IIIBESBE19 PHAT TRIEN KHA NANG SUY LUAN CHO SINH VIEN TRONG DAY HOC TOAN CAO CAP ThS Nguyfin Thl Dunn, ThS Di Phi Nga Hpc vi^n Cdng ngh$ Buu chinh Viin thdng SUMMARY The paper presents the types of re[.]
Trang 1IIIBESBE19
PHAT TRIEN KHA NANG SUY LUAN
CHO SINH VIEN TRONG DAY HOC TOAN CAO CAP
ThS Nguyfin Thl Dunn, ThS Di Phi Nga
Hpc vi^n Cdng ngh$ Buu chinh - Viin thdng
SUMMARY The paper presents the types of reasoning (deductive and inductive) and the issues that students need to consider when deducting For example: Understanding the rules of logic, combining both types of deductive, using diatoms, Identifying common mistakes while reasoning The above strategies will help students have good deduction and limit the mistakes as a result, students 'deductive skills can be Improved
Keywords: Reasoning, deductive Inductive, advanced mathematics
Ngdy nfifin bdi: 15/9/2015; Ngdy duyft bdi: 16/9/2015
L Md diu: Suy lugn la thdnh phan quan trgng
trong suy nghT cfla mSi ngucri Qud Uinh suy lufn s6
tgo ra thdng tin mdi tfl nhQng dilu dd biet, nd giflp
Ich cho vifc gidi qi^lt cdc bdi todn, cdc van dk trong
cufc sing Vifc rSn luyfn khd ndng suy lufn la mpt
trong nhfhig bifn phdp dl cd thl phdt uiln tu duy
phdn tfch, tu d\xy hf thing, tu duy sdng tgo, Bdi
vilt nhim giflp sinh viSn hilu rd hem ve cdc dgng suy
lufn, cdc kieu suy lufn sai lim thudng gfp cfla hg khi
gidi bdi tfp Todn cao cip Tfl do, smh viSn cd thl bilt
lya chgn vd vfn dyng suy lufn tit hem
n Khdi nifm vd cdc dgng suy lugn
2.1 Khdi ni^m suy lupn: Suy lufn Id hinh thflc
cfla tu duy, dk rflt ra mft phdn dodn mdi tfl mgt hay
nhilu phdn dodn dd cd Cdc phdn dodn dd cd gpi Id
tiln de Phdn dodn mdi dugc rflt ra gpi la ket lugn cua
siQ* lufn Cdch thflc rflt ra kit lugn tfl tiln dl gpi la
lgp lufn
MSI suy lufn dupc bieu dien duin dgng mpt
mfnh dk kdo theo md tiSn dl Id rapt mfnh dl ho$c
hfi cfla nhieu mfnh de: Pi,P2f;p„=^q (cdc p^ Id
cdc tiln dl, q Idkltlufn, / = 1,2 n )
Cd hai dgng suy lufn co bdn: Suy lufn dien djch
vd sity lufn quy ngp
2.2 Suy lupn diin dich
2.2.1 Khdl nlim: Mpt sl tdi lifu dd dua ra khdl
nifm vl suy lufn diln djch nhu sau:
"Suy lufn dien djch (cdn gpi Id suy dien) Id suy
lufn theo mft qi^ tde thda mdn dilu kifn: Nlu tiln
dl A dflng thi kit lugn B dflng Ki hifu: — " ([5], U
13) ^
"Suy lufn dien djch Id suy lufn vdi each % lufn di tfl tien de phdn dnh hieu biet chung din kit
ludn phdn dnh hilu bilt riSng" ([ 1 ], tr 39) Vi d\i.: Con
ngudi khdng thl sing dugc den 200 tuoi Ong Hodng
Id ngudi, vfy dng iy khdng the sdng dugc den 200 tuoi
2.2.2 Mgt so vdn de cdn chu y trong SIQ> lugn diin djch
Dilu kifn cin vd dfl de suy lufn dat tdi kit lufn dflng id phdi xuit phdt tfl nhQng tien de ehan thyc vd qud Uinh lfp lufn phdi dflng ddn, nghTa Id tuan theo cdc quy tdc Idgic (cdc phdp lien kit Idgic mfnh dl, phdp phfl djnh jugng tfl, lugt d^ng nhit' ) NhQng vin dl ndy sinh vien cin tham khdo Uong cac t^ lifu
vl Idgic todn Chdng hgn, xdt raft vdi quy tdc sau:
1) Mfnh dl p ^ 9 chi sai khi p dflng, q sai
2) Mft cdng thflc rafnh dl dupc gpi la hdi^ dflng nlu nd ludn nhfn gia Uj 1 vdi mpi thl hifn ciia cdc bien mfnh de cd trong cdng thurc
Nlusuylufn Pi,p2,—P„=>q IdmfnhdShdng
dflng thl ta ndi suy lugn do hgp Idgic Ngupc 1^, nSu
P\>p2>—Pn =^ 9 khdng phdi Id mfnh dl hdng dflng
thi suy lufn dd khdng hgp Idgic
3) Vxe D,S(,x) • » (axe A^OO) 3x e D,S(x) o (Vx e A5(x))
4) Luft ding nhit (Trong qud Uinh suy lufn, mgi tu tudng diu phdi ddng nhit vdi chinh nd, nlu khdng tuan thfl qiiy luft ding nhit thl sS din din lgp lufn Iflng cflng, sai lira trong suy nghi)
Trang 2LiSn hf Uong hgc tip todn cao cip, ta thdy:
1) Mfnh dl "NIU chudi sd ^ u „ hfi ty thi
B=l
lim u^ = 0 " chi sai khi tdn tgi mgt chuoi so J^ "
ndo dd hfi ty nhung lim u„ ^ 0
2) Suy lugn ( ( ^ =>9)^p)^q la hgp Idgic
p
1
1
0
0
Th|t v|y
Q
1
0
1
0
p = > ?
1
0
1
1
xet bang gia tri cllSn li, ta cd:
( P = > ? ) A P
1
0
0
0
((p=>g)Ap)=>5
1
1
1
1
Tu ket lugn ndy, ta thdy neu m^nh de p^g
dflng vd mfnh de p dflng thi suy ra q dflng (Dilu
niy thudng dupc dp dyng mgt cdch hlln nhien trong
cdc suy lufn md khdng cin kilm tra lgi bdng gid trj
chdn If)
Ching hgn: Ta biet ring "Neu hdm so / ( x ) lien
tyc USn [o,b] thi / ( x ) khd tich trSn [a,6]", vd
"hdm so / ( x ) = xlnx liSn tyc tren [l,e]"- Vgy suy
rahamso / ( x ) = xlnx khd tfch tren [\,e]
3) Tfl djnh nghla: "Hdm sd / ( x ) dgt cyc dgi tgi
XQ nlu tin tgi mpt lan cdn C^(X(,) ndo dd cfla XQ
saocho V x e f ^ ( x o ) , / ( x ) < / ( x o ) "
Ap dyng cdng thflc
3xeD.S(x) o (Vx e D,S(^) ^ ta cd:
/ ( x ) khdng dgt cye dgi tgi XQ nlu vdi mpi lan
cfn Ogix^) cfla Xp, VxEqj(xo),/(x)^/(xo).Lgi
dp dyng cdng thflc V x e A 5 ( x ) •» (HX e £>,5(x)),
tasuyra:
/ ( x ) khdng dgt cyc dgi tgi x,, neu vdi mpi
ldn cfn Cigixo) cfla x^,, Hxeqj(xo), / ( x ) > / ( x o )
4) Cdc Ifp lufn sau Id khdng dugc vl khdng
tudn tiieo luft ding nhit:
"Cd iy da dogt gidi hai lin, mft lin tgi SEA
Games va mpt tan t ^ Sin^wre";
"Tgp X = (0,2] cd vd sd phan tfl ^ A' khdng
bj chfn"
Nhgn xet: Ngodi viic dp dgng cdc quy tdc Idgic, ngudl ta cdn ket hgp dp d\tng cdc dgng sa do de qud trinh suy lugn duac rd rdng mgch lgc han Chdng hgn, xet vi dg sau
Vi dy: Tfl cdc tiln de: "Tit cd nh&ng ngudi nudi
ong Id nhd hda hgc", "Cd vdl nhd hda hpc Id nhgc sT', suy ra "vdi ngudi nudi ong la nhgc sT" Suy lufn ndy
cd dflng khdng?
Gldl: Dung so dd, ta thay cd thl xdy ra trudng
hgp sau:
(o: ngudi nudi ong; h: nhd hda hgc; n: nhgc SI) Vfy suy lufn USn khdng dflng
2.3 Suy lupn guy ngp
Suy lugn qi^ ngp Id suy lufn khdng dya theo mpt quy tdc ndo, kit lufn thudng dugc rflt ra trSn co
sd xem xet nhflng tmdng hgp rieng
2.3.1 Suy lugn quy ngp hodn todn
Suy lugn quy ngp hodn toan Id phdp suy lufn nhim rflt ra kSt lugn chung ve tat cd cdc tm&ng hgp
cy the dd dupc xet dSn
Chdng hgn, phucmg phap quy ngp todn hgc Id mft dgng suy lufn qi^ ngp hodn todn
Suy lugn quy ngp khdng hodn toan (cdn ggi Id suy lufn nghe cd If)
Si^ lugn quy ngp khdng hodn todn Id mgt logi
suy lufn quy ngp, Uong dd kit lugn rflt ra dya trfn sy
xet khdng diy dfl cdc tmdng hpp riSng, do vfy kit lugn chi mang tinh chit dy dodn, gid thuylt M^c dfl dgng suy lugn nay cd thl dua din nhihig kit lugn khdng dflng, nhung nd giflp md rfng hilu bilt cfla ta d mft mflc dp ndo dd Hon nOa, nhihig dy dodn, gid thuylt cd thl Id ggi ^ ban diu dl dua din nhihig chflng minh dflng ddn
Cd mft dgng dfe bift Uong suy lufn quy n ^ khdng hodn todn, dya Uen sy tuong ty, gpi Id suy lufn tuong ty
Cildng hgn, vdi cdc vd cflng bd, tfl nhan xft:
Trang 3ap~a^py khix->Xfj,
>a + fi~af+fif khlx-^XQ
[fi~/Ji khix-*x^
ta nghT den dieu tuoi^ ty
{a~a^ kltlx-*XQ
P~P, khix^XQ
Kit lufn ndy khdng dung trong mpi
tmdng hgp (Chdng hgn, no khdng dflng khi
a = x-x^; a,=x + x^; fi=fit=-x'> Xo=0).Tuy
nhiSn, ta thiy nd dflng trong nhieu trudng hgp khac,
vfy cd thl iQri dyng dilu nify de dua ra gid thiflt vd
tfl dd ed cdch gidi l>di todrt
Vf dy: Xdt sy hfi ty cfla tich phan suy rpng
Se -e
Phdn tich:
e ' - l ~ x khlx-*0* vd e " ' - l - - x JtWx-^0*
Nhu vfy, ta dodn ring cd thl
Ue' -1) - (e"' -1)] ~ 2x khi x^O* hay
( e ' - e " ' ) ~ 2 x khix-*0\ j ^ ^6^ each gidl bdi
todn Id nhu sau:
Gldl: Cgc diem: x = 0
Dl thiy — Id hdm so duoi^ USn (0,1],
e ' - e '
khd tich tren mpi dogn [a, 1] vdi a > 0
lim : — = lim — = lim ::- = 1
'•^ye'-e~' 2x) i - t o ' e ' - e x->^e +e
md J— phdn kl nSn tich phan dd cho phan ki
^2x
Vf dy: Khi hpc d phan tich phan tm Idp, sinh
vifn d3 bilt ring:
+ jjdxdy = S (S Id dif n tich miln D)
D
+ N I U miln D cd tinh dli xflng qua tryc Ox vd bilu
tiiflc cfla fix,y) IS doi vdi y tiii JlfU,y)<ixdy =0
D
Nhu vfy, khi hgc tich phan ba ldp, hg cd thl suy
lufn tuong ty (dua ra gid thuyet) ring
+ jjjdxdydz = V {V Id till tich mien V)
va bilu thflc cua f(x,y,z) le ddi vdi z tiii
Id lllfix,y,z)dxdydz = 0
Nhan xdt: "Moi quan hf gifla quy ngp vd dien dich Id moi quan hf kl thfla, tgo tien de, bo sung
va ho trg cho nhau trong qud trinh nhfn thflc" ([1],
U.47) Quy ngp cung cip phdn dodn, ldm co sd cho dien dich Dien djch giflp kilm tra tfnh dflng din cfla suy lugn quy n ^ , dilu nay cd thl dua din kit lufn phdn dodn la dflng, ho$c nlu bdc bd thl sS d^t ra phdn dodn quy ngp mdi, giflp quy ngp sau din gin vdi ban chit hifn tupng hem
Bk phdt Uiln khd ndng suy lufn, sinh vifn nen
thudng xuyfn kit hp^ sfl dyng cd hai dgng suy lufn
USn, ddng thdi phdi chfl f dk khdng mdc phdi nhQng
sai lim Uong suy lufn 4 Mft s6 vi dy vl suy lufn khdng dflng cfla sinh vien trong hpc tfp Todn cao cip Trong khi gidi bdi tfp todn cao cip, mft sl sinh viSn cdn cd nhflng sai lim trong suy lufn Chdng hgn:
+ Tfl mfnh dl dflng p^q, c6 q dflng, si^ ra
p dung
+ Sai khi vilt phfl djnh cfla mft mfnh dl, sai khi suy lufn vdi mfnh dl cd lugng tfl phi biln, lupng tfl tin tgi
+ Sai khi vfn dyng suy lufn quy ngp khdng hodn
+ Sai khi khdng chfl ^ din luft ding nhit Sau day Id mft vdl vf dy vl cdc sai lim nfu Uen
T«j ^L ui i J-, ^ x*sin— Idil x^O
Vi dy: Cho hdm so / ( x ) = x
A khix = 0
TimAdl / ( x ) khd vi tgi jc = 0
Gidi: Dl / ( x ) khd vi tgi x = 0 * ! cin
dilu kifn fix) lien tyc tgi x = 0, nghTa Id
lim/(x) = /(x„) = ^ Tacd lim/(x)=limx^sin-=D
Vgy vdi A~0 thi / ( x ) khd vi tgi jc = 0
* B ^ gidi sai vl cho ring hdm s l lifn tyc tgi
x = 0 thl sS khd vi tgi x = 0 Sai lim ndy cd till do sinh vien khdng nhd chinh xdc djnh If, hofc do suy
lugn kilu: Tfl mfnh de dflng p=>q, cd q dimg,
suy ra p dflng
Trang 4Vf dy: Xet sy hgi ty cfla chuoi sd y\— ^
^^[2 +n
Gidi;Tacd lim =lim— = 0 = * ^
hfity
• Sai lim tuong ty d vi dy tren, sinh vien cho
rdng lim u„ = 0 => ^ u„ hfi ty
Vfdu: Tim gidi hgn: vi = lim
cot*x—;-'^\ ^)
Gidl:
, , fcos^x I^ , f'cos'x 1 1 , -sin^x ,
-4=lim—1 r p l i m — j r-
=lim—j—=-l-'-^^sinx r J '-^\_sinx sinxj '-« smx
* Ldi gidi sai khi vilt
, fcos^x 1 ^ , f cos^x I I ^ ,
hm —5 r =lim -7-; r-;- tuc la suy
'-*<»\^smx X ) '-"l^sin^x sin x j
lugn kieu
a ~ a, khi x->Xfj
P- 0X f^' X-> XQ
I:
2,
( 6 bii aky dip 6n diing Ik A = ~)
Vfdv: Tim cvc trj ciia iiams6 f^x,y) = x* +/
Glai: C 6 / ; = 4x'; / ; = 3 /
H t a s6 c6 m?t dilm t4i lljn 14 M, (0,0)
T^M,(0,0) c6s'-rl=0
Trong mpi ISn c§n ciia Mo(0,0), x^t cfic diem
mx,0) Tac6 / ( M ) - / ( W „ ) > 0 V^ylltasddSt
cilctieut?iM,(0,0) v i / ( M , ) = 0
• Sai 1 ^ c6 till vl chua iiiiu rd dinti ngliTa cvtc
tri, cijua llilu r5 ve cSc liriTng tii "t^n t^i", "mpi",
iip$c da dflng suy l u ^ quy n^p ]di6ng liofin toan
Vf dv: HSy ciiijmg minli dinh li Cauchy
(Cho f, giAckc him s6 iien tyc trSn [a, b];
kh4viti«n(a,i);
g'(:t)*0 v6i VATS (a,*).Khi d6, t6n t?i
Vilyi cre(a,A)saocho —-T- r T =
-77T'-Gidi: Do / lien tyc tren [a, b]; khi vi tren (a, b) nen theo dinh li Lagrange, t6n t^i c G (a,b) sao cho nb)-na) = /XcXb-a)
Tucmg ty, tdn tgi ce{a,b) sao cho gib)-gia) = gXcXb-a)
fib)-fia) fjc) g{b)-g(a) g'(c)
* Sai lira trong chflng minh d ddy Id dd khdng
chfl ^ din lugt ding nhit Sl c trong bilu thflc
f{b)-f{a) = fXcXb~a) vd s l c trong bilu tiiflc gib) - gia) = gX^Xb - a) Id khdc nhau
K £ T LUAN Vifc hieu rd vl cdc dgng si^ lufn sS giflp mSi
ngudi bilt vgn dyng SIQ* lufn tit hon Be cd nhflng
suy lugn dflng vd nhanh, ngu&i hpc cin nim vttng cdc quy tie Idgic, hilu cdc djnh nghTa, djnh li mft cdch chlnh xac, chu ^ dieu kifn cln vd dfl trong mSi djnh
If, liSn hf gifla cdc cdng thflc Idgic vdi cdch suy nghT
dl hilu hon Hai dgng suy lufn dien djch vd qi^ ngp
cd mdi quan hf vdi nhau vd deu quan trpng trong sy sdng tgo "Suy lufn quy ngp khdng hoan todn" mang lgi hifu qud cao trong qud trinh ^di quylt vin dl, Tuy vdy, chi nen dp dyng suy lufn ndy dudi dgng gid thifet, cin kilm tra lgi sy chfnh xdc cfla kit lufn trong tflng tmdng hpp cy thl
Tdi lifu tham khdo
1 Phan Dung (2012) Tu duy Idgich biin chung
vd hi thing, NXB Dgi hgc Quic gia TP HCM
2 Dy dn Vift- Bl, Bf Gido dye vd Ddo tgo
(2000) Dgy cdc kS ndng tu duy, Hk^^u
3 LS Bd Long, Gido trinh Dgi sd, NXBTTVTT,
2010
4 Chu cim Tho (2015) Phdt triin tu duy thdng qua dgy hgc mon todn a trudng phd thdng, NXB Dgi
hgc su phgm, Hd Nfi
5 Nguyin Anh Tuin (2012) Gido trinh Logic todn vd lich sir todn hgc, NXB Dgi hgc Su phgm,
HdNfi