Đại số đại cương hoàng xuân sính

xac dinh mot anh xa tit X dgn Qua dinh nghia mkt anh xa, cluing ta they rang khai niam Emit xa la khdi niam m6 rang cue khai niam ham s6 ma ta gap trvang ph6 thong.. Prong dinh nghia an

HOANG XUAN SINH

DAI SO DM CMG

(Ti] IS Idn thit tim)

NHA XUAT BAN GIAO DUC

0th, Erich nhi&n xuat ban:

Chu tich FIDQT hem T6ng Giam ddc NGO TRAN AI Ph6 T6ng Giarn diic hem Tdng hien tap VU DUONG THVY

Bien tap kin ddu vd tat ban :

TRAN PHUONG DUNG

Bien tap la thuat :

BPI CHI HIED

Lol NO! DAU Cho xugt ban Ian Chit nh&t

Tap II hau nhu ride ldp ddi vai MI) I, man khOng ke den met s6 khai niem nhu Junin vi, pile!) the, ma trdn , ma melt dOi khi de cap den Thy (My, ve mat ngOn nga ua ki hieu cua

ii thuylt tap hop, tap II trung thanh tun tap I

O day cheat& tai khong lam vide thaylt minh tang chuong, ccic ban doc co the xem bdn thuylt minh chuong Dinh DO ad cao cap cim BO GM° dye Sau mai § cita ding chuang, ban doe c6 nhang bhi tap de, hada hilu situ a thuyet hon va ran luyen

hi nang tinh town, hod° hieu rdng li thuylt han, cal mat so khai niem mai dua vim trong bai tap M6i van de duce nhac led thi duoc chit thich chuong, tilt, mac can van a da duce dua vao, chang han ch V §3, 2 co nghia la van a do da nOi din 0 chuong v, tilt 3, mite 2 Nan van a duce nhdc Lai cling chuong cal van de clang xet thi chi' chit thich tilt ,va mac ;

cling tilt thi chi chit thich mac Thing cling mot tilt, cite dinh nghia, bd de, dinh li duoc ddnh 36 bang 1, 2, 3 Cu& cling dl xay Ming mat gulf, trinh Dai el cao cap tang ngity cang Jett han, chang tai rat mong ban dye viii long chi bait nhctng thilu sot_han !thong trinh khoi cart man sach nay Xin cam an PTS Btu Huy Hien 6 td Dai ad elm khoa Than

trttang Dai hoc SW pham Ha NO 2 dd c6 nhieu clang gap lie phan Bat tap

HaHai ngay 13-1-1972 HOANO XI/AN SINN

3

LOI NOI DAU Cho Inuit ban litn thu hai

Thy nhiiu Mn Nha xudt ban Giao dqc dP nght chung tot cho tai ban cuan sach Dai ad cao cdp tap II, nhung chung tot

dd tit chili ui dd co cudn Dai s6 lib S6 hoc caa giao su Ngo Tittle Lanh Nhung trong qud trinh day hoc, cluing tdi thdy coon Dal sd cao cdp tap II duce sink uien Mc trubng DO hoc

Su phqm dung dl On chi, con sink Men cac trubng Cao clang

Su phew' lai clang nhu tai lieu chink khda, cho nen chung tesi nhan lei Mt Nhit xudt ban Gieto due cho in lqi coon Bach nay Tong cudn setch tai ban chung tOi dd lam Mc Idea sau : 1) Chaa lai mat s6 cluing minh hay phat biltt dinh li cho ngan ton han, hay khOng thita

2)- Cho them §3 trong chuang I, nOi so luoc ui cite tien di elta thuylt tap hop, mat dieu can thilt cho ngubi gidng yet cling Mn thilt cho sink vien co tri to mb khoa hoc

3) Them vi du, bai tap ve vanh chink va vanh Oda, hai loci yank d6ng vat 'tre quan trong trong S6 hoc

Chang tot Iran trong cam on cdc.bqn clang nghiep dd co ahung 9 kiln (long gdp va Nho xudt ban Gido due dd nhieu Ian a ngh‘ cho tai ban

Ha NO new 21-12-1994 HOANG XUAN SINE

CHUONG I

§1 TAP HOP VA ANH XA

1 Khai niom tap hqp

NhUng vat, nhung dai tuong twin hoc duoc to tap do mat

tinh chat Chung nao do thinh lap nhUng tap hap Day khong

phiti la mat dinh nghia ma la mat hinh anh true quan cita khai niam tap hop IA thuygt tap hap trinh bay a day la mat It thuygt so cap theo quan digm rigay thc

Ngttai to ndi : Tap hop the hoc sinh trong mat lap, tap hop the lap trong mat trtamg, tap hap N the so to nhien, tap hop

Z eac so nguyen, tap hop Q cite s6 huu ti, tap hop R cac sti time, tap hop C the so phtte

Cite vat trong tap hop X goi la the phan td eim X Kt hiau

x C X doe la "x la mat phgn tit tha X" haat "x thuae X P Phit dinh cim x E X ki higu la x X

'Pa him hai tap hap A va B bang nhau va vigt la A = B kbi va chi khi mai phgn tit thuac A thi thuae B va dim nghia la the quan ha x E A va x E B la twang throng Nhu vay A = B khi va chi khi chting chtla the phan td y nhu nhau

2 BO Win caa mat top hqp

Dinh nghia 1 Gid su A va B la hai tap hap Ta hi hieu

A C B quan he sau day

voi mai x, x E A keo theo x e B

Ndi met each khac, quan he A C B cd nghia la mpi Win tit Ma A dau thuec B

Quan he A C B la quan bao ham, doe la "A chilit trong

B", hoac chat* A", hoac "A la mot be phan cita B" hoac "A

la mot tap hop con dim B" va ngudi ta cling vigt B 7 A Phil dinh vim A C B vigt la A Q B hay B A A

Dinh li 1 Quan he bon ham c6 the tinh chat sau :

(i) Cdc quan he A C B vie B C C keo theo quan he A C C (ii) Mudn co A = B can nit cla a 5 A C B va.B C A Cluing mink (i) Ta hay lay met phan tit thy g x E A Vi

A C B nen x E B Nhung B C C nen x E C fly vol mai x,

x E A keo theo x E C, tut la A C C

(ii) than nhien n

Thttang met be plan A Ma met tap hap B dutqc xac dinh beri met anti chat C nao dd, ma mpi phan tit dm tap hop B

thda man tinh chat C sa la phalli tit ciut tap hem A Ta ki hieu nhu sau

A ={xEBI x cd tinh chat C

va doe IA : "A la tap hop tat ca Mc phan t8 x E B ma x cO

drib chat C "

Vi du - Xet tap hop Z Mc ad nguyen va be phan A the ad

nguyen chgn, ta vie%

A = {xEZI x chia hgt cho 2 }

A - B={xEAlx0B}

gel la higu cart tap hop A oh tap hap B Ngu B c A thl hieu

A - B g9i la phan bit dm tap hqp B trong tap hap A va can ki hieu la CAB

Dinh Ii 2 Gid sit' A vd B lit nhung 60 phdn caa mot Sp hop X, the' thi

(i) X - (X - A) = A

(ii) Cite quan h@ A C B od X-B C X- A lit Wang duong Cluing mink (i) Tap hop X - (X.- A) gem cAo phtin to x E X sao cho x e X - A, We la gam the phan tit x E X sao cho

x e A

(ii) Gia sit ACB.VI quan he x E A ken theo quan ha x E B, nen quan ha ± B keo thee x e A, hie la quart ha x E X - B keo theo quan he x E X - A Dao lai gia sitX-BCX-A The' thi bang If Juan twang to nhu tren ta cd X - (X - A) C

non fly, to cd the' coi X - X va Y - Y chila cat phan tit

y nhu nhau vi chting chano ed phan to nao ea (xin thing col day la mat chitng minh)

Tap hop X - X = 0 khong ph6 thuac van tap hop X, vi If

do do, ta goi nd la tap hop ring Tap hap nay khong c6 mat phan tit nao ca

Ito rang ta c6 0 C X vai mot tap hap X va tinh chat nay

dac trung tap hop rang

5 Tap hqp mot, hai phOn bY

Gia sit x la mat vat Thg thi cd mat tap hap ki hiau {x} chi

gam c6 mat phan to la x Mat tap hop thuac loaf de got la tdp hop mot phan S

Bay gib giA sit x va y la hai vat phan Mat Thg thi cd mat

tap hop ki lieu {x, y} chi gam cd hai plan ta la x via y Mat

tap hop thuae lotti d6 goi la tap hop hai phan

Ngubi ta cling djnh nghta nhu vay tap hop ha ban phan

tit Cite tap hop da ding vol Sp hop rang got la cdc top hap han, con cac tap hop Mac goi la ale Op hap u(5 han

6 Top hop cac 60 ph(in mitt mOt top hqp

Gia sit X la mat tap hop, the t) phan ciut X lap thAnh mat

tap hop Id hiau P (X) va goi la tdp hop cdc bd ph4n cda X

Tap hop thy bao gib cling S it nhat mat phtin tv, d6 la X

'Pa S cluing mirth duac rang, ngu X la mat Sp hop him

han gam n phan tit thl P(X) la mOt Sp hop him han gom 2" phan

tit Nhu vay cac Sp hop 0, P(0), P (P (0)), P (P (P (0))),

P (93 (P (P (0)))), P (P (P (P (0))))) then the W S 0, 1,

2, 2 2 rt- 4, 2 4 = 16,- 2 16 = 65536 phan tit Tit Sp hop 0

chitng ta da - thanh lap nhang tap hop cri nhigu pit to data mdc trong this° to ta Itheng dgm duac

7 71th de cat cUa hai top hqp

Gia sit x vA y la hai vat, tit hai vat nay ta thanh lap mat vat thtl ba ki Mau (x, y) va goi la c4p (x, y) Hai Sp y) va

bt, u) la bang nhau khi va chi khi x = u va y = v Dar Mat

ta c6 (x, y) = (y, x) khi va chi khi x = y, digu nay Si Ian thd

to ma ta vigt hai vat cim mat cap IA can thigt

Ta ro thg mb rang khai niam cap nhu sac Gia sit So ba

vat x, y, z, ta Ott

(x, y, z) = ((x, y),

va got (x, y, z) la mat bd ba Milan S

(x', y', z) = (x", y", z") can va da la

Thvg fly ((x', y'), z') = ((x", y"), z") trong during vdi

(x', y') = (x", y") va z' = z", fly Wring duong voi x' = x", Gang vay, cho bOn vat x, y, z, t ta dat

(x, y, z; t) = ((x, y, z), t)

Ira ta ggi (x, y, z, t) la mat bt) bdn

Dinh ngliia 3 Cho hai tap hop X NIA Y; tap hop cac cap

(x, y) vat x E X vb y E Y goi la tich de the tha X uti Y va

Cac phan tii oda X x YxZ lit the be ba (x, y, z) vet x EX,

y E Y, z E Z Cling nhu vay, the phan to the X x Y x Z x T

la the ba bOn (x, y, z, t)voixeX,yEY,zEZ,tET Cu& ding nen X la mat tap hop, ta dat

X2 =XxX,X3 =XxXxX,X4 =XxXxXxX,

8 Hop va giao ctia hai tap hop

Dinh nghia 4 Gift sit X th Y la hai tap hop Ta ggi la hop mitt X ud Y tap hop ki hien X U Y g6m eac phan tit hoae thug°

X hotm thuae Y, nghia la

z E X U Y trong during vbi z E X hoe z E Y

Ta can e6 thg ndi X U Y gam eac phan tit thuac it nhat mat trong hai tap hop X va Y

Dinh nghia 5 Gia sit X vet Y la hai tap hop 1k ggi la .giao tha X ut2 Y tap hop ld hiau X U Y g6m the phfin ta vita thuac

X vita thug° Y, nghia la

z E X fl Y Wong during veil z E X va z

Ngttbi ta bao hai tap hop X va Y la /thong giao nhau hay

roi nhau khi X fl Y = 0, nghia la khi X vb Y lcheng th phan tit chung nao

9

Rd rang to eat the quan ho

XnYcXvkY,XUYD.XvkY

Ngoai ra, gia sit Z la mot tap hop thy y, mu6n cho Z C X

va Z C Y, can va du coz E X vb z E Y voi Inca z E Z, nghia

la z E X n Y, tilt litzcxn Y Nhu vAy X n Y la tAp hop

Ion nhat trong tat ca the tap hop Z vita chda trong X vita chile trong Y Cling vay, 'math Z chda ca X va Y can va du la Z chda X U Y ; nhtt the' X U Y la tap hop be Witt dada ca X

C, chAng hen x e B Vtiy x E A 11 B, tilt la x E (A n B) U (A n C) Dao lei gia sat x E (A 11 B) U (A fl C), lieu do co nghia la x thuOc it nhat mat trong hai tAp hop A n B, A n C,

chinghanx EA n B, tae lax E A vax EB,vAyx EA va

x e (B U C) do do x E A n (B U C)

Ta ehting minh cOng tilde thd hai caa (Ili) Gia thx EA U

U (B n C), dieu d6 cat nghia la x thuSe it nhat mot trong hai

tap hop A,BnC, chAng han x E A \ray x E A U B Ira xEAUC,titclaxE(AUB)(1(AUC).NauxEBnC thi ta cdx€BviixeC, the la x E A U B va x EAU C,

\ray x E (A U B) n (A U C) Dao lai, gia x E (A U B)

n (A U C), diau do cd nghla la xEA U B va xEA U C

x EA U B cd nghia la x thuac it nhat mat trong hai tap hop

A, B NauxElithixEAU(BilC).NanxtZAthixE B,

vavixEAUC,nenxEC.VityxEBDCvadoddxGAU

U (B n C)

• (iv) Gia sit x E X - (A U B) Dieu dd ea nghia la x e X

va x E X - B, tic lit x e (X - A) tl (X - B) Dtto lai gilt sit

x E (X - A) 11 (X - B), dieu do c6 nghla la x E X - A va

x (it A U B, tile la x e X - (A U B) D61 via tang thdc thd hai dm (iv) ta co the" Ching minh Wong tp, hoac gp dung tang tilde thd nhat mitt (iv) va dinh II 2 (i) ne'u A, B C X Ta xat, via A, B C X

Voi myi x E N ta hay chia x cho m vA (Woe mat s6 du kJ bleu

la f(x), S6 f(x) thuOc Zm Turing ling

xac dinh mot itnh xa td It d6n R

3) Gift sit X = (1, 2}, Y = (a, b, c}

Thong dng

1 I— c

2 1—n a xac dinh mat anh xa tV X dgn Y

xac dinh mot anh xa tit X dgn

Qua dinh nghia mkt anh xa, cluing ta they rang khai niam Emit xa la khdi niam m6 rang cue khai niam ham s6 ma ta gap trvang ph6 thong Cac ham sd ma ta gap 6 trttang phd thong

la nhfing anh as ma - ngugn Ira dich la tap hop Mc sd thtte It

hoac nhf.tng b0 phan cue lt, va 56 ?Tx) Wang ling yeti s6 x lit mat bigu thdc dai ad hay mat bigu tilde throng gist, chAng han :

f(x) = 21c2 — x + 3 hay f(x) = 5sinx

Prong dinh nghia anh xa ta that' cite tap hap nook' va filch

khOng nhat - that lit nhitng tap hop s6 va phan to f(x) toting ttng voi x lei cang khong phil la mat bigu thdc dal s6 hay

luting giac !

Trong Giat doh cluing ta thuting co railing bai tam ye ye d6 thi eta met ham se 6 day doing ta cling hay dinh nghia

da thi eta met anh

Dinh nghia 7 Gia su f : X -• Y BO phan l eta X x Y gem Se cap (x, f(x)) vet x E X goi IA do 'thi oda dnh xe Nhu vay, cho met anh xa f : X -• Y, ta dune met ybe phan

I' eta X x Y cri tinh chat : vbi moi x E X, c6 met vit chi met cap, co phan to that nhat IA x, thuOc r Dim lai, cho met be phan P eta X x YS tinh chat d6, thi r cho ta met anh xa

f : X Y ma d6 thi lit F Cho nen lived ta d6ng nhat anh

xa f vat d6 thi eta S la met be phan eta doh de St X x Y

Dinh nghia 8 Gia au f : X -> Y la met anh xa da cho, x

la met phan tay f eta X- A la met be phan thy 9 eta X,

B la met be phan thy g eta Y TM thi nguoi ta goi :

- pa) la dnh mitt x bdi f hay gid tri can anh xa f tot di dm x

- f(A) = { y E Y I tantaixEAsscho f(x) ei- y} la dnh Cela A bdi f

(B) ={ x E XI f(x) B ) la too dnh todn phan caa

B ben f

Dac HO vet b E Y, r ({b}) {xEXI f(x) = b }

don gian ki hieu ta vidt f I (b) thay elm ft ({b}) va goi la tao

anh town phan eta b bbi f Mei phan tit x E f (5) goi la met tao (ink eta b bel f

Ki hieu f(A) la met diau lam clang vi f(A) chi S nghia khi

A E X RO rang ta c6 f(0) = 0 vdi moi f Ta chdng minh da

Bang the quan he-:

- A C hi(A)) obi moi b0 pia A act X

- B t(rim)) ubi moi b0 phGn 3B oda Y

Nhung ta khong cci quyen, trong cat quan he ay, thay the dilu bao ham bang dau citing Unit Chang han, trong vi do 2)

13

ctla muc 9, neu lay A = thi ta cd ritiotil = {-1, 1} va

B = {-1, 1} thi ta co fir l (B)) = {1}

11 Dan anti - Toan anh - Song anh

Dinh nghia 9 Anh xa f : X —> Y la met don anh nau Vol moi x, x' E X, quan he f(x) = f(x') keo theo quan he x = x' hay x m x' keo theo f(x) # fix') ; hay vai moi y E Y e6 nhigu nhgt mot x E X sao cho y = f(x) Nailed ta can goi met don anh

f : X , 1 7 Ia met anh xa mot ddi mat:

Vi du 1) Xet anh xa

Dinh nghia 10 Ta bao mot anh xa f : X s Y la mat tam anh niu f(X) = Y, nci mat each line, Mu Si moi y E

it nlifit mat x E X sao cho y = fix) Nguai ta con goi mat than anh f: X — Y la mat drift xa fit X len Y

Cac anh xa trong cac vi du 1) va 2) la nhung than anh

Dinh nghia 11 Ta bao mat anh xa f Y la mat song anh hay mat anh xa mat d6i mat tit X len Y, Mu ud vita la

don anh vita la town anh, ndi mat each 'Khoo Mu vdi mai y E Y

co mat S chi mat x E X sao cho y = fix)

Chang hon anh n (tong nhat 1, la mat song anh via mai X

Dinh li 4 Cid sit cho

f:X—>31,g Y Z, h Z—>T Thi thi

h(gf) = (hg)f

Ta bdo phep nhan ode arch sa c6 tinh chit kit hap

Chzing mink Ta cd Si mai x E X :

Dinh nghia 13 Gia sit f : X Y va g Y X In hai anh

xa sao cho

gf = lx Ud fg = l y Th6 thi g goi la mOt dnh xr) ngucc clic) f

Tit dinh nghia ta suy ra f cling 121 mOt anh xa %edge ohs g

Dinh sau day cho ta bi6t khi nao mot anh xa cd anh xa tigurele

Dinh li 5 Anh xa f : X —> Y cO mOt anh xa ngitqc khi va chi khi f la mr)t song dna

Chang mink GU sit f cd mOt anh xa nguroc g Y X

Theo dinh nghia 13, ta cd

f la mOt don anh Bay gio gia sit y la mOt phan td thy

g cala Y Dht x = g(y) C X trong clang thdc f(g(y)) = y, ta ddoc

y = f(x) V4y f la mOt to/in anh

Dito lai gia sit f la mOt song anh Quy Sc oho thong ling

v6i m6i y e Y phan tit duy nhat coa f l (y) xac dinh Wit anh

xa g : Y X va ta thay ngay

gf = l x va fg = l y • Nhu vityf:X—n Ycci mOt anh xa notoc khi va chi khi f

la song anh, va trong trddng hop dd ta cd mOt anh xa ngdgc

g : Y —*X cua f xac dinh b61

y 1-r g(y) = x, sao cho fix) = y

Ngoai anh xa nmierc nay, f can co anh xa ngutoc nao khac

khOng ? rlh co

Dinh li 6 GM set g : Y X uh g' Y X Zd hai anh x¢ nguye cast f: X Y The thi g g'

Cluing mink Ta c6

gf = 1 x fg' = 1 y

Tit do

g = g y = g(fg) = (gf)g' l xg' = g' • Nhv yay nab f : X —1, Y co anh xa nguac thi anh xa ngttac

la duy nhat, xac dinh bed

y x, arta x la phtin td duy nhat caa f 1(y)

Do lam clang ngathi ta cling ki hieu phan td duy nhat x cita

1(y) bang f ya do dci ngmai ta Id hieu anh xa

la anh xa ngttoc oda f, bang f 1 Vi f la anh xa ngtotc oda

g = f , can anh xa f goi la cat and rOng elk g tren tkp hop

A

X

17

= xy

Ta bac the phan tU xa , x13 , xy thanh lap mOt hp nhung phan td cda X dupe dank s6 bdi tap hop I, Id hiau la (xa)a E can tap hop I gal la tap hop chi s6 Nau the xa , x13 , xy la

nhiing tap hop thi ta gal (xa), c la nt9t hp tap hqp dcinh s6 bdi tap hop I Ndu cac phan tit dm X la nhUng ba phan cim mat tap hop E, tire la ta ed xa , xp , xy C E, thi ta got

(xa)a e / la met hp nhang by phan mia tap hop E

Thoc re cluing ta da thay vide danh ad trttoc day rat Trong

&tat dal chung ta thuong xet nhung day s6 Utile n o, u 1, u2, Digit &I co nghia lit ta da danh sd bang clic s6 to nhien 0, 1,

Vi du Gia sat I = { 1, 2, 3 ), X = { a, b } va do dci

15 Dap, giao, tich de cite mot hp tap hqp

15 day, chang ta hay mo rang the phep toan hop, giao, tich

ra mat s6 thy 3: flitting tap hop

Dinh nghia 15 Gigt sit (2(a)aei la mat h9 tap hop lb Ail

la hop cua hp d6, va kt lieu bang U Xa tap hop the x sac

a E

cllo x thuac It nhat mat tap hqp cua 119 (Xa)aef

Dinh nghia 16 Gigt sii (21a)aci la mat h9 tap hop Ta goi

la giao edit hp c/6, va Id hiau bang r) X , tap hap die x sao

ezel

cho x thuac tat ca cite tap hop cart h9 (Xa)aei

Dinh nghia 17 Gilt sit (20aei lgt mat 119 tap hop va

X = U X la hop cna ho dd Ta goi la tich d8 cac cua hp

a E /

(Xa)ae 1 !va U hieu bang 11 Xa tap hop can 119 (xa)aer nhilng

ael

phan to cart X sao cho xa E X a via mot a E L N6u cdc tap

hop 2c, den bang mat tap hop A, till tich d0 cite cua 119

(Vac/ goi IA idy thita 'de cot b4c I cua hlp hap A va ki lieu

la

Trong trueng hop I = (1, 2} ta lai tim thity hop, giao, tich

de the cua hai tap hop

Vi du 1) /Cat ho tap hop (In)ne N danh s6 bat cite s6 tO

nhien 0, 1, 2, v6i

In = (0, 1, n},

n E N (1 I = In = {0}

neN

2) Lay thita de cite RN la tap hop clic day s6 Rule

(n o, , 14 )

19

BAI TAP

1 X& tap hap {A 1 , A 2, , ma aid phan to A t , A 2, ,

la nhitng tap hap Chang minh ed it nit& mOt tap hop A i kliOng chtla mOt tap hop nao trong cac tap hap con lai

2 Cluing minh to S A - (A - B) = B khi va chi khi B C A

3 GiA sit X 14 mOt tap hop cd n phan tit va r la mOt s6 to nhien, 0 C r C rt Tinh :

a) S6 cac b6 phan Se X ed r phan

b) S6 cac phan to cila

4 Bi4u than hinh hoc cac tap A x B voi

a) A= {x ER I 14X4 3}

B = R, tap hap cac s6 thitc

b) A = B = Z, tap hop the s6 nguyen

5 Bigu din hinh hoe tap hop X em the diem (x, y) caa mat pang dg cac S clang (x, x) ved 0 x C 1 hoc S clang (x, x + 1) v6i x 3 0

6 Chdng minh •

a) A UB =A khi va chi khi B C A

b) A f1 B = A khi va chi khi A C B

9 Tap hop

1 G= {( x x —1)1 x R, x

cd th6 coi nhu da thi caa mat anh xa the Tao ? Bleu diOn hinh hoc tap hop dd

10 GM wit f : X -> Y la mat anh xa A va B la hai bQ phan

dm X, C va D is hai bQ phan cua Y Chung minh

f ad phdi la don anh toan anh, song anh kitting ?

12 GM saf:X-.17 vag:Y->Z1a hai anh xa vah = gf

la anh xa tich oda f va g Chung minh :

a) INI6u h la don anh thi f la don anh, lieu them f la than anh thi g la don anh

b) Neu h la toan anh thi g la than anh, nen them g la don anh thi f la toan anh

13 Cho anh xa f : X -> K Chung minh f la mat don anh khi

va chi khi cd mat anh xa g : Y -> X sao cho gf = 1x (X m 0)

14 Cho anh xa f : X -> Y Chung minh f la mat than anh khi va chi khi cd mat anh xa g : Y X sao cho fg = l y

15 Cho ba anh xa f : X Y va g, g' : U -> X Chting minh : a) Neu f la don anh va fg = fg', thi g - = g'

b) bieu veil moi g, g' ma fg = fg' keo theo g = g', thi f la

mat don anh

16 Cho ba anh xa f : X -> Y va h, Y -> Z Chung minh Yang neu f la mat than anh va hf = h'f thi h = h' Nguac lai

21

la mat song anh

19 Gia sa (Adaei ut mot ho !Ailing Ith pilau tha mat tap hap X, B la mat tap hap thy y Chang minh

a) U Aa D A„ vbi moi a e I

a) C6 met song anh tit Horn (X, Y) den YX

b) Neu Y chi c6 hai phtin to thi e6 met song anh tit

Horn (X, Y) den P(X)

c) Ttif a) va b) hay suy ra neu X ad n phan tif, thi P

co 2" phtin tit

§2 QUAN HE

1 Chian hai ngoi

Trong §1, 9 thong ta da dua vao khai niem anh xa Met

anh xa f : X Y cho thong dng voi mei phan to x E X met phtin t0 y = f(x) e Y Nhu vay,cac phtin to x, y c6 met.quan

he viii nhau, quan he dri la y = f(x), hay ndi met each khae, quan he do IA (x, y) e G, yeti G la dd thi tha f Vag plat

hon, ngt.tai ta nghi den met each ghep cap nhang phtin to curt

X yea nhung phan to caa Y de thanh lap met be phan caa

X x Y, va goi each ghep cap do la met quan he hai ngei Dinh nghia 1 Gia sit X Ira Y la nhilng tap hop Met quan

he hai ng0i t0 X dgn Y la met be phan S cua tich de the

X x Y Ta bao, vai hai plain to a E X va b e Y , rang a c6

quan he S yea b ngu va chi neu (a, b) E S Ta viet aSb D5 thi G aim met anh xa f X Y cho ta met vi du ve quan he hai ng6i, va tong throng hop nay ngutoi ta vial b = ;(a) dui khOng vie% aGb Met anh xa cho ta met quan he hai ng6i,

nhung dao lai khang dung (§1, bai tap 5)

23

Nhu \Tay, mat anh xa cho ta mat quan ha hai ngbi dac Mat Ngoai quan h@ hai fled quan trong nay, Wan h9c con cd hai loaf quan ha hai ngdi quan trong nita, do la quan he tudng during va quan he thti to

2 Quan ha Wang Wang

Dinh nghia 2 GM sit X la mat tap hop, S IA mat b0 pan oda X x X The thi S goi la mat quan he Wang duong trong

X ngu va chi ngu cac digu kien sau day Woo man :

1 (Phan xa) Vdi moi a E X ; aSa

2 (D6i ming) Vdi moi a, b E X ; ngu aSb, thi bSa

3 (Bac cam) Vdi moi a, b, c E X ; ngu aSb va bSc, thi aSc Ngu S IA mat quan he tuong duong, thi nguoi ta Hwang ki hiau S bang - va thuong doc aSb (a b) la "a Wong during veil b"

Vi du 1) Dgu bang thudng dung trong 86 hoc thong Uniting

We s6 thtic lit mot vi du quen thuac ye quan he Wong throng

'Prong trudng hop dd tap hop S la dyeing thAng y = x cua mat phAng de can R 2

2) gh xet quan he &Mg du mod 5 chang han trong s6 hoc

Hai so nguyen tn, n gal la clang du mod 5 ngu m - n chia hgt

cho 5 RO rang quan he nay la mat quan he Wong during trong

Z Ta hay ki hiau bang C(i), i = 0, 1, 2, 3, 4 tap hop cac s6

nguyen tag duong vol i

C(i) = {5x + i I x E X},

4

thg thi moi s6 nguyen thuac U C(i) va C(i) fl C(j) = 0 voi

o

i # j, i = 0, 4 ; j = 0,1, , 4, 'lb se Wily We digu kin

taring to nhu vay cho mat quan he Wing during thy Y

3) 1k xat tap hop X cac vecto trong knang gian coa Hinh hoc giai tich 1k Wm mat vecto a co quan he S voi vecto p khi

va chi khi a ding hudng, ding chit, ding m6dun vdf j3 Quan

he S r6 rang lit mat quan he Wong duong rIlt cling ki hiau

bang C(a) tap hop cac vectd tudng during veil a, the thi C(a)

chang qua la mat vectd to do

Dinh nghia 3 GM sit S la mat quan he toeing during trong

X Va a E X Tap hop

C(a) = Ix E X I xSal • gal lit lap Wang duang cast a den obi quan he Wong duang S

VI S la phan xa nen a E C(a)

Ta thy tac khac rang C(a) co cac tinh chat sau :

(I) C(a) r 0

(ii) x, y E C(a) keo theo xSy

(iii) x e C(a) va ySx keo theo y E C(a)

136 de 1, Vbi hai phan tit bat Id a Os 5, ta den co hails C(a) C(b) = 0 hods C(a) = C(b)

Cluing minh Gia sit C(a) n C(b) # 0 Ta se &sang minh

C(a) = C(b) Goi c mat digm thuac C(a) fl C(b) Ta co cSa

va cSb, va do Grill chat clai ring in bac cam, nen a E C(b)

Do do vai moi x e C(a), tdc la yea moi x Mang duang poi a,

to dew ed x E C(b), tue la C(a) C C(5) Thong to, ta cluing minh C(b) C C(a) fly, ta c6 C(a) = C(b) n

Tit be de trail ta suy ra ngay C(x) = C(a) vai moi x e C(a)

Dinh nghia 4 Ta boo to 'awe hien mat sit chia lop tren rnOt tap hop X khi ta chia nd thanh nhiing Ina phan A, B, C, khan 0, r ii nhau tang del mat, sao cho moi phan tit ciao X thugs mat trong cat 60 phan do

Dinh fi 1 Gid sit' X la mat tap hop, S la mot quan he Wang duong trong X Thd thi cite lop Wong duang phan biOt aria X dee vOi S thanh lap mot sit chia lop tren X

Oning minh That vey, yea moi x E X, ta ed x C C(x) Con

hai lap tieing dung phan biet la rbi nhau thi do 136 de 1 n

Nhu vay cho met quan he Wong dining S trong mat tap hop X, ta dude mat sit chia lap tren X, do la vise chia X

thanh cac lap Wong &tong Dinh li sau day cho ta thay dao lai sung

Dinh li 2 Gid su ta co mot sy chia 16p tran mot tap hop

X ail A, B, C, la cac ba phlox caa X do su chia lap The thi

co met quan ha twang duong duy nhat S trong X sao cho cac lop twang duong caa X clai adi S la Sc ba phan A, B, C,

Viec chUng minh dinh li tren, xin dash cho doe gia, xem nhu bai tap

Dinh nghia 5 Gia atl X la mat tep.hop, S la mat quan he Wang duong trong X Tap hop cac lop twang duong phan biet cila X ddi vol S gni la tap hap thuang act X trail quan he

twang duong S va duac ki hieu la X/S

Vi du Xet quan he &Mg du mod n trong SO' hoc (n la mat s6 nguyen throng oho trtidc) Hai s6 nguyen x, y gal la dang du mod n ndu x - y la bai coa n Rd rang quan he nay la mat quan he tuang diking trong Z Tap hap thuang cua Z Hen quan

he ddng du mod n ed n Idp Wang during :

• C(0), C(/), , C(n- 1)

voi C(i) = fnx ilxe = 0,1, n - 1

3 Quan h0 thii tir

Dinh nghia 6 Gab sit X la mat tap hap, S la mat ha phan cfm X x X The thl S dune goila mot quan he thil ta

trong X (hay ngubi ta con gal S la mat quan he thit to gida cac phan tQ mia X) ndu va chi ndu cac didu Man sau day thda man

1 (Phan xa) Vol moi a E X : aSa

2 (Phan ddi ring) Vdi mai a, b E X ndu aSb va bSa, thi

a = 6

3 (Bac eau) Vol mai a, b, c E X ; aSb va bSq thi aSc

NgUdi ta bao mat tap hap X la 84 this to ndu.trong X co' mat quan he thu to

Vi dy 1) Quan he E trong tap hop cac s6 to nhieh N la mat quan he thu to

2) Quan he chia het trong N : ngded ta Id hieu al b va dee

"a chia het b"

3) Quan he bao ham C gifta cac be phan eta met tap hop X Trong vi du 1), vol a va b try y to luon cd a g 6 Ileac

a NgUdi ta gel met quan he that to nhu thy la town phan

Trong vi du 2) khong plied ta hien luen cd al b hoac It( a ved a,

b thy y, clang Ilan vai a = 2 VA b = 3 Dieu dd cling thy ra vat vi du 3) Ngudi ta ban I vat C la tinting quan he thd to

60 ph4n

Neu S la mot quan he thd tut trong X, thi rived ta thudng

Id hieu S bang g (bat chub° ki hieu quan he thil to thong tinging eta s6 nguyen hay s6 flute) va doe a g b la "a be hon

b" Ngubi ta coi b ?•.- a la clang nghia vet a g 6 va doe la "6

Ion hem a" Ngdoi ta con viet a < b (hay b S a) quan he

va a # 6" va doe la "a thtte su be ban b" hay "6 thAlc ski lon hon a"

Dinh nghia 7 Gia su X lit met tap hop sap thd W Met phgn tit a e X gei lit phdn to t6i ties (phan to t61 dal) eta X

neu quan hex # a Ix a) kat) theo x = a

Vi d, 1) Trong tap hop cat se to nhien thve stt len lam I, sap that to theo quan he I (quan ha chia het), the Olin tit tOi tigu la the s` nguyen

2) Trong tap hop cat he theta doe lap tuyen tinh eta khong gian vecto IV sap tint W theo quan he bao ham C, cac ha

\recto gem rt \recto la t61

3) Tap hop cac se thuc, ved quan he that to thong thudng, khong cd plign to t6i dai cling khong cd phan to tei tidu Dinh nghia S Girl sit X la met tap hop sap that tu Mot phgn tit a E X get la phlin tti be nit& titan to lan nhgt) coo

X nen, vet mot x E X, to cd a E x( x g a)

Neu met tap hop X sap thd to cd met phtin tit be nhat a thi a la phtin to be nhgt duy nhgt That thy gib sit 5 6 la phtin to be nhgt, ta suy ra a < b va 6 # a, We la a = 6

Cling nhan get nhu thy del veil phan ta len nhgt

27

Vi dzt.• 1) Tap hop the s6 to nhien sap thit tp theo quan he

eq phtin tit be nhgt IA 1 va OHM to Ion nhat IA 0 Neu sap

• OM W theo quan he this to th6ng thuiting, tap hop eac so to nhien ea phgn to be nhgt IA 0 va khOng c6 phan tit lon nhgt 2) Gia sit X la mot bq phan khac rang thit P (E), tap hop the ba phan elm mat tap hap E Mu X act phan W be nhgt A

(Ion nhgt) ddi obi quan he bao ham, thi A ellang qua la giao

(hop) eim the tap hop thuac X DM) lai, neu giao (hop) the tap hop the X 1p.i thuiic X, thi dO la phtin to be nhat (Ion nhat) eaa X DO.c bi0t, 0 IA phg.» tit' be nhgt va E la phan to len nhat cua P (E)

3) Tap hop the s5 thoc, vbi quan ho UM to thong thuong, khOng ca phan to be Mt) Ming khOng ca phan tit IOn nhgt

Dinh nghia 9 Ta bao mat tap hop X IA sap Mit tit t6t trau nti la sap thu to va nau moi ba phan khae rang ella X ca mat phan tit be nhgt

Vi dp Tap hop the so to nhien N vei OM to thong thuOng

la sap Hui to tot

BAI TAP

1 GM sb f : X —> Y IA mat anh xa, S la ba phan aim X x X

Om the cap (x, x) sao cho f(x) = flx it e

a) Chung minh S la mat quan he Wang during trong X b) Xet tap hop thurang XIS va anh xa

nghia la f = TP •

c) Chang minh f la don anh va trong twang hap f la toan anh thi 7 la song anh

2 Xet tap hop the s6 nguyen Z va tap hop N' cac s6 tor nhien khac 0 GQi S la quan ho trong Z x the dinh bai

(a, b) S (e, d) khi va chi khi ad = be

Chang minh S la mat quan ho tuang thong

3 Gia th S la mat quan he hai ng6i the dinh trong tap hop cac s6 nguyen Z bai cac cap (x, y) vol x, y nguthn va x + y 16 Chang minh :

a) S khOng phai la 116 thi Mut mat anh xa tit Z dth Z

b) S khong phai la mat quan Wang (Wang

c) S khang phai la mat quan ha dui to

d) Nth d61 gia that mat ehtit bang each cho x + y chain, thi the' nth ?

4 Gia sit X la mat tap hop va T la mat qtian ho hai ng6i phan xa, dot 'tang trong X Th hay xae dinh mat quan ha hai ng6i S trong X nhu sau : thy khi va chi khi cd x t = x,

x a = y sao cho x 1 Tx2, x2 Tx3, , xn_ i Tx n Chang minh

a) S la mat quan he Wang diking va T C S

b) Val moi quan he tuang dttang H sao cho T C H thi S C H

5 Xet quan ha hai ngai trong tap hop cac s6 thtic R xae dinh bai tap hop X (*1, bai tap 5)

a) Hay b6 sung X d6 cd mat quan he hal fled phan xa, d6i xang T (tat nhien b6 sung it nil& !) Mau din hinh hoc T

b) CO T, hay xay dung quan he Wang throng S nhu bai tap 4) Brett din hinh hoc S

6 Gia th X la tap hop cac ham s6 kha vi xac dinh trail tap hop cac s6 tithe R GiA sa S la quan ha the dinh bai ySz khi

va chi khi dyldx = dzldx vai mai x E R S cd phai la quan

ho Wang ding khOng ?

29

7 Cho X la khong gian ba chigu thong thuang va 0 la mat digm c6 dinh dm X 'Prong X ta xac dinh quan he S nhu sau : - PSP' khi va chi khi 0, P, P' thAng hang

a) S cd phai la quan he thong (thong trong X khong ?

b) S ad phai la quan -ha tuong <Wong trong X - {O} khangl

Ngu phai, the dinh Mc lop Wong throng

8 Gia sit X lit mat tap hop va S la mat quan ha thil to trong X Chung mirth rang quan he T trong X the dinh bed khi va chi khi bSa thing la mat quan ha thil to trong X

9 Gia sit X lit mat tap hop va S la mat quan ha tuong diking trong X Chung minh S khong phai la mat quan ha tilt/

ta

10 Xet tap hap X = Nn (n ?- 1), vai N la tap hop the s6

to nhien 'Prong X ta xac dinh quan he S nhu sau :

(a 1 , , O) khi va chi khi (a 1, .; = (b 1 ,

host cd mat chi se i (i = 1,2, , n) sao cho ¢ t = b i ,

a i-i bi-1, a.< bi - -

Chung minh S la mat quan h@ Ulu to toan phtin

11 Gia sit f la mat don anh tit mat tap hop X cten tap hop the se tg nhien N va S la mat Tian he trong X xac dinh nhu sau :

xSx' khi va chi khi f(x) s fix)

Chung minh S la mat quan he tht/ W than phan

12 Cho hai tap hap X va Y Gal ft (X, Y) la tap hop the anh xa tit the ba phan elm X den Y, nghia la neu f E t 0C,

thi f IFt mat anh xa cd ngudn la mat ba phan dm X va cd dich

la Y Xet quan ha S trong 'V (X Y) the dinh nhu sau :

fSg khi va chi khi g la ma rang cim anh xa f

a) Chung minh S la mat quan ha thu tg

b) Tim to phtin to UR tido, tai dai, be nhgt, ldn nhgt cda (1• (X, Y) ddi vai S

13 Chung minh nth a la phtin tit be nhat (len nhat) cua

mot tap hop X del thi mot quan he thti td S, thi a la phein

to tei lieu (t6i dal) duy nhat eim X

14 Cluing minh nth X sap Hill ttt t6t thi X sap thu toan phan

15 Cluing minh cac tap hop trong the bid tap 10) va 11)

la sap tint to tot

§3 SO LUOC VE CAC TIEN DE CUA Li THUYET TAP HOP

1 Mb dgu

Nhu ta da ndi a dam chuong, li thuyet tap hop ma ta da trinh bay la mat If thuyet so cap theo quan diem new this Bay giO chting ta hay gioi thigu so Moe the tien de dia li thuyet tap hop Ban disc mthn tim hieu ki ea the tham khao nhieu sach, ehang ban "Li thuyet tap hop" cua Bourbaki

2 KW nidm nguyen thily

Chung ta khong Binh nghia cha tap hop : ta goi nd la khai niem nguyen thdy Khdi niem thuec yap &toe ki higu nhu ta

da biet bang e, cling la met khai niem nguyen they

Thy hai khai niem tap hop va thuoc vao kheng doge dinh nghia, nhung ehting to lai td an dinh the quy the : dieu gi ta

ed the lam duoc va diet' gi ta khOng the lam dude vai hai khai

niem dd DO la bay tign de &la Zermelo - Fraenkel, ma chting

ta them vao cal thtl tam, tien clg Chan

Theo iruyen thong, cac ties de dd throe cho dtith ten va tint

to sau : tien de awing tinh, tien de tuyin 1ga (hay con goi tien de nOi ham, hay tien de chi re, hay tien de tach), tien de cap, tien de hop, tien de tap ling Sc b0 ph4n, tien de ue han

31

(hay tien de the so tg nhien), tien de ehon Ira sea Ming la tien

da thay the Tien de mei cimg con g9i la tien de thay chi chi tham dv vao li thuyet tap hop khi dua vao kited niem quy nap sieu han va s6 hoe tin/ tut Tien de dd disc Fraenkel dua vao nam 1922 He thong tien de (khong ad tien de chant thanh lap

li thuygt tap hop eua Zermelo - Fraenkel Mat vai trong can tien de dO n6i len nhiing tinh chat it nhieu hien nhien khi ta phat bigu cluing trong ngen ngit thong thyong

Triton het cluing ta hay xac dinh the So lit met vat ban toan hoc Bang the chit (16n, nhe, la tinh, hi lap, v.v.:), the ki hieu tha logic kinh dign, hai ki hieu E va =, va the Su ngoac, chMig ta cci the vigt bat ki van ban toan hoe nao (ma chting

ta cling gel lit Gong thus, phat bleu, menh de, khang dinh, v.v ) can thuygt tap hop

Chung ta se gel la ph& bigu toan hoc hinh that cal ma ta dude bang each 4 dung the quy the sau

1 ° ) x E y, A = B, trong dO x, y, A, B la Mating chit thy y,

la nhang phat bigu Cac plaid Mu del g9i la Sang tong thdc

so cap hay nguyen ta

2°) Mu P vet Q da la Sang phat bleu, the thi

(1P), (P A Q), (1) v Q), (P Q), (P ti Q)

la nhang phat bigu

3° ) Mu P la met phat bigu, the thi (B x) (P(x)), (Vx) (P(x))

Chzi y : - Gia sit X, A, B la nhfing tap hop Chung ta da

Si rang quan he X e A dot la X thuee A, hay X la met phan tit Ma tap hop A Thong thuang nivel ta hay vigt quan he do

dual clang x e A, nhung chi chi la met att lam dung each vie% trong khi x la met tap hop cling nhtt A

4 Tien de tuyan Ipa hay nai ham

(VA) (3B)(Vx)ix E A A P(x)) x E B]

True gide, dieu dd cd nghla cho met eeng that P(x) trait met tap hop bin x, ton tai met tap hop B, ma the phan tit la cat ph&n tit dm A, ed tinh chat P(x) (nghia la lam cho c6ng than P(x) dung) Tap help B Mc dd ducic xac dinh duy nhat bai tion

de quang tinh Trot& khi phat bleu Hen de tuyen leta duoi clang

da cho, nhieu nha town hoc da nghl rang chi can met cang that

la do de xac dinh met tap hop :

la tap hop the vat lam cho thug thile P(x) dung

Nam 1901 Bertrand Russel khaim pha noel ta cd the suy

ra met mau than t8 (4.1) bang each xet the vat hi:tang thuec chinh nd That tray, xet cting thite x x KM dd ehting ta mu& xet tap hop the tap hop khang thuec chinh nd Gift sit

JO hop dd ten tai fly ta ed the vial

(3 BRVx)(x E B » x & x) Lay x = B, ta di den thug thite :

BEB4-13 (4 B

ye mat lich sit, chinh nghich li dd da d&n tai Ong thtic hien nay tha Hen de tuyen Ma

Nghich 11 B Russel doa trait thing thtic x x, dua cluing

ta toi dinh li eau day

Dinh li 1 - Khdng ton tot tap hop ma cdc vOt act n6 14 cdc tap hqp

Chang mirth That fly, gia sit met tap hop A nhu tray ten tai Theo tien de tuyen Loa, ta c(5 the xac dinh tap hap B ?is; :

B=jseAlx0x)

Theo dinh nghia dm A, tap hop B la met phfin tit eim A The thi ta c6 B E B nen va chi neu B B, diet' nay man

than vi P va 1P khong the thing thoi dung (nguyen tge khAng mAu thugn),I ♦

Th vita they Id hieu clang (4.1) khOng MI& thief chi ra mOt tap hop Nhung H hieu d6 lai rat Hen De mixt chits Mt den

do, ta dva tho mOt tit nguyen fluty m6i, do IA khai niem ChOng ta Me (4 1) kf hieu lop the tap hop x thee man tfnh chat P(x) Vi du :

fx I x = x}

la lop tat cd cat tap hap Lac do ta se n6i den 16p cac nh6m, 16p cite khang gian vecto tren mat trubng

5 Tien de cop

(Va) (Vh) (3 c) (Yx) [x E c 4-4(x =aVx=b)]

Trite ghic dieu, do co nghia cho hai tap hop a va b cd met tap hqp c co Olen tit IA a vb b Ica chi Chung Tap hop do la duy nhgt theo flan de quang tinh

cap (khong xep tint thanh lap beli a va b, tap hop Id hieu

fa, b} xite Binh MI Hen de cap Ngtati ta gal la don to thanh lap Mi tap hap a, tap hqp 4a, a} = {a} chi cd tap hop a IA plain to duy nhgt

cap sap dui hi the a (thing thtt nhgt) va tha b (dung thd hai), cap thanh lap bai don to {a} va cap {a, b} Cap d6 Id hieu :

(a, b) = { fat, ta, b}

6 Tien de hqp

(VA) (3 B) (Vx)fx E B »(3 C) ((C eAnxe C))]

nye gide diet' cO nghia cho met tap hop A ma the !Men tit gam nhung tap hop ki hieu C, c6 mOt tap hop B ma the phgn tit IA the tap hop the van met trong the tap hap C ciut 130 (= tap hap) A Tap hop dcl IA duy Hi& theo Hen d@ quang filth

ya goi la hop eim the tap hop cim be A va Id hieu

U C hay U A hay U ,{C I C e CEA

7 Tien de tap hqp hoe be phen

(VA) (3 B)(VXUX E B »X C A) Trirc Mac ed nghia cho mat tap hop A, cci mat tap hop B

ma the phgn tit la the ha phan ciaa A Tap Imp &I la duy nhgt

theo nen de aiding tInh va goi la Op hop Sc b0 plz4n can t6p

hop A, ki hiau P (A)•

he dci is mat tap hop khong reng

Dieu do tong c6 nghia rang mei ho khong rang (Xi ) ; e nhitng tap hop khong reng c6 mat ham cher', nghia la mat anh

xa p xac dinh trong I sao cho voi e I, T(i) E

Paul J Cohen da cluing minh rang flan de chum la clac lap del yea cac Hen de coo li thuygt Zermelo - Fraenkel Truck d6

Gadel da chting minh nett II thuyet tap hap cart Zermelo -

Fraenkel la khong matt than, n6 se khong mau thugn neu ta

them van den de chon

'Prong bail giang Dai sg, chting ta hay dung b6 de Zorn va dinh II Zermelo (thu tar tat) Tien de ellen Wong dttong ved cac phat bieu d6

B6 de Zorn - Moi t4p hop E khOng rang sap thzi ikt quy

nap co it nhtft mot phan td t6i dot

9 Tien de ye hen

Djnh nghia 3 - VOi MO tap hop x, ta get la eel ke tiip

ctla x, tap hqp co the philn tit la cac phgn tit eua x va tap

35

hap x Vay dd la hop cua tap hop x va don to {x} Nd &roc

ca cac s6 to Mien do ? Tien de vd han se tra ]di eau MI dd

Tien de u6 hen la nhu sau :

(3 A) [ 0 E A A (Vx) (x E Aix` e Troc giac dieu dd c6 nghia din tai mot tap hop chda 0 va chda cal lid tiep cita m61 phan tit cua no

Ta cd thd do ding chting minh dinh sau day :

Dinh li 2 - Co met tap Op be nhat, ki hieu N, cd the tinh chat sau

(1) 0 E N,

(ii) Voi moi x e N, x+ E N

Tap hop N goi la tap hop the s6 to nhien

10 Tien de thay the

[(Vx) BY) (Vy) ((x E A) A Six, y) e y G Y)]

CI-WONG 11

§1 NIJA NHOM

1 Phap twin hal ngOi

Lam Dai sd, chu ygu la tinh WEIL ma vi du dign MS la ban

phep town dim S6 hoc s cap Tilde that ma Si lam mat phep

toan dai s6 tren hai phan td a, b cua Ming mat tap hop X IA cho Wang dug Si cap (a, b) mat phdn tit the dinh eta tap hop X Ndi mat each khac, cho mat phep toan 41 sd la cho mat anh xa to X x X dgn X

Dinh nghia 1 Ta goi la pile], tam hai ngOi (hay con gal

St IA phep San) trong mat tap hop X mat anh xa f to X x X

dS X Gig tri f (x, y) eta f tai (x, y) goi la edi hap thttnh cita

x Ott y

Cal hop thanh cua x va y thutmg ddoc ki hiau bang each vigt x va y theo mat thd tut nhat dinh Si mat dau 4c trong chc phep toan dat gifta x, y Trong the dflu ma ngdgi ta hay

dung tot nhigu nhgt, la cac dau + va ; dal vai dgu ngurdi

ta thliong quy tallc be di ; vgi cac dgu do, cal hop thanh eta

x va y duos vial x + y, x . y hay xy Mat SS toot hai ng6i

ki hiau bAng dau + goi la phep tong, cal hap thanh x + y lac

dd goi IA tong oda x va y ; mat MIS toan hai ng6i ki hiau bhng dau goi la phep nhan,cai hop thanh x y hay xy lac dd goi la tech eta x va y Ngubi ta can dung cac ki hiau x o y,

x * y, x T y, x 1 y dg chi cai hop thanh cua x va y

37

Vi du : 1) Thong tap hap N the s6 4t nhign, phep ph4p nhan la nhUng phep toan hai ngtoi ; cal hop thanh eta x

va y E N bai the phdp toan dd ki hieu theo tint to bang

'x + y, xy Phep hop thanh xY, la mot phep than 2 ng61 trong

tap W = N— {0}

2) Phep trit khong phai la mot phep San hai ng8i trong N, nhung la ma phep twin hai ngti trong tap hop Z the sd nguyen 3) Tich anh xa la mOt phep toan hai ngti trong tap hap the

anh xa tit ma tap hop X don chinh no

4) Tich ngoai the vecta a A ft cua liinh hoc giai tf ch la mot

phep toan hai ned trong tap hop can vista S ding met didm

g6C 0 dm khong gian 3 Sian thong thubng

5) 'rich ma trail la ma pile') toan hai ngti trong tap hap the ma tran sting cap n

Sau day, trong the If luan tang 'pat, ta se via cai hop thanh

tha x va y la xy n8u khong cd li do nao khign ta phai via khan Dinh nghia 2 MOt 130 phan A chit X goi la do dinh (d6i vbi phep toan hai ngoi trong X) nat va chi Su xy E A Si moi x, y E A

Phep toan hai ngai * the dinh trong 130 pan do dinh A bai

quan 110 x * y = xy voi moi x, y E A got la eat thu hap van

A tha phep than hai ng6i trong X Ngubi ta can Si rang * la

thuang ki hiau phep toan earn sinh nhu phep than dot X Dinh nghia 3 MOt phdp toan hai ng6i trong mat tap hop

X goi la kit hop ndu va chi n8u ta cd

(x y) z = x (y z) vai moi x, y, z e x ; la giao hoar Mu $ chi Su to S

xy = yx

x, y E X

Phep °Ong va phdp nhan trong tap hop the s8 to nhien N

la kat hop, giao hoan ; nhdng AO mu hda Ichtong kat hop :

(25 2 # 2(12) , cling kh6ng giao hoiin : 2 1 # 1 2 Tich anh xa trong

vi du 3) la ket hop, kheng giao hoiin (nett X cep nhieu hon mat

phgn tit)

Dinh nghia 4 Gia sit de, cho mat phep than hai ngOi trong

mat tap hdp X MCA phan to e caa X goi la mat don t4 trai

eau phep toan hai ng6i neu va chi nen

ex = x voi rayi x e X Thong tu, mat phan tit e dm X goi lit mat don

ui phdi cua phep toan hai net nett va chi nett

xe = x vdi nagi x E X Trong trudng hop mat phg.n tit e cita X vita la mat don vi trai vita la mat dan vi phdi, thi e goi la mat don 0, hoar mat phtin to trung lOp dm pilot) toan hai ngdi

Trong vi du 1) s6 0 la phan to trung lap cua phdp Ong, so`

1 la phan to trung lap ciut phep nhAn va IA don vi phAi cua pile)) mu hda Trong vi du 3), anh xa thing nhat lit phan tit trung lap Tich ngoiti edc vecto trong vi du 4) khong cd don vi tram ciing khong cd dan vi

Dinh li 1 Niu mot phep town hai ngdi trong mat tap hop

X co met don vi trai e' ties mat don 0 phdi e", thi e' = e"

Cluing mink Xet tich e'e" trong X VI e' lit don vi trai nen e'e" = e" MM khac, vi e" la don vi OW nen e'e" = e' Ta suy ra e' = e"

Mat ha quA tdc khAc la

114 qua Mot phep town hat- ngoi co nhiM nhdt mot plain

Cac vi du 1) (trit phep mil Ma), 3, 5) trong 1 la nhAng vi

du ve nita nhdm, va hon nits vi nhdm My) be phan do dinh

39

A ciut mot nita nhdm X eting voi phep than cam sinh trot' A

la mot nita nhdm g9i la nita nhOrn con cos rola nhdm X

Trong mot nit° nhdm X, not:* ta ki hi6u gia tri chung eiut hai v6 cda tang tilde

va g9i lit tich eta n phan t* x i , , xi,

Tinh chat chinh eta eac phep Wan hai ngei k6t hop la dinh

li sau day :

Dinh Ii 2 ((huh II k6t hqp) Gta sa x i , x2 , xn lit rt (n a 3)

phan td (phan biet hay khong) mia mat nue: nhdm X, the thi

Dinh nghia 6 Trong rant naa nhdm X, lay thha n (rt la mot to nhien khac 0) caa mot phan to a E X la tieh cat) n phan tit bang a, ki hidu a" Ta co" the quy tae (do dinh li 2)

a" = +", (an )" = arm