Chng 1. Tp hp nh x - Quan h - S phcChng 1: TP HP NH X - QUAN H - S PHCBi 1: Khi nim v tp hp, tp hp con, cc php ton trn tp hp__________________________________________________________1. Tp hp:1.1Khi nim: Tp hp l mt khi nim nguyn thy, khng c nh ngha, m c hiu mt cch trc gic nh sau: Mt tp hp l mt s qun t cc i tng c cng thuc tnh no ; nhng i tng ny c gi l cc phn t ca tp hp .V d: - Tp hp cc sinh vin ca mt trng i hc. - Tp hp cc s nguyn t.Ta thng k hiu tp hp bi ch ci vit hoa nh A, B, C,, X, Y, Z, v cc phn t ca tp hp thng c k hiu bi mt ch ci vit thng a, b, x, y. ch phn t a thuc tp hp A, ta vit a A v c l a thuc A. Nu b khng phi l phn t ca A th ta k hiu b A v c l b khng thuc A. V d: - l tp hp cc s t nhin - l tp hp cc s nguyn- l tp hp cc s thc- l tp hp cc s hu t-{1; 2;3} S l tp hp cc s nguyn dng nh hn 4. - Tp rng l tp hp khng c phn t no.K hiu: . V d: tp hp cc s thc m bnh phng ca s bng 1 l tp rng. 1.2 Cch xc nh mt tp hp: Mt tp hp c th c xc nh bng cc cch nh:- Phng php lit k: Mt tp hp c th xc nh bng cch lit k ra ht cc phn t thuc tp hp . Phng php ny ch dng i vi tp hp hu hn.V d: A = {1; 3; 4; 5; 7}- Phng php ch ra thuc tnh c trng : Mt tp hp c th nhn bit bng cch ch ra thuc tnh ca i tng v da vo thuc tnh ny ta c th bit phn t no c thuc tp hp ny hay khng.V d: { | } B M OM r l tp hp cc im nm trn mt cu tm O bn knh r.{ | 3} C n n Ml tp hp cc s t nhin chia ht cho 3. 1.3 S bng nhau ca hai tp hp:nh ngha: Hai tp hp A v B c gi l bng nhau khi v ch khi mi phn t ca A u l phn t ca B v ngc li mi phn t ca B u l phn t ca A. Khi ta vit A = B.T nh ngha mun chng minh A = B phi chng minh cc iu sau:- Nu x A th x B i s tuyn tnh 1 11AAAChng 1. Tp hp nh x - Quan h - S phc- Nu x B th x A 2. Tp hp con:2.1 nh ngha: Cho hai tp hp A v B. Nu mi phn t ca tp hp A u l phn t ca tp hp B th khi ta ni tp A cha trong B, hay tp A l tp hp con ca tp hp B. K hiu: A B V d:- - Tp hp {1; 3} l tp hp con ca tp hp {1; 2; 3}- Tp hp cc tam gic u l tp hp con ca tp hp cc tam gic. 2.2Tnh cht: - Vi mi tp hp A th A A ;- Vi mi tp hp A th A ;- Nu A B v B C th A C (tnh cht bc cu);- Nu A B v B A th A B .2.3 Tp cc tp con ca mt tp hpCho A l mt tp hp, k hiu ( ) P A l tp cc tp con ca tp A. Nu A c n phn t th P(A) s c 2n phn t.V d: A = {a} khi ( ) { , } P A a A = {a, b, c} th ( ) { ,{ },{ },{ },{ , },{ , },{ , },{ , , }} P A a b c a b a c b c a b c 3. Cc php ton trn tp hp3.1 Hp ca cc tp hp3.1.1 nh ngha: Cho A v B l hai tp hp ty , ta gi tp hp C gm cc phn t thuc t nht mt trong hai tp A, B l hp ca hai tp A, B. K hiu: C A B hoc{ | A B x x A hoc } x B Biu Venn:V d: Nu nh ngha A, B, C l cc tp nh sau: { | ( ) 0} A x f x v { | ( ) 0} B x g x th { | ( ). ( ) 0} C x f x g x . Khi C A B 3.1.2 nh l: Vi A, B, C l cc tp no khi i) Nu B A th A B A ;ii) Vi mi tp hp A th A A v A A A ; iii) A B B A ; iv) ( ) ( ) A B C A B C .3.2 Giao ca cc tp hp3.2.1 nh ngha: Cho hai tp A, B ty . Ta gi tp hp C gm cc phn t thuc c hai tp hp A, B l giao ca hai tp hp A, B. K hiu: { | } C A B x x A v x B Biu Venn: i s tuyn tnh 1 2ABBAABBAChng 1. Tp hp nh x - Quan h - S phc3.2.2 nh l: Vi A, B, C l cc tp hp ty th ta c cc khng nh sau:i) Nu B A th A B B . Vi mi tp hp A th A v A A A ;ii) A B B A ;iii)( ) ( ) A B C A B C .3.2.3 nh l: Cho A, B, C l cc tp ty khi :i) ( ) A A B A ;ii) ( ) A B B B ;iii) ( ) ( ) ( ) A B C A B A C ; iv) ( ) ( ) ( ) A B C A B A C .3.3 Hiu ca hai tp hp3.3.1 nh ngha: Cho hai tp A, B ty . Ta gi tp hp C gm cc phn t thuc A v khng thuc B l hiu ca tp A v tp B. K hiu: C = A\B hoc \ { | v } A B x x A x B Biu Venn: 3.3.2 nh l: Vi A, B, C, D l cc tp no , khi :1. \ A B khi v ch khi A B ;2. Vi A, B bt k th \ A B A ;3. Nu A B v D C th \ \ A C B D ;4. Nu A B th vi tp C bt k ta c \ \ C B C A .3.4 Phn bNu B A th A\B c gi l phn b ca B trong A, k hiu ( )AC B hay( ) { | }AC B x A x B . Thc cht phn b ( )AC B l hiu A\B vi iu kin B A nn mi tnh cht lin quan n phn b c suy ra t tnh cht ca php hiu A\B.3.4.1 nh l: Vi cc tp A, B, C ty ta c-\ ( ) ( \ ) ( \ ) A B C A B A C ;- \ ( ) ( \ ) ( \ ) A B C A B A C .Cng thc i ngu De Morgani s tuyn tnh 1 3A B\ A BChng 1. Tp hp nh x - Quan h - S phc-( ) ( ( ))A i A ii iC B C B U I ;-( ) ( ( ))A i A ii iC B C B I U.Ta c th pht biu phn b ca hp bng giao cc phn b, phn b ca giao bng hp cc phn b. 3.5 Hiu i xng ca A v B:K hiu: ( \ ) ( \ ) A B A B B A Biu Venn: 3.6 Tch Descartes ca cc tp hpGi s a v b l hai i tng bt k, t hai i tng ny ta thnh lp i tng th ba k hiu (a; b) v gi l cp (a; b). Hai cp (a; b) v (c; d) c gi l bng nhau khi v ch khi a = c v b = d. Nu a b th cp (a; b) v (b; a) c coi l khc nhau.3.6.1 nh ngha: Tch Descartes ca n tp hp 1 2, ,...,nA A Al tp hp gm tt c cc dy sp th t 1 2( ; ;...; )na a atrong 1 1 2 2, ,...,n na A a A a A .Ta k hiu tch Descartes trn l 1 2...nA A A . Nu 1 2...nA A A th tch Descartes ca chng c k hiu lnA . 3.6.2 V d: Cho1{ ; } A a b , 2 3{ ; }, {1; 2} A c d A . Khi :1 2 3{( ; ;1), ( ; ;1), ( ; ; 2), ( ; ; 2), ( ; ;1), ( ; ; 2), ( ; ;1), ( ; ; 2)} A A A a c a d a c a d b c b c b d b d 3.6.3 Nhn xt: A B khi v ch khi A hoc B .Nu A B th ' ' A B A B khi v ch khi ' A A v ' B B .Bi 2: Khi nim c bn v nh x - Cc nh x c biti s tuyn tnh 1 4A B\ A BChng 1. Tp hp nh x - Quan h - S phc______________________________________________________1. nh x:1.1 Khi nim: Cho hai tp hp X v Y. Mt quy tc tng ng f mi phn t x X vi mt v ch mt phn t y Y c gi l nh x t tp X vo tp Y. K hiu: : f X Y hoc fX Y .Phn ty Y , tng ng vi phn t x X qua nh x f, khi , x c gi l to nh ca y v y c gi l nh ca x qua nh x f.Ngoi ra, X c gi l tp ngun (min xc nh), Y cn c gi l tp ch (min gi tr) ca nh x f.1.2 V d: - Hm s y = x 1 l nh x t tp s thc vo - Hm s lg y x l nh x t +vo - Php tng ng mi s x +vi mt s y sao cho 2x y khng l nh x v vi mt gi tr 0 x > ta s c hai gi tr ca y l: y x v y x u tng ng vi x. - Php tng ng: f sao cho 1( )1f xx+khng phi l nh x v vi 1 x th khng c y tng ng vi x cho. 1.3 nh ngha: B phn A ca tp X c gi l n nh i vi nh x f vi : , ( ) f X Y a A f a A .1.4 nh x bng nhau: nh ngha:Cho f v g l hai nh x t X vo Y. nh x f c gi l bng nh x g nu f(x) = g(x) vi mi x X .Nu vi mi x X u c ( ) f x a vi a l mt phn t xc nh ca Y, th ta ni f l mt nh x khng i, hay nh x hng s. Nu X = Y v ( ) , f x x vi mi x X th f c gi l nh x ng nht ca X. K hiu 1X.Nhn xt: Hai nh x f v g l bng nhau khi v ch khi chng c chung tp ngun v chung tp ch v , ( ) ( ) x X f x g x .2. nh v to nh:2.1 nh ca mt tp hp:a) nh ngha: Cho nh x : f X Y v A l mt tp con ca X. Tp con ca Y gm nh ca tt c cc phn t ca A c gi l nh ca tp A qua nh x f.K hiu: f(A). Hay, ( ) { ( ) | } f A f x x A .Khi , ( ) , ( ) y f A x A y f x .b) nh l: Cho nh x: f X Y . Vi hai tp con ty A v B ca X ta c: ( ) ( ) ( ) f A B f A f B v ( ) ( ) ( ) f A B f A f B .(Sinh vin t chng minh nh bi tp.)2.2 To nh ca mt tp hp:i s tuyn tnh 1 5Chng 1. Tp hp nh x - Quan h - S phca) nh ngha: Cho nh x : f X Y v U l mt tp con ty ca Y. Tp con ca X gm cc phn t x X sao cho ( ) f x U c gi l to nh ton phn ca U qua nh x f .K hiu:1( ) f U. Khi ,1( ) { | ( ) } f U x X f x U v 1( ) ( ) x f U f x U .b) nh l: Cho nh x: f X Y . Vi hai tp con bt k A, B ca Y th -1 1 1( ) ( ) ( ) f A B f A f B ;-1 1 1( ) ( ) ( ) f A B f A f B .(Sinh vin t chng minh nh bi tp nh).3. Cc loi nh x c bit3.1 n nh:3.1.1 nh ngha: nh x : f X Y c gi l mt n nh nu vi hai phn t khc nhau 1xv 2x bt k ca X th1 2( ) ( ) f x f x . Ni cch khc, f l mt n nh nu mi phn t ca tp ch ch c ti a mt to nh trong tp ngun. T nh ngha trn, chng minh f l mt n nh ta chng minh:1 2 1 2, , x x X x x th1 2( ) ( ) f x f x .Hoc 1 2 1 2, , ( ) ( ) x x X f x f x th1 2x x .3.1.2 V d: nh x : f xc nh bi 2( ) f x x khng l n nh v f(1) = f(-1) = 1.nh x : f xc nh bi 1( ) f nnl mt n nh v vi hai s t nhin khc nhau m, n th 1 1n m.Nu A E , nh x nhng chnh tc : Ai A Ex xal mt n nh c gi l n nh chnh tc t A vo E. 3.2 Ton nh: 3.2.1 nh ngha: nh x: f X Y c gi l mt ton nh nu f(X) = Y. Ni cch khc : f X Y l ton nh nu vi mi y Y u tn ti x X sao cho f(x) = y. Ton nh : f X Y cn c gi l nh x ton nh t X ln Y. T nh ngha trn, chng minh f l mt ton nh th ta cn chng minh , y Y x X sao cho f(x) = y.Nhn xt: Ni cch khc mt nh x: f X Y l ton nh khi v ch khi mi phn t ca Y c t nht mt to nh trong X. 3.2.2 V d: nh x : f xc nh bi cng thc ( ) cos f x x khng l ton nh v tn ti s 2 m khng c x cos 2 x . Tuy nhin nu xt nh x g t tp s thc vo on [-1, 1] th g l ton nh. 3.3 Song nhi s tuyn tnh 1 6Chng 1. Tp hp nh x - Quan h - S phc3.3.1 nh ngha: nh x : f X Y c gi l song nh nu n va l n nh va l ton nh. chng minh mt nh x f l song nh th ta phi chng minh f l n nh v f l ton nh, hoc chng minh rng y Y tn ti duy nht x X sao cho( ) f x y .3.3.2 V d: nh x ng nht 1 :XX X l mt song nh.nh x 2: fx x a khng l song nh v n khng phi l ton nh (cng khng l n nh).Nhn xt: Mt nh x bt k t E vo E gi l mt hon v ca E. V d:Cho : 2fx x a v :2 12gyyy' aKhi f l n nh khng l ton nh. g l ton nh khng l n nh.(Sinh vin t kim tra.)3.4 Tch cc nh x:3.4.1 nh ngha: Cho hai nh x : f X Y v : g Y Z . nh x : ( ( ))h X Zx g f xac gi l nh x tch ca hai nh x f v g. K hiu h g f ohay h = gf.Nhn xt: Theo nh ngha ta ch xc nh c tch gf khi tp ch ca f cha trong tp ngun ca g.Nu: f X X v : g X X th ta c th xc nh c tch fg v tch gf, tuy nhin gf c th khc vi fg, hay tch ca hai nh x khng giao hon. V d: Nu f v g l hai v d cho trn th Ng f Id o nhng : 1f gxxx'o a3.4.2 nh l: Cho: f X Y , : g Y T v : h T U th h(gf)=(hg)f.3.4.3 nh l: Gi s : f X Y v : g Y T l hai nh x v : h gf X T . Khi :a) Nu f, g l cc n nh th h l n nh;b) Nu h l n nh th f l n nh;c) Nu h l n nh v f l ton nh th g l n nh;d) Nu f, g l ton nh th h l ton nh;e) Nu h l ton nh th g l ton nh;i s tuyn tnh 1 7Nu y chnNu y lNu y chnNu y lChng 1. Tp hp nh x - Quan h - S phcf) Nu h l ton nh v g l n nh th f l ton nh.3.4.4 H qu: Gi s : f X Y v : g Y T l cc song nh th gf cng l song nh. 3.5 nh x ngc3.5.1 nh ngha: Gi s : f X Y v : g Y X l hai nh x tha:1Xgf v 1Yfg th khi g c gi l nh x ngc ca nh x f.V d: nh x 3: fx x ac nh x ngc 13: yfy aTrong trng hp cc hm, khi nim nh x ngc chnh l khi nim hm s ngc. 3.5.2 nh l: nh x : f X Y c nh x ngc khi v ch khi f l song nh. Nu f l song nh th 1f cng l song nh. 3.5.3 nh l: Inh x ngc ca mt nh x l duy nht.3.5.4 nh l: Nu : f E F v : g F G l nhng song nh, th : g f E G o l song nh v ( )11 1g f f g o o3.6 Thu hp v thc trin (hoc m rng) nh x:3.6.1 Thu hp nh x:Cho X v Y l hai tp hp v : f X Y l mt nh x, gi A l mt tp con ca X. Khi thu hp ca f vo A l nh x k hiu l |Af xc nh bi:| : ( )Af A Yx f xa3.6.2 Thc trin (m rng) nh x:Cho X v Y l hai tp hp v : f X Y l mt nh x, X l tp hp sao cho ' X X . Khi , m rng ca f trn X l nh x : ' g X Y sao cho , ( ) ( ) x X g x f x .V du:*:sin fxxx aKhi , nh x g l mt m rng ca f c xc nh bi: :sin 0 1 0gxxxxx' aNhn xt: g l m rng duy nht ca f v lin tc ti 0. Bi 3: Php th i s tuyn tnh 1 8Chng 1. Tp hp nh x - Quan h - S phc_____________1. Khi nim: Php th trn mt tp hp X l mt song nh t X ln chnh n. Khi X l tp c n phn t th php th trn X gi l php th bc n. Khng mt tnh tng qut ta ly tp n phn t l X = {1, 2, , n} Tp hp cc php th ca tp {1,2, , n} c k hiu l nS. Thng thng ta k hiu mt php th nh sau:1 2 ...(1) (2) ... ( )nn _ ,Ta c th xem ( (1), (2),..., ( )) n nh l mt cch sp xp th t ca tp {1, 2, , n} nn ta c th vit gn php th di dng ( (1), (2),..., ( )) n . V l mt song nh nn cc phn t (1), (2),..., ( ) n dng di u khc nhau do chng l mt hon v ca n phn t 1, 2, , n. Vy, mi mt hon v xc nh mt php th bc n nn s cc php th bc n bng s cc hon v ca tp c n phn t v bng n!.Nu php th ch i ch hai phn t i < j cho nhau v gi nguyn cc phn t cc v tr cn li th php th c gi l chuyn v. K hiu: (i, j). 2. V d: Nhm cc php th3Sgm cc php th sau: 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3; ; ; ; ;1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1 _ _ _ _ _ _ , , , , , ,Trong : 1 2 3(1 2)2 1 3 _ , l mt chuyn v. Cho 1 2, ,...,ka a al cc phn t khc nhau ca {1, 2, , n}. Php th gi nguyn cc phn t khc 1 2, ,...,ka a a v tha mn 1 2 2 3 1 1( ) , ( ) ,..., ( ) , ( )k k ka a a a a a a a c gi l mt vng xch di k. K hiu: 1 2( , ,..., )ka a a. Khi ta c 1 2 1 2 1 1( , ,..., ) ( , ,..., ) ( , )k k k ka a a a a a a a o. Vng xch di 1 chnh l php th ng nht. Vng xch di 2 gi l php chuyn v hoc (chuyn tr).V d: Cho php th ( )1 2 3 4 5 6 7 81 7 2 6 3 5 47 6 5 1 4 3 2 8 _ , l vng xch di 7. Hai vng xch 1( ... )mf i i v 1( ... )kg j j gi l c lp nu 1 2 1 2{ , ,..., } { , ,... }m ki i i j j j .V d: Vng xch ( ) 1 2 3 4 f v ( ) 5 6 7 g l hai vng xch c lp. Nu f, g l cc php th bc n th tch , f g g f o o(l tch ca hai nh x), nh x ngc 1f v nh x ng nht 1Xcng l php th bc n. Nhn xt: Php nhn cc vng xch c lp c tnh cht giao hon. V d: Cho cc php th bc 4 f v g nh sau:1 2 3 42 4 3 1f _ , v 1 2 3 44 3 2 1g _ , khi :i s tuyn tnh 1 9Chng 1. Tp hp nh x - Quan h - S phc1 2 3 4.1 3 4 2f g _ ,; 1 2 3 4.3 1 2 4g f _ , v 11 2 3 44 1 3 2f _ ,3. nh l: Mi php th bc n khc php th ng nht u phn tch c duy nht (khng k th t) thnh tch cc vng xch c lp di ln hn hoc bng 2. 4. H qu: Mi php th u phn tch c thnh tch cc chuyn v. V d: ( ) ( )1 2 3 4 5 6 7 8 91 8 2 4 3 5 9 6 78 4 1 3 9 7 5 2 6f _ ,5. Du ca php th5.1 nh ngha: Ta gi cp ( ) {1, 2,..., } i j n l mt nghch th ca php th nu i j tri du vi ( ) ( ) i j (hoc i < j v ( ) ( ) i j >).Php th vi s nghch th chn (l) c gi l php th chn (l). Du ca nhn gi tr bng 1 nu l php th chn v bng 1 nu l php th l. Ta k hiu du ca php th l ( ) sign .V d: Xt php th 1 2 33 2 1 _ , th cp (1 3) l mt nghch th v 1 < 3 nhng (1) 3 (3) 1 > . y l mt php th l.Xt php th 1 2 3 44 3 2 1 _ , c cc cp nghch th l (1 4) v (2 3) nn y l mt php th chn. 5.2 Tnh cht:- S php th chn bc n bng s cc php th l bc n v bng!2n.- Tch ca hai php th c cng tnh chn, l l php th chn. Tch ca hai php th khc tnh chn l l php th l. - Khi phn tch mt php th thnh tch ca cc chuyn v th s cc chuyn v tham gia trong tch l s chn hay l ty theo php th l chn hay l. 5.3 nh l: Vi mi php th nS ta c1( ) ( )( )i j ni jsigni j < .Nhn xt: 1(1 ) 1, ( ) ( )XSign Sign f Sign f vi minf S . 5.4 nh l: Vi mi ,nS ta c( ) ( ). ( ) sign sign sign .Bi 4: Quan h hai ngi_______________________i s tuyn tnh 1 10Chng 1. Tp hp nh x - Quan h - S phc1. nh ngha: Cho X l mt tp hp, ta ni S l mt quan h hai ngi trn X nu S l mt tp con ca tch Descartes 2X .Nu hai phn t a, b tha ( ; ) a b S th ta ni a c quan h S vi b. Khi , thay v vit ( ; ) a b S ta c th vit l aSb.2.V d: - Quan h chia ht trong tp hp s t nhin. - Quan h bng nhau.- Quan h ln hn. 3. Mt s quan h thng gp:3.1 Quan h tng ng:3.1.1 nh ngha: Mt quan h hai ngi trn tp X c gi l quan h tng ng nu n tha cc tnh cht sau:i) Phn x: xSx, vi mi x X ,ii) i xng: Nu xSy th ySx, vi mi , x y X .iii) Bc cu: Nu xSy v ySz th xSz vi mi , , x y z X .Khi trn tp X xc nh mt quan h tng ng, khi thay v vit xSy ta thng k hiu x y :.3.1.2 V d: - Quan h bng nhau cc tp hp s ; ; ; ... l mt quan h tng ng v tha cc tnh cht phn x; i xng; bc cu. - Xt trong quan h S xc nh bi 2 2xSy x y x y l mt quan h tng ng.- Gi X l tp cc ng thng trong mt phng, quan h cng phng ca hai ng thng bt k trong mt phng l quan h tng ng. (Ch : Hai ng thng c gi l cng phng l hai ng thng song song hoc trng nhau.) - Quan h vung gc gia cc ng thng trong mt phng khng phi l quan h tng ng v khng tha tnh phn x. - Quan h chia ht cho trong tp hp s t nhin khng phi l quan h tng ng v khng c tnh cht i xng. - Quan h nguyn t cng nhau trn tp hp s t nhin khng l quan h tng ng v khng c tnh cht bt cu. V d (2, 3) = 1; (4, 3) = 1 nhng (4, 2) 1 .Cho S l mt quan h tng ng trn tp X v x X . Ta gi tp hp ( ) { | } S x y X y x : l lp tng ng ca x theo quan h tng ng S. Khi ta c:- ( ) S x v ( ) x S x .- ( )x XS x XU.- , x y X th hoc S(x) = S(y) hoc ( ) ( ) S x S y . T tnh cht trn ta nhn c mt phn hoch ca X qua cc lp tng ng S(x). Tp hp tt c cc lp tng ng ny c k hiu l X/S v gi l tp thng ca X qua quan h tng ng S. 3.2 Quan h th t:i s tuyn tnh 1 11Chng 1. Tp hp nh x - Quan h - S phc3.2.1 nh ngha: Mt quan h hai ngi S trn tp X c gi l quan h th t nu quan h c cc tnh cht: phn x, bc cu v phn i xng (tc l nu xSy v ySx th suy ra x = y vi mi , x y X ).Nu tp X c mt quan h th t b phn S th ta ni X l mt tp c sp th t bi S. Ta thng dng k hiu ch mt quan h th t b phn. Vi hai phn t , x y X , nu x c quan h vi y ta vit x y (c l x b hn hay bng y) hoc vit y x (c l y ln hn hay bng x). Khi x y th thay cho x y (hayy x ) ta vit x < y (hay y > x) v c l x b hn y (hay y ln hn x).Quan h th t trong X c gi l quan h th t ton phn (hay tuyn tnh) nu vi mi , x y X ta u c x y hoc y x . Mt quan h th t khng ton phn gi l quan h th t b phn (hay tng phn).3.2.2 Cc phn t c bit. Quan h th t tt.Cho X l tp c sp th t bi v A l mt tp con ca X.Phn t a A c gi l phn t b nht (ln nht) ca A nu vi mi x A th a x (x a ).Phn t a A c gi l phn t ti tiu (ti i) ca A nu vi mi , , ( ) x A x a x a a x a x . Phn t 0x X c gi l cn di (cn trn) ca A nu vi mi 0 0: ( ). a A x a a x Quan h th t trong X c gi l mt quan h th t tt nu mi tp con khc rng ca X u c phn t b nht. Khi , X gi l c sp tt bi .V d: a) Cho X l mt tp hp, trn P(X) ta xt quan h bao hm . Ta chng minh c y l mt quan h th t b phn trn P(X). Ngoi ra, nu X cha t nht hai phn t x y th quan h th t trn khng phi tuyn tnh (hay quan h th t ton phn) v {x} khng so snh c vi {y}.b) Quan h th t thng thng trn tp hp cc s nguyn l mt quan h th t tuyn tnh, nhng khng phi quan h th t tt v khng phi mi tp con khc rng ca u c phn t b nht. V d: Tp {..., - 2, -1, 0} khng c phn t ti tiu.c) Quan h chia ht trn tp hp s t nhin l mt quan h th t b phn, nhng khng phi l quan h tuyn tnh.d) Quan h th t thng thng trn tp hp s t nhin l mt quan h th t tuyn tnh, hn na y cn l mt quan h th t tt. Vi phn t b nht l phn t 0, nhng khng c phn t ln nht. e) Trong tp cc s t nhin ln hn 1, sp th t theo quan h chia ht cc phn t ti tiu l cc s nguyn t. 3.3 Cc nguyn l tng ng:3.3.1 Tin chn: Vi mi h khng rng ( )IX cc tp hp khc rng , X I u c mt nh x :If I XUsao cho ( ) f X vi mi I .3.3.2 Nguyn l sp tt: Mi tp hp khng rng u c th c sp tt (tc l tn ti mt quan h th t tt trn tp ).i s tuyn tnh 1 12Chng 1. Tp hp nh x - Quan h - S phc3.3.3 B Zorn: Cho X l mt tp khng rng c sp th t bi . Nu mi tp con A ca X c sp ton phn bi , u c cn trn th X c phn t ti i. Bi 5: S phc______________i s tuyn tnh 1 13Chng 1. Tp hp nh x - Quan h - S phcS phc c s dng gii phng trnh bc hai 20 ax bx c + + khi 24 0 b ac < . Ta xt tp hp sau: { | , } a bi a b + , trong i c gi l n v o tha mn 21 i .Trong tp hp ny, ta xc nh hai php ton nh sau:Php cng +: Vi mi , a bi c di + + th( ) ( ) ( ) ( ) . a bi c di a c b d i + + + + + +Php nhn .: Vi mi , a bi c di + + th( )( ) ( ) ( ) a bi c di ac bd ad bc i + + + +Nhn xt: Nu z a bi + v ' ' ' z a b i + th ' ( ') ( ') z z a a b b i + v2 2 2 2'z ac bd bc adiz c d c d+ _ _ + + + , ,1. nh ngha: Tp hp vi hai php ton nh trn c gi l trng s phc.K hiu: . Mt s dng a + bi, vi 21 i c gi l s phc.Nhn xt: Ta gi biu thc dng c = a + bi l dng i s ca s phc c trong a l phn thc, k hiu l Re( ) c v b c gi l phn o ca s phc c, k hiu Im( ) c. Khi , c a bi c gi l s phc lin hp ca s phc c. Im( ) 0 c c . Nu 0 c v Re( ) 0 c th ta ni c l s thun o. S 2 2. c c a b +c gi l modul ca s phc c (hay cn gi l chun ca s phc c). K hiu: |c|.Hai s phc a bi + v c di + c gi l bng nhau nu a = c v b = d. V d: 2 3 z i + th 2 3 z i v | | 4 9 13 z + 1. Mt s tnh cht ca s phc:Vi mi s phc v th:a) t t v . b) c) . d) | | | | .e) | | | || | . f) Nu 0 th 1 1 v g) | | | | | | + + (bt ng thc tam gic)3. Biu din hnh hc ca s phc:Ta xt mt nh x t tp s phc vo mt phng ta Oxy sao cho mt s phc a + bi ng vi mt im c ta (a; b). Khi , ta ni im (a; b) l nh ca s phc a + bi cn s phc a + bi l to nh ca im (a; b). nh ca mt s thc a nm trn trc honh Ox, mt s thun o bi c nh nm trn trc tung Oy. Do , ta gi Ox l trc thc v trc Oy l trc o cn mt phng Oxy l mt phng phc. V mt hnh hc s phc lin hp c a bi chnh l nh ca s phc c = a + bi qua php i xng qua trc thc. i s tuyn tnh 1 14Chng 1. Tp hp nh x - Quan h - S phcHnh: Biu din dng i s ca s phc trn mt phng phc4. Dng lng gic ca s phc: Cho s phc a bi + , khi | | l khong cch t im (a; b) n gc ta O. Hnh: Dng lng gic ca s phcVi 2 2| | 0 a b r + > th cos a r v sin b r suy ra 2 2cosaa b + v 2 2sinba b + trong l gc nh hng to thnh gia tia Ox v tia i t gc ta O n im (a; b). Khi , c vit di dng:(cos sin ) r i + vi 0 | | r < . y gi l dng lng gic ca s phc . Ta thy rng, v tr im (a; b) trong mt phng hon ton xc nh bi modul | | r v gc nh hng . Gc nh hng c gi l bin ca s phc v k hiu l arg( ) . Gi tr arg( ) c th nhn bt k gi tr no khc 0, vi quy nh hng dng ca gc nh hng l hng ngc chiu kim ng h. Do , nu hai gc hn km nhau 2 , k k th chng cng xc nh mt s phc. V d: Tm dng lng gic ca s phc 1 i +Gii: Ta c 2 21 1 2 r a b + + , suy ra 1 1cos , sin2 2 Nn ta chn c 4 . Vy 2(cos sin )4 4i +. Ngoi ra s phc a bi + cn c th biu din tng ng vi mt s phc khc l (cos sin )ae e b i b + . Suy ra, nu l mt s thc th cos sinie i + . V vy, nu s phc c vit di dng lng gic (cos sin ) r i +th c th c biu din di dng khc l ire , trong r l modul v l bin ca . Cng thc Euler: cos2i ie e + , sin ,2i ie ei .5. Ly tha - Cng thc Moivre:i s tuyn tnh 1 15Chng 1. Tp hp nh x - Quan h - S phcDa vo dng lng gic ca s phc ta c th thc hin mt cch d dng php tnh nng ln ly tha ca mt s phc da vo cc cng thc sau:Vi hai s phc 1 1 1 1(cos sin ) r i + v 2 2 2 2(cos sin ) r i +, khi :1 2 1 2 1 2 1 2. (cos( ) sin( )) r r i + + +. T cng thc ny ta c th suy ra trng hp 1 2 th 2 2(cos 2 sin 2 ) r i + . Bng quy np ta c cng thc tng qut, gi l cng thc Moivre tnh ly tha ca mt s phc(cos sin )n nr n i n + hoc nu c vit di dng ire th n n inr e . V d: Cho 1 3 z i + . Khi , dng lng gic ca z l: 2 cos sin3 3z i _ + , suy ra2 cos sin3 3n nn nz i _ + , hoc c th vit 32iz e v 32n n inz e 6. Khai cn - Cn bc n ca n v:6.1 Khai cn bc n:6.1.1 nh ngha: Cn bc n ( 1) n ca s phc l tp hp tt c cc s phc tha mn phng trnh nx . Vic tm tp hp y c gi l vic khai cn bc n ca s phc .Gi s cn khai cn bc n ca s phc (cos sin ) r i + th ta cn tm s phc (cos sin ) p i +sao cho n . p dng cng thc Moivre ta tm c np r v 2 kn +.T suy ra: 2 2cos sin , 0, 1nk kr i k nn n + + _ + ,V d: Khai cn bc 3 ca s phc sau:3 33 cos sin4 4i _ + ,Gii:Ta c 3 33 32 24 43 cos sin3 3k ki _+ + + ,Vi k = 0 th 302(cos sin )4 4i +Vi k = 1 th 3111 112 cos sin12 12i _ + ,Vi k = 2 th 3219 192 cos sin12 12i _ + ,6.1.2 Cn bc n ca n v:i s tuyn tnh 1 16Chng 1. Tp hp nh x - Quan h - S phcTa c 1 cos 0 sin0 i + nn 2 21 cos sin , : 0, 1nk ki k nn n + . K hiu 0 1 2 1, , ,...,n l cc cn bc n ca n v. V | | 1, : 0, 1kk n nn trong mt phng phc cc s k nm trn a gic u n cnh ni tip trong ng trn n v. V d: Cc cn bc 6 ca 1 l012345cos 0 sin 0 11 3cos sin3 3 2 22 2 1 3cos sin3 3 2 2cos sin 14 4 1 3cos sin3 3 2 25 5 1 3cos sin3 3 2 2ii ii iii ii i + + + + + + + + Sinh vin hy biu din cc cn bc 6 ca 1 ln ng trn n v. 7. nh l c bn ca i s: Mi a thc bc ln hn hoc bng 1 vi h s phc u c nghim phc. Hay Trng cc s phc l trng ng i s.Tm tt chng:Chng ny gii thiu mt s kin thc nn tng ca Ton hc bao gm: Khi nim tp hp, nh x, quan h, mt s ni dung v php th v s phc. Sinh vin cn tham kho thm ph lc 1, gii thiu mt s ni dung c bn ca logic ton nhm bc u lm quen vi cu trc cc mnh ton hc v mt s phng php chng minh mt mnh ton hc. y l nhng ni dung c bn hc tt cc mng kin thc sau ny.Khi hc xong chng ny, sinh vin phi tr li c cc cu hi sau:1. Tp hp l g? C nhng cch no xc nh mt tp hp? Chng minh tp con v chng minh hai tp bng nhau nh th no?2. nh x l g? Cch chng minh mt php tng ng l nh x? Mun chng minh mt nh x l n nh, ton nh, song nh nh th no?3. Quan h hai ngi l g? C nhng loi quan h hai ngi c bit no thng gp trong Ton hc? Cc khi nim v tnh cht lin quan n loi quan h ny?4. Php th, cc php ton trn cc vng xch v cch phn tch biu din php th thnh tch cc chu trnh c lp? 4. Biu din lng gic ca s phc nh th no? Cch tnh ly tha, khai cn ca mt s phc?Bi tp:i s tuyn tnh 1 17Chng 1. Tp hp nh x - Quan h - S phcA. V tp hp1. Chng minh vi mi tp A, B, C ta lun c:a) \ ( \ ) A A B A B ;b) ( \ ) ( ) \ ( ) A B C A B A C ;c) ( \ ) ( \ ) \ ( ) A B A C A B C ;d) ( \ ) ( \ ) ( ) \ ( ); A B B A A B A B e) ( \ ) A B A f) \ \ ( ) ( ) \ A B A A B A B B 2. Cc ng thc sau, ng thc no ng? a) \ A A ;b) ( \ ) \ \ ( \ ); A B C A B C c) ( \ ) ( ) \ ( ) A B C A B A C .3. Chng minh rng:a) ( ) ( ) ( ) ( ) A B C D A C B D ;b) ( ) ( ) ( ) A B C A B A C c) ( ) ( ) ( ) A B C A B A C d) ( \ ) ( ) \ ( ) A B C A B A C e) ( ) ( ) ( ) ( ) A C B D A B C D 4. Tch Descartes c tnh cht kt hp khng? V sao?5. Gi s X l tp c n phn t v r l s t nhin khc khng b hn bng n. Tnha) S cc tp con ca X gm r phn t;b) S cc phn t ca P(X).B. V php th1. Tm tt c cc php th ca mi tp sau v xc nh du ca mi php th: a)3{1, 2, 3} X b)4{1, 2, 3, 4} X 2. Cho cc php th sau:i) 1 2 3 4 5 66 5 4 3 2 1 _ , ii) 1 2 3 4 5 64 1 3 6 5 2 _ , iii) 1 2 3 4 5 63 4 1 2 6 5 _ ,a) Vi mi php th trn hy xc nh du ca n, tm php th nghch o v du ca php th nghch o .b) Tnh v 3. Chng minh rng:a) Mi php th bc n (n>1) u c th phn tch thnh tch cc chuyn tr dng (k, k+1) trong 1 k n < .b) Mi php th bc n (n>1) u c th phn tch thnh tch cc chuyn tr dng (1, k) trong 1 k n < . i s tuyn tnh 1 18Chng 1. Tp hp nh x - Quan h - S phc4. Chng minh rng mi php th chn u c th phn tch thnh tch cc vng xch di 3. 5. Xc nh du ca cc php th sau:a) 1 2 32 3 1 _ ,b) 1 2 3 43 2 4 1 _ ,c) 1 2 3 4 52 1 5 3 4 _ , d) 1 2 3 4 54 3 2 5 1 _ ,6. Tnh 2 2 1, , , , trong cc trng hp sau:a) 1 2 3 4 5 1 2 3 4 5,2 1 5 3 4 3 5 2 4 1 _ _ , ,b)1 2 3 4 1 2 3 4,3 4 2 1 4 1 2 3 _ _ , ,7. Tm s nghch th ca php th sau, t suy ra u l php th chn, u l php th l:a) 1 2 3 4 55 3 2 4 1 _ ,b) 1 2 3 4 5 6 7 8 91 9 6 3 2 5 4 7 8 _ ,c) 1 2 ...1 ... 1nn n _ ,8. Cho l mt php th thuc nS, chng minh rng 1( ) s ( ) sign ign C. V quan h1. Cho X l tp cc im trong khng gian v O l mt im c nh ca X. Trong X ta xc nh quan h R nh sau:PRP khi v ch khi O, P, P thng hng.a/ R c phi l quan h tng ng trong X hay khng?b/ R c phi l quan h tng ng trong X\{O} hay khng?2. Trong tp cc s nguyn xc nh cc quan h R v T nh sau:a R b khi v ch khi a + b la T b khi v ch khi a + b chn.Hy xt xem cc quan h trn c nhng tnh cht g?3. Cho tp 0 X . Trn tp ( ) P Xcc tp con ca X xc nh cc quan h P, Q, R, S nh sau: P B Q B A\B = AA R B A BA S B A B=A A B AA Hy xt xem nhng quan h trn c nhng tnh cht g?4. Trn tp s thc cho quan h T nh sau: aTb nu 2 2a b Chng minh T l mt quan h tng ng.E. V nh x1. Trong cc nh x t X vo Y sau, nh x no l n nh, ton nh, song nh. Trong trng hp song nh, hy tm nh x ngc. a. , (0, ), ( ) cot X Y f x arc x b. X = [1; 2], Y = [1, 7], 2( ) 3 3 f x x x + c. , ( ) 3 4 | |; X Y f x x x i s tuyn tnh 1 19Chng 1. Tp hp nh x - Quan h - S phcd. 22 4, [0;5], ( ) ;1xX Y f xx x + +e. X = (-1; 0) 1, ( ) ln .1xY f xx+ _ ,2. a ra v d v nh x f, g sao cho a. gf tn ti nhng fg khng tn tib. gf v fg u tn ti nhng khc nhau.3. Cho : f X Y l nh x, A v B l cc tp con ca X, C v D l cc tp con ca Y. Chng minh:1 1 11 1 11 1a. ( ) ( ) ( );b. ( ) ( ) ( );c. ( ) ( ) ( );d. ( ) ( ) ( );e. ( \ ) ( ) \ ( );f. ( \ ) \ ( ).f A B f A f Bf A B f A f Bf C D f C f Df C D f C f Df X A f X f Af Y C X f C 4. Cho nh x : f A B . Chng minh:a) f l n nh khi v ch khi vi mi tp hp X v vi mi cp nh x : , ' : g X A g X A th fg = fgsuy ra g = g.b) f l ton nh khi v ch khi vi mi tp hp Y v vi mi cp nh x : ; ' : h B Y h B Y th ' hf h f suy ra h = h.5. Gi s : f X Y l nh x v ; A X B Y . Chng minh:a) 1( ( )) f f A A v 1( ( )) f f B B ;b) 1( ( )) f f A A , vi mi A X khi v ch khi f l n nh. c) 1( ( )) f f B B , vi mi B Y khi v ch khi f l ton nh.6. Cho nh x 2: fx x a . Hy tm:a) nh ca cc on [-1, 1]; (-2; 1]b) To nh ca cc on [-1, 1], [1, )7. Cho nh x : f bi 3( ) 24 2 f x x x +a) Xc nh ( ) f ;b) Cho A = [-1; 1], xc nh 1( ) f A.8. Cho hai nh x : f A C v : g B D . Gi h l nh x tha: :( ; ) ( ( ); ( )), ( , )h A B C Dh a b f a g b a b A B Chng minh rng:a. Nu f,g l n nh th h l n nh;b. Nu f, g l ton nh th h l ton nh;c. Cc mnh o ca hai mnh trn c ng khng?i s tuyn tnh 1 20Chng 1. Tp hp nh x - Quan h - S phc9. Gi s Xl tp hp cc tam gic, 0X l tp hp cc ng trn trong mt phng.a) Quy tc cho tng ng mi tam gic vi ng trn ngoi tip tam gic c phi l nh x t Xn 0X khng? Ti sao?b) Quy tc cho tng ng mi ng trn vi tam gic ni tip trong n c phi l nh x t 0X n Xkhng? Ti sao?F. S phc1. Tnh cc biu thc sau:3 353a. (2 )(3 ) (3 2 )(4 );(5 )(7 6 )b. ;3c. (2 ) (2 ) ;(1 )d. (1 )e. , .ni i i ii iii iiii n+ + + ++ ++ + +2. Tm cc s thc x, y tha mn phng trnh sau:a. (2 ) (1 2 ) 1 4 ; i x i y i + + + b. (3 2 ) (1 3 ) 4 9 . i x i y i + + + 3. Tm dng lng gic ca s phc sau:a. 5;b. 2;c. -3i;d. 1 + i;e. 1 i;( )f. 3 ;g. 1 2 3 .ii + 4. Bin i v dng lng gic tnh cc biu thc sau:10001503024a) (1 ) ;b) (1 3) ;c) ( 3 ) ;3d) 12 2iiii+++ _+ + ,e) 121 31ii _ + ,5. Tnh cc gi tr sau theo cosv sina) cos5b) sin 7c) cos nv sin ni s tuyn tnh 1 21Chng 1. Tp hp nh x - Quan h - S phc6. Hy gii cc phng trnh sau trn 22222a. ;b. 3 4 ;c. 12 ;d. 5 4 10 0;e. (2 7) 13 0.x ix ix ix x ix i x i + + + + + 7. Vit di dng lng gic nhng phn t ca tp hp sau:6834. ;. 8 2(1 );. 1;. 4;a ib icde. 31 ; i +f. 32 2i g. 38 243ii+h. 472(1 3) i 8. Biu din trn mt phng phc cc tp hp sau:a. { | | 3};b. { | 1 | 2};z zz z i + { }3c. 1 | | 2, arg(z)4d. | 1| 1,| 1 | 1z zz z z i < ' ;