQ vo;o fo"ke gSa] tc fd {2, 4, 6}, osQ vo;o le gSaA Reprint 2024-25 6 xf.kr iz'ukoyh 1-1 1. fu/kZfjr dhft, fd D;k fuEufyf[kr laca/ksa esa ls izR;sd LorqY;] lefer rFkk laØked gSa% (i) leqPp; A = {1, 2, 3,..., 13, 14} esa laca/ R, bl izdkj ifjHkkf"kr gS fd R = {(x, y) : 3x – y = 0} (ii) izko`Qr la[;kvksa osQ leqPp; N esa R = {(x, y) : y = x + 5 rFkk x < 4}}kjk ifjHkkf"kr laca/ R. (iii) leqPp; A = {1, 2, 3, 4, 5, 6} esa R = {(x, y) : y HkkT; gS x ls} }kjk ifjHkkf"kr lac/ a RgSA (iv) leLr iw.kk±dksa osQ leqPp; Z esa R = {(x, y) : x – y ,d iw.kk±d gS} }kjk ifjHkkf"kr laca/ R- (v) fdlh fo'ks"k le; ij fdlh uxj osQ fuokfl;ksa osQ leqPp; esa fuEufyf[kr laca/ R (a) R = {(x, y) : x rFkk y ,d gh LFkku ij dk;Z djrs gSa} (b) R = {(x, y) : x rFkk y ,d gh eksgYys esa jgrs gSa} (c) R = {(x, y) : x, y ls Bhd&Bhd 7 lseh yack gS} (d) R = {(x, y) : x , y dh iRuh gS} (e) R = {(x, y) : x, y osQ firk gSa} 2. fl¼ dhft, fd okLrfod la[;kvksa osQ leqPp; R esa R = {(a, b) : a ≤ b 2}, }kjk ifjHkkf"kr laca/ R] u rks LorqY; gS] u lefer gSa vkSj u gh laØked gSA 3. tk¡p dhft, fd D;k leqPp; {1, 2, 3, 4, 5, 6} esa R = {(a, b) : b = a + 1} }kjk ifjHkkf"kr laca/ R LorqY;] lefer ;k laØked gSA 4. fl¼ dhft, fd R esa R = {(a, b) : a ≤b}, }kjk ifjHkkf"kr laca/ R LorqY; rFkk laØked gS ¯drq lefer ugha gSA 5. tk¡p dhft, fd D;k R esa R = {(a, b) : a ≤b3} }kjk ifjHkkf"kr laca/ LorqY;] lefer vFkok laØked gS\ 6. fl¼ dhft, fd leqPp; {1, 2, 3} esa R = {(1, 2), (2, 1)} }kjk iznÙk laca/ R lefer gS ¯drq u rks LorqY; gS vkSj u laØked gSA 7. fl¼ dhft, fd fdlh dkWyst osQ iqLrdky; dh leLr iqLrdksa osQ leqPp; A esa R = {(x, y) : x rFkk y esa istksa dh la[;k leku gS} }kjk iznÙk laca/ R ,d rqY;rk laca/ gSA Reprint 2024-25  laca/ ,oa iQyu 7 8. fl¼ dhft, fd A = {1, 2, 3, 4, 5} esa] R = {(a, b) : |a – b| le gS} }kjk iznÙk laca/ R ,d rqY;rk laca/ gSA izekf.kr dhft, fd {1, 3, 5} osQ lHkh vo;o ,d nwljs ls lacaf/r gSa vkSj leqPp; {2, 4} osQ lHkh vo;o ,d nwljs ls lacaf/r gSa ijarq {1, 3, 5} dk dksbZ Hkh vo;o {2, 4} osQ fdlh vo;o ls lacaf/r ugha gSA 9. fl¼ fdft, fd leqPp; A = {x ∈ Z : 0 ≤x ≤12}, esa fn, x, fuEufyf[kr laca/ksa R esa ls izR;sd ,d rqY;rk laca/ gS% (i) R = {(a, b) : |a – b|, 4 dk ,d xq.kt gS}, (ii) R = {(a, b) : a = b}, izR;sd n'kk esa 1 ls lacaf/r vo;oksa dks Kkr dhft,A 10. ,sls laca/ dk mnkgj.k nhft,] tks (i) lefer gks ijarq u rks LorqY; gks vkSj u laØked gksA (ii) laØked gks ijarq u rks LorqY; gks vkSj u lefer gksA. (iii) LorqY; rFkk lefer gks ¯drq laØked u gksA (iv) LorqY; rFkk laØked gks ¯drq lefer u gksA (v) lefer rFkk laØked gks ¯drq LorqY; u gksA 11. fl¼ dhft, fd fdlh lery esa fLFkr fcanqvksa osQ leqPp; esa] R = {(P, Q) : fcanq P dh ewy fcanq ls nwjh] fcanq Q dh ewy fcanq ls nwjh osQ leku gS} }kjk iznÙk laca/ R ,d rqY;rk laca/ gSA iqu% fl¼ dhft, fd fcanq P ≠ (0, 0) ls lacaf/r lHkh fcanqvksa dk leqPp; P ls gksdj tkus okys ,d ,sls o`Ùk dks fu:fir djrk gS] ftldk osaQnz ewyfcanq ij gSA 12. fl¼ dhft, fd leLr f=kHkqtksa osQ leqPp; A esa] R = {(T1, T2) : T1, T2 osQ le:i gS} }kjk ifjHkkf"kr laca/ R ,d rqY;rk laca/ gSA Hkqtkvksa 3] 4] 5 okys ledks.k f=kHkqt T1 ] Hkqtkvksa 5] 12] 13 okys ledks.k f=kHkqt T2 rFkk Hkqtkvksa 6] 8] 10 okys ledks.k f=kHkqt T3 ij fopkj dhft,A T1, T2 vkSj T3 esa ls dkSu ls f=kHkqt ijLij lacaf/r gSa\ 13. fl¼ dhft, fd leLr cgqHkqtksa osQ leqPp; A esa, R = {(P1, P2) : P1 rFkk P2 dh Hkqtkvksa dh la[;k leku gS}izdkj ls ifjHkkf"kr laca/ R ,d rqY;rk laca/ gSA 3] 4] vkSj 5 yackbZ dh Hkqtkvksa okys ledks.k f=kHkqt ls lacaf/r leqPp; A osQ lHkh vo;oksa dk leqPp; Kkr dhft,A 14. eku yhft, fd XY-ry esa fLFkr leLr js[kkvksa dk leqPp; L gS vkSj L esa R = {(L1,L2) : L1 lekUrj gS L2 osQ} }kjk ifjHkkf"kr laca/ R gSA fl¼ dhft, fd R ,d rqY;rk laca/ gSA js[kk y = 2x + 4 ls lacaf/r leLr js[kkvksa dk leqPp; Kkr dhft,A Reprint 2024-25 8 xf.kr 15. eku yhft, fd leqPp; {1, 2, 3, 4} esa] R = {(1, 2), (2, 2), (1, 1), (4,4), (1, 3), (3, 3), (3, 2)} }kjk ifjHkkf"kr laca/ R gSA fuEufyf[kr esa ls lgh mÙkj pqfu,A (A) R LorqY; rFkk lefer gS ¯drq laØked ugha gSA (B) R LorqY; rFkk laØked gS ¯drq lefer ugha gSA (C) R lefer rFkk laØked gS ¯drq LorqY; ugha gSA (D) R ,d rqY;rk laca/ gSA 16. eku yhft, fd leqPp; N esa] R = {(a, b) : a = b – 2, b > 6} }kjk iznÙk laca/ R gSA fuEufyf[kr esa ls lgh mÙkj pqfu,% (A) (2, 4) ∈ R (B) (3, 8) ∈ R (C) (6, 8) ∈ R (D) (8, 7) ∈ R 1.3 iQyuksa osQ izdkj (Types of Functions) iQyuksa dh vo/kj.kk] oqQN fo'ks"k iQyu tSls rRled iQyu] vpj iQyu] cgqin iQyu] ifjes; iQyu] ekikad iQyu] fpg~u iQyu vkfn dk o.kZu muosQ vkys[kksa lfgr d{kk XI esa fd;k tk pqdk gSA nks iQyuksa osQ ;ksx] varj] xq.kk rFkk Hkkx dk Hkh vè;;u fd;k tk pqdk gSA D;ksafd iQyu dh ladYiuk xf.kr rFkk vè;;u dh vU; 'kk[kkvksa (Disciplines) esa lokZf/d egRoiw.kZ gS] blfy, ge iQyu osQ ckjs esa viuk vè;;u ogk¡ ls vkxs c