Đại số đại cương hoàng xuân sính

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
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
vfleAUB,
tticlaxE2Cvax(I6Avaxit
B. VayxEX-A
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
xEX-
B,
tdclaxE/CvaxAvaxcEB.
Vayxe.Xvit
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
A
n
B (X - (X - A))
n(X-(X-
B)) =
= X- ((X - A) U (X - B))
Vity


X
- (A
n
B) = (X - A) U (X - B).
n
9.
Anh
xa
Dinh nghla 6. Gilt sit
X
v& Y la hal tap hop da cho. Mat
dnh
xit f tit X din
Y la mat quy Sc cho Wang ting vat mai
phtin tit
x
dm
X
mat phan tit
'tan
dinh, Id hieu
f(x)
caa Y. Ta
viat
f

f
: X Y hayX—>Y
x


f(x)

x

f(x)
Tap hop
X
gni la
ngu6n hay mi.en the d nh
va tap hap Y
g9i la
dich hay mien gid tri
coa anh xit
f.
Vi du. 1)
Xet tap hop N cat s6 to nhien va tap hop Z
s6 nguyen khang am nhe han mat sic nguyen during da cho m.
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 Z
m
Turing ling
x J—
n
f(x)

xac dinh mat anh xa
f
:
N
2) Xat tap hop the s6 thvc R. Tutting
ling
x 1
n
x
2

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.
4) Gift sit
X
=
Y

=
(1,
2,
31.
Taring ling
1
" 3
2
2
3 I—. 1
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) = 21c
2

— 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 !
12
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 y
be 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.
10. Anh


MTh
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 f
t
({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
3
B 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
f
R —> R
x


x
3

Ile rang
f
la met don anh, vi ngu. x Ira y la nhitng s6 time
thi quan he x
3
= y
3
keo theo
x = y.
Thing le lay
R,
ta lay C thi anti
xa

f : C

C
x

x
3
khang phat la don anh nita, vi got
t
o ,

s
2

la ba gia tri
can bee ba cua don vi, ta cd
2)
GM sit
X lit
met tap hap, anh xa
—
n
x x
got lit
anti
xa dong nheit
cita
X H hidu 1,,
hose e
x
Hign nhien
l
x
la don
3)
Cho X C Y. Anh xa
j
:
X
Y
x t—*j(x)
= x
goi la
don anh chinh hic

tit
X
den Y. Ta cd thg cd nhieu don
Ards tii
X
den Y, nhung don anh j goi la chinh Sc vi n6 duce
fly dung met each
to
nhien.
14
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.
12. Tich anh xa
Dinh nghia 12. Gib. sit cho
f: X—>Yvag Y Z.
Anh xa

X Z
x

gr(ln)
goi la
tich oda anh xa f ud (nth xa g, Id hibu gof,
hay van ta.
0
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 :
(h(gf)) (x) = h(gf(x)) = h(g(f(x)) =
= (hg)(f(x)) = ((hg)f) (x).
n
Do do ta ki hiOu
h(gf) = (hg)f
bang
hgf va
goi la tich ena
ba anh xa
f, g, h.
Chia y rang
Mu
f : X —>
Y la road anh xa bat ki thi ta
f
i
x
= 1
Y1 ft
15
Dinh nghia 13. Gia sit

f : X Y va g
Y
X In
hai anh
xa sao cho
gf = l
x
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
gf = l
x
va fg =
1
y
.
tdc la
g(f(x)) = x
ved mot
x
Xat quan hO
fix) = f(x),
ta suy ra

x = g (f(x)) = g (f(x)) = x'.
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
16
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

x
g' = 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
ri
nen
f= (f
1
)
-1
Ta cd 1X
1
= l
x
.
He quit.
Cho hai song anh f : X Y u4 g Y —1. Z. The
thi gf : X —> Z Id mkt song
anh.
Cluing mink.
Ta cd
(gf) (f'g')
=

p
i
)

=

glyg
_i
=
(fi
g
-i) (gf) =

=

= .
5
0

fly

= (gf
13. Thu
lima va ma rOng anh
xa

Dinh nghin 14. Gib. sd
f
:
X

Y la mot anh xa va Ala
mQt be phan dm
X,
anh xa
g : A Y

x 1—
n
g(x) = f(x)
goi la cat
thu hep ella anh xa f Mt° kJ, phkti
A ytt Id
hieu
la
g = f ,
can anh xa
f
goi la cat
and rOng elk g tren tkp hop
A
X.
17
14. Thp hqp chi
s6
Gia sit / la mat tap hop toy
g
khge Tong ma cat phan to
dole ki hieu lit a, /3, y va f la mat anh xa
f: I
'Pa Id hieu
f(a) = x,,
tip) = x
fi

= xy
Ta bac the phan tU x

a ,
x
13
,
x
y

thanh lap
mOt hp nhung
phan td cda X dupe dank s6 bdi tap hop I,
Id hiau la (x
a
)
a E

can tap hop
I
gal la
tap hop chi s6. Nau the x
a
, x
13
, x
y
.
la
nhiing tap hop thi ta gal (x
a
),
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 x
a
, x
p ,
x
y
C
E,
thi ta got
(x
a
)
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

,. u
2
,
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
9
3
(X) = { 0, { a },

b
},

a,
b } }, va
f :
r
—P(X)
1 •—
n
A
i
= f b 1
2


A
2
= {
a,
b
3

A
3
=
a,
b }
/chit vay ha (A
i
),
e i
gam ba tap hop
A
1
, A
2
, A
3
trong do
A
z
=
A
3
.

18
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 X
a
tap hop the
x sac
a
E
cllo
x

thuac It nhat mat tap hqp cua 119 (X
a
)
aef
.
Dinh nghia 16.
Gigt sii (21
a
)
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 (X
a
)
aei

Dinh nghia 17. Gilt sit (20
aei
lgt mat 119 tap hop va
X = U X

la hop cna ho dd. Ta goi la
tich d8
cac
cua hp
a E /
(X
a
)
ae
1
!va U
hieu bang 11 X
a
tap hop can 119 (x
a
)
aer
nhilng
ael
phan to cart
X
sao cho x
a
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
(V
ac/
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
I
n
= (0, 1,


n},
tha thl

U I
n
= N
n E N
(1 I = I
n
= {0}
.
neN
2) Lay thita de cite R
N
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.
c)
A UQ = A.
d)
A
n

0.
7.
Tap hop X 6 bai tap 5) S pilaf la d6 thj cua mOt anh
xa tit
R
dgn
R ?
8.
Tap hOP

G = {(x, 4.1 x <

U
{(x,
0) I x 3 0} CO
phial la d6 thl atla mOt anh xa tit
R
den
R ?
Bigu (Ban hinh
hoc tap hdp dd.
9.
Tap hop
1

G= {(
x.
x
—1)1
x

R, x
20
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
a)
f(A U B) = f(A) U f(B).
b)
f(A n B)
C
f(A) fl f(B).
c)
U
D) =
ft(C) U riay.
d)
f'(C
n

=
ri(c) fl rkm.
e)
f(X - A)
D
f(X) - f(A).
ri(Y
-
C)
= X
- r
{
(C).
11.

Gia sit n la mat sd tp nhien cho triteic,
f la
mat anh xa
tit tap hop cat se to nhien
N
den chinh nd duoc xac Binh bdi
n - k new k < n
f(k) = {n + k nen k an.

f ad
phdi la don anh toan anh, song anh kitting ?
12. GM
saf:X 1
7
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
= 1
x
.
(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 t
hi
h
= h'. Nguac lai
21

n5 veil moi
h,

to th h f

.
= h'f
keo theo
h = h'

la mat
than anh.
17. Chang minh nau c6 mat song anh tit
X
d6n Y va mat
song anh
tit X dgn Z,
thi th mat song anh to Y dan
Z
18. Cluing minh rang mu6n cho mat ba phan G tha tich 66
the
X
x Y la d6 thi dm mat anh xa tit
X
deli Y thi can
va
du la anh xa (phep chit') :
G X
(x,

x
la
mat song anh.
19. Gia sa (Ad
aei

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
el
b)
n
/l
a
C

vbi
MO
a
E I.
aef
a)
B
n


A
G
) U
(B
n
A
a
).

aEl

aEl
d)
B U
(n
A„)
= n
(B U Ad.

a E

a el
e)
X
— (U .4
a
) =
n

-


aEl

aEl
0 X
(n

=
U
(X
—A
a
)

aEl

aEl
20. GM
siti:X—>Ylit
mat anh xa
(A
a
)
ae
la mat ha
nhfing ba phan efia
X,
(B
fl
)

fi e j
IA mat ha nhftng ha phan
cila
Y.
Cluliag minh
a)
AU
A
a
) =
U M
a
).
ael

aEl
b)
An
Aa
)
C
n
f(A„).
aef

aEl
22
c)
ri(u
B1) =

U
ri(s
fl
).

fie/

fie/
d)
r(n

=
n

fie/

13E/
21. Cho hai tap hop X.vA Y
.
Ta ki hieu bang Hom
(X,
Y)
tap hop tat ca the anh xa tit
X
den Y. Chung mirth
a)
C6 met song anh tit Horn
(X,
Y) den Y
X

.
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
24
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