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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 Sad ban in : HOANG DIEM Sip cha : PHONG Off BAN (NXB GIA0 DUC) 512 21/320 - 05 Me se: 7K108T5-TTS GD - 05 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 4 CHUONG I TAP HOP VA MAN HE §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 }. 3. Hiau cOa hai  hqp Dinh nghia 2. Cho hai tap hop A va B thy  .P hIP 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 C A B. 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 X - (X - B), tilt la then (1), A C B. n 4. Tap hqp rang Gia sit X la mat tap hop, X cling la mat ba phan dm X digu do cho phep ta xet tap hap 0 = X - X got la bo phan rang caa X x e 0 co nghla la x e X vb. x X. R6 rang khong c6 mat phan tit x nao cita X Lai c6 filth chat dd. Tap hop X - X = 0 khong pho thuac vao UV hop X. Nth each khac, ta ed X- X= Y- Y Arth so) X, Y. 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 ki hieu bang X x Y. Khai them tich da the cci thg ma rang cho trtiOng hop nhigu tap hop. /46u X, Y, Z, T la nhitng tap hop, ngubi ta dinh nghia XxYxZ=(XxY)xZ XxYxZxT=(XxYxZ)xT 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 X 2 =XxX,X 3 =XxXxX,X 4 =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 r ang 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 14n Y. Dinh Ii 3. Poi cot AV lit(t) A, B, C tit( X tuy 9, to co : (i) Tinh chat giao hoan , AnB=Bn A, 'A UB=BU A. (ii) Tinh chat kit hop A n (B n  (A n B) 11 C, A U (B U C) = (A U B) U C. (iii) Tinh chcit phan phdi A n (B U  -» (A n B) U (A C), A U (B n  = (A U B) fl (A U C). (iv) Gang that Do Mo6c-gang X - (A U B) = (X - A) n (x- B), X - (A n B) = (X - A) U (X - B). Ching minh. (i) va (ii) hidn nhien. 1 1k hay chdng minh tong thde that nhat cue (iii). Gia thx EA n (B U C), didu do c6 nghia la x E A va x Dandle it nhat mot trong hai tAp hap B, 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 10