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GIAI TICH 1 - CHUONG 1 1
TOAN GIAI TCH 1
ay la cac slides bai giang mon Toan Giai Tch 1 danh cho sinh vien nam th nhat Khoa Toan-Tin, trng ai hoc Khoa Hoc, ai hoc Quoc Gia ThanhPho Ho Ch Minh, nien hoc 2007-2008. Bai giangnay c soan theo quyen : Giao Trnh Toan GiaiTch 1, cua GS Dng Minh c, Nha xuat banThong Ke, 2006.
DNG MINH C

vn thc tin
m hnhton hc
kt lunton hc
TON HC V THC TIN
din giikt lun
CHNG MOTTAP HP VA LY LUAN C BAN

Mot van e co the giai quyet bang cac bc sau : dung toan e mo hnh van e : lam ro va gon hn,
dung cac phng phap toan e giai quyet bai toantrong mo hnh.
dien giai ket qua toan hoc bang ngon ng thc tien
Th du1. Gia mot cuon tap la 3.000$, qu tai tr chco 3.500.000$, hoi co the mua c bao nhieu tapcho hoc sinh ngheo?
Chung ta mo hnh van e nay nh sau: so tap muala mot so nguyen ln hn hay bang 1, so tien co thechi tra ch co the la cac so t 1 en 3.500.000, neuso tap mua c la n th so tien phai tra la 3.000n.

Chung ta mo hnh van e nay nh sau: so tap muala mot so nguyen ln hn hay bang 1, so tien co thechi tra ch co the la cac so t 1 en 3.500.000, neuso tap mua c la n th so tien phai tra la 3000n.Chung ta thay trong mo hnh nay khong con cac
van e rac roi nh : qu t thien, tap v, tien bac vahoc sinh ngheo. Va van e bien thanh : tm so nguyen n ln nhatsao cho 3000n 3500000.Dung ky thuat lam toan thong thng, bai toan trthanh tm so n ln nhat sau cho n 1166,66.
Vay ta co li giai la 1166 quyen sach.

Th du 2. Chung ta co hai he thong onhiet o : Celcius va Fahrenheit. Nhieto e nc ong bang la 0o C va 32o F, va Nhiet o nc luc bat au soi la100oC va 212oF. e lam mot nhiet ke dung trong nha,
chung ta phai lap bang ke cac so otrong he Fahrenheit tng ng vi cacso o t -20 en 70 cua he Celcius,
at C va F la so o nhiet o cua mot vat trong heCelcius va he Fahrenheit. Ta biet: C=0 khi F=32, va C=100 khi . Ta phai tnh F tng ng vi cactr gia C t -20 en 70.
C F
0 32
100 212
C F

at C va F la so o nhiet o cua motvat trong he Celcius va he Fahrenheit. Ta biet: C=0 khi F=32, va C=100 khi . Ta phai tnh F tng ng vi cac tr giaC t -20 en 70.
Ta e y
Vay hay
0 32100 0 212 32C F
32180 100
F C 1810 32F C
C -20 -15 -10 -5 0 5 10 15 20 25 30 35F -4 5 14 23 32 41 50 59 68 77 86 95C 40 45 50 55 60 65 70F 104 113 122 131 140 149 158
C F
0 32
100 212
C F

A. TAP HP
Th du : trong bai tnh so cay phai trong doc theocac con ng, ta phai tm li giai trong tap hp cacso nguyen dng
Trong viec mo hnh nh cac th du tren, chung tacan quan tam en mot vai so nguyen (ch khongphai tat ca cac so nguyen). Trong cac van e khaccung vay, ta phai quan tam en mot so s vat cochung vai tnh chat nao. Mot tap the mot so cac svat nh tren c goi la mot tap hp, va cac s vato c goi chung mot ten la phan t cua taphp o .

Cho mot tap hp E va mot phan t x cua E (ay x co the la mot so, mot iem hoac mot dlieu), luc o ta noi x E .
Th du : Trong cac bai toan ve cac chuyen ongchung ta quan tam en cac yeu to thi gian, van tocva khoang ng di chuyen, cac yeu to nay buocchung ta phai xet tap hp cac so thc.
Dung ly thuyet tap hp chung ta co the dien ta dedang mot so s viec trong toan hoc. Ngoai ra chungta co the khao sat cung mot luc mot so van ekhac biet nhau bang cach s dung cac khai niem vetap hp va anh xa.

Th du. e xet cac nghiem cua phng trnhx3 + 4x2 - 5 = 0,
Ta xac nh tap hp E = x : x3 + 4x2 - 5 = 0.
Ta co cac tap hp thong dung nh tap hp cac so nguyen dng = 1,2, 3,....., tap hp cac so nguyen =....,-3,-2,-1,0,1,2,3,.. , tap hp cac so hu t = : m va n , tap hp cac so thc , tap hp cac so phc = x+iy : x va y trong , tap hp trong la tap hp khong cha phan tnao ca
mn

Ta thng mo hnh tap hp cac so thc nh la taphp cac iem tren mot ng thang D. So 0 cgan cho mot iem A tren ng D, mot so thcdng x c gan cho mot iem M nam pha benphai A tren ng D vi khoang cach AM = x, vamot so thc am y c gan cho mot iem N nampha ben trai A tren ng D vi khoang cach NA =-y
A MN
xy 0

Nam 1881, ong John Venn (nha toan hoc ngiAnh) e xuat viec mo hnh mot tap hp X nh motphan A cua mat phang gii han bi mot ng cong.
Ta gan cac phan t cua X nh la cac iem canh dau trong mien A . Tuy nhien nhieu luc ta cmo hnh X nh mien A, ma khong can anh daucac iem c gan trong A .
AX

Mo hnh tap hp nh ong Venn lam gian n nhieubai toan, th du mot mien A trong mat phang co themo hnh mot tap hp X co vai phan t hoac tap hpco rat nhieu phan t nh .
ay chung ta thay toan hoc nhn s vat theo nhieucach, neu theo mot cach nao o, X va ch cnhn theo y ngha tap hp, th chung co the c ois nh nhau va mo hnh nh nhau!Chung ta se thay nh tnh ong nhat hoa nhng sviec khac nhau nh vay, trong toan co the co cackhai niem chung cho cac s vat o nh : phan giao, phan hoi cua cac tap hp .

F = x : x A hoac x B , F la phan hp cua A va B va ky hieu la A B.
Cho hai tap hp A va B. Ta at
E = x : x A va x B , E la phan giao cua A va Bva ky hieu la A B
A B

at X va Y la cac o th cua cac ham so y = cos xva y = sin x , vi x [0,6]. Luc o XY la tap hpgom cac iem A , B, C, D, E va F. Cac iem chungcua cac ng thng c goi la giao iem.
50 6
y x=cos y x = sin
A C
D
B
E F
X
Y

Thi du : at A = {x : sin x = 0} vaB = {x : 2x2 + x - 1 = 0}.
AB la tap hp cac nghiem cua he phng trnh
2
sin 0,2 1 0.
xx x
AB la tap hp cac nghiem cua phng trnh
(2x2 + x - 1 ) sin x = 0

Cho hai tap hp A va B. Ta atG = x : x A va x B . Ta ky hieu G la A \ B .
A \ B

nh ngha. Cho hai tap hp A va B. Ta noi
A bang B neu va ch neu A B va B A ,luc o ta ky hieu A = B.
A cha trong B neu vach neu moi phan t cuaA eu thuoc B (luc o tanoi A la tap con cua B vaky hieu A B)
A va B rinhau neu va chneu A B = f,
A B
BA

Neu A B, ta goi B \ A la phan bu cua A trong B.
Cho A la mot tap hp, ta at P (A) la tap hp tatca cac tap hp con cua A.
AB B \ A
Th du : A = { 2 , a , }, luc o
P (A) = { ,{2},{a},{},{2,a},{2, }, {a, },{2,a, }}

Th d . Gi A la tp hp tt c cac linh kin trongmt ca hang may tnh trong mot ngay nao o. Motmay tnh c lap rap bang cac linh kien nay co thecoi nh mot tap con cua A, hay la mot phan t trongP(A). at M la tap hp cac may tinh c lap rap vaban ra trong ngay hom o. Luc o M la mot tap con cua P(A).
Th du. at A = {0,1,2, . . .,9}. Luc o {1,9,2,4} lamot tap con cua A, nhng so 1924 khong phai lamot tap con cua A.

e khao sat thiet ke he thong may lanh trong giangng nay, chung ta o nhiet o tai mot so v tr tronggiang ng nay (goi A la tap hp cac v tr o) tai motso thi iem t 7.00 gi sang en 6.00 gi chieu trongmot ngay nao o. Luc o chung ta quan tam cung motluc en hai tap hp : A va [6,18] (cac thi iem ma tao nhiet o). Ta mo hnh viec nay bang toan nh sau.
nh ngha. Cho A va B la hai tap hp, ta at tchcua A va B la ho tat ca cac cap (x,y) vi moi x Ava y B va ky hieu no la A B.Th du: A = { 2 , } va B = {@,#,&}, luc oA B = {(2, @), (2, #), (2, &), (, @), (, #), (, &)}B A = {(@, 2), (@, ), (#, 2), (#, ), (&, 2), (&, ) }

Th du: A = { 2 , } va B = {@,#,&}, luc oA B = {(2, @), (2, #), (2, &), (, @), (, #), (, &)}B A = {(@, 2), (@, ), (#, 2), (#, ), (&, 2), (&, ) }
2
#@
( , )@ ( , )# ( , )&
( , )@ 2 ( , )# 2 ( , )& 2
&ABA
B2
#
@
&
( , )2 &
( , )2 #
( , )2 @
( , ) &
( , ) #
( , ) @

Th du: C = { m , n } va D = {a,i,o}, luc oD C = {(a,m), (a, n), (i, m), (i,n), (o,m), (o,n) }C D = {(m,a), (m,i), (m,o), (n,a), (n,i), (n,o)}
an
im om
a
am
i
in
o
on
m
n
C
D m
ma
mi
m o
na
ni
no
n
a
i
o
C
D

Th du: C = { 1 , 2 } va D = {-1,-2,-3}, luc oCD = {(1,-1), (1,-2), (1,-3), (2,-1), (2,-2), (2,-3)}DC = {(-1, 1), (-1,2), (-2,1), (-2,2), (-3,1), (-3,2) }

Neu B = A, ta thng ky hieu A A la A2. Luc o A2 la ho tat ca cac cap(x,y) vi moi x A va y A, ta phai luy trong trng hp nay la (x,y) cothe khac (y,x), th du nh M = (1,2) khac N = (2,1) trong 2.
a b
c
d[ ]x[ d]a,b c,
Dung bieu dien theo tch Descartes
a b
c
d
( )a,c
( )b,d
1 2
1
2
0
M
N

Co hai bai toan c ban lien quan en tap hp : xacnh mot tap hp va chng minh tap hp nay chatrong mot tap hp khac. Chung ta xem cac phngphap thong dung sau ay dung e giai quyet cacvan e nay .
A.1. Xac nh mot tap hpe xac nh mot tap hp E ta co cac phng
phap sau :
Liet ke tat ca cac phan t cua E
nh ngha lai tap hp E mot cach gian

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