Cuoc Vien Chinh Vao Vuong Quoc Toan Hoc

Th cm n 1

NG PHC THIN QUC NGUYN TH THY VY

NG PHC THIN PH NGUYN TH KIM ANH

LI NGUYN THNH CNG NGUYN NGC MINH CHU

(Cng vi tp th sinh vin v ging vin

khoa Ton hc, trng i hc Khoa hc, Tp. HCM

khoa S phm Ton, trng i hc S Phm, Tp. HCM)

Th cm n 2

Cun sch ny l mt quyn sch c bit v n c vit gn nh min ph, nhng li c s u t

thi gian, cng sc v lc lng nhn lc cc ln. Nu khng c s gip ca cc bn sinh vin

khoa Ton trng i hc Khoa hc Tp. HCM, v khoa s phm Ton trng i hc S phm Tp.

HCM, c l quyn sch ny s khng bao gi hon chnh. Ti xin cm n cc bn hc sinh, sinh vin

sau y gip ti dch, nh my, nghin cu, v sa li sai cho nhng phn khc nhau trong cun

sch ca ti trong cc chng trnh nghin cu t nm 2012 n nay:

Phn Dch thut v nh my

Kim tra Nghin cu

mi

Bt ng thc Nguyn Th Thanh Ho Trn Hu c

Xc sut v

Thng k

Nguyn Trn Tho Vn, Mai Th Ngc Huyn

Ton ri rc

S hc Cao S Tin

Logic Dng Th Dim Hng

L thuyt th

Gii tch tnh ton Hunh Minh Phng, V Th Nh Trang,

V Hong Trng

Hnh hc

i s Nguyn Th Thu Trang, Nguyn c Duy,

Trn Th Qunh Hng

Phng trnh vi phn

on Thanh T

Thng tin ton hc

Nguyn Hng Tuyt Trn, Phm Th Tuyt Anh

Chng ti cng xin cm n hai ngun ti nguyn chnh m chng ti s dng trit cho vic

hon thnh cun sch ny:

- Bch khoa ton th Wikipedia.org (ting Anh): http://en.wikipedia.org/wiki/Main_Page

- Trang web tnh ton cao cp WolframAlpha.com: http://www.wolframalpha.com/

Li ni u 3

Quyn sch khng phi l mt cun tiu thuyt Cuc vin chinh vo vng quc Ton hc

m l mt cun sch Ton tng hp ton b kin thc t trung hc c s cho n sau i hc. Vi mc ch l ti liu tham kho, nghin cu, so snh kin thc cho hc sinh, sinh vin v l ti liu h thng ton b kin thc mt cch y v chnh xc nht cho ging vin.

Quyn sch gm khong 90 chng trong 11 phn vi khong 600 trang sch. Mi chng trong sch ny c nghin cu v c trnh by rt chi tit, hp l thnh cc mc nh c th c tt nht ngay c theo cch th ng (c t u n cui) v vic sp xp hp l cng gip chng ti c th tit kim nhiu trang giy nht c th. Ngoi ra, sch cng c nhiu hnh nh, biu , v d, minh ha, v cc thut ng bng ting Anh cho mi t c nh ngha c gi d tra cu v tm thm ti liu nu cm thy cn thit.

Quyn sch ny s c dng h tr (khng phi thay th) cho cc mn hc: gii tch 1, 2, 3, 4, gii tch hm, gii tch thc, gii tch phc, gii tch phi tuyn, gii tch s, i s 1, 2, i s i cng, i s hin i, i s ng iu, l thuyt xc sut, l thuyt thng k, ton ri rc, hin ang c ging dy ti khoa Ton hc trng i hc khoa hc, Tp. H Ch Minh.

V, nu bn thy vi phn ca cun sch ny ging vi mt phn ca nhng ti liu ang c trn mng internet, hay trong nhng quyn sch khc th xin ng ngc nhin. Chng ti khng phi l ngi to ra mi th! Chng ti ch c gng gip cc hc sinh, sinh vin v ging vin c th hc v dy mt chng trnh tin b hn, cao cp hn, y hn v hon thin hn. Nhng, d th no i chng na, chng ti cng rt bit n mi ti nguyn m nhiu ngi trn th gii gp nht li v chia s chng ti c th hon thnh cun sch ny. V vy, xin cm n tt c!

Ti bt u vit nhng trang sch u tin vo khong thng 07/2011. Khi , ti mi ch va hon thnh chng trnh dnh cho sinh vin nm I, khoa Ton Tin hc, trng i hc Khoa hc, Tp. H Ch Minh, h c nhn ti nng. Hm nay ngy ??/??/???? quyn sch cng c hon thnh. Mc d ht sc c gng trong qu trnh bin son v c gng kim tra trong nhiu nm, nhng y l mt quyn sch vit theo mt hng nghin cu mi m nn chc chn cn nhiu khim khuyt. Rt mong c s gp t bn c.

Mi chi tit xin lin lc vi ti theo cc cch sau:

Email: [email protected]

Facebook: [email protected]

Chat: [email protected];

[email protected]

S in thoi: 093 250 8 350

Xin cm n cc bn s dng quyn sch ny.

Tc gi ng Phc Thin Quc

Mc lc 4

Trang ba: Cuc vin chinh vo vng quc Ton hc ......................................................................... 1

Th cm n ............................................................................................................................................ 2

Li ni u ............................................................................................................................................. 3

Mc lc .................................................................................................................................................. 4

Phn A: L thuyt s hc ....................................................................................................................... 8

Chng 1: Cc con s ........................................................................................................................ 9

Chng 2: Cc php chia ht ........................................................................................................... 33

Chng 3: Ton hc gii tr ............................................................................................................. 36

Chng 4: L thuyt s hc i s ................................................................................................... 38

Chng 5: L thuyt s hc gii tch ............................................................................................... 39

Chng 6: L thuyt s hc t hp .................................................................................................. 44

Chng 7: L thuyt gii m ........................................................................................................... 45

Phn B: Ton ri rc v t hp ............................................................................................................ 46

Chng 8: Logic .............................................................................................................................. 47

Chng 9: Mnh , i s Boole, tp hp ...................................................................................... 57

Chng 10: Quan h ........................................................................................................................ 73

Chng 11: Hm s v ni suy ........................................................................................................ 79

Chng 12: Cng ngh m .......................................................................................................... 104

Chng 13: Xc sut ...................................................................................................................... 113

Chng 14: L thuyt th .......................................................................................................... 114

Chng 15: L thuyt th t (cy v dn) ..................................................................................... 115

Phn C: i s ................................................................................................................................... 116

Chng 16: S phc - Quaternion ................................................................................................. 117

Chng 17: Hm lng gic Hm hyperbolic ............................................................................ 122

Chng 18: Phng trnh i s .................................................................................................... 142

Chng 19: i s tuyn tnh ........................................................................................................ 159

Chng 20: L thuyt nhm .......................................................................................................... 177

Chng 21: L thuyt vnh ............................................................................................................ 178

Chng 22: L thuyt trng v min nguyn .............................................................................. 179

Chng 23: i s tru tng ....................................................................................................... 180

Chng 24: Cu trc i s ............................................................................................................ 181

Mc lc 5

Chng 25: L thuyt phm tr ..................................................................................................... 182

Chng 26: i s giao hon ......................................................................................................... 183

Chng 27: i s ng iu v i ng iu ............................................................................. 184

Chng 28: L thuyt biu din ..................................................................................................... 185

Chng 29: Lut thun nghch ....................................................................................................... 186

Phn D: Gii tch ................................................................................................................................ 187

Chng 30: Gii hn Lin tc ..................................................................................................... 188

Chng 31: Logarithm super-logarithm ..................................................................................... 202

Chng 32: o hm Cc tr ...................................................................................................... 207

Chng 33: Tch phn vi phn .................................................................................................... 217

Chng 34: Dy v chui .............................................................................................................. 255

Chng 35: Gii tch phc ............................................................................................................. 299

Chng 36: Gii tch hm .............................................................................................................. 318

Chng 37: Gii tch hm nhiu bin ............................................................................................ 319

Chng 38: Gii tch thc .............................................................................................................. 327

Chng 39: Gii tch iu ha gii tch Fourier ......................................................................... 328

Chng 40: Gii tch bin phn ..................................................................................................... 329

Chng 41: L thuyt tensor .......................................................................................................... 330

Chng 42: Phng trnh vi phn .................................................................................................. 331

Phn E: Khng gian ........................................................................................................................... 340

Chng 43: Khng gian vector ...................................................................................................... 341

Chng 44: Cc loi khng gian vector c bit ........................................................................... 342

Chng 45: Khng gian topology .................................................................................................. 345

Chng 46: Cc loi khng gian topology c bit ....................................................................... 360

Chng 47: Topology i s .......................................................................................................... 371

Chng 48: Topology hnh hc ..................................................................................................... 372

Chng 49: L thuyt nt (knot theory) ........................................................................................ 373

Chng 50: Nhm Lie ................................................................................................................... 374

Chng 51: Khng gian vector topology ....................................................................................... 375

Chng 52: Khng gian o ....................................................................................................... 376

Chng 53: Khng gian hm v khng gian dy ........................................................................... 386

Phn F: Hnh hc ............................................................................................................................... 388

Chng 54: Hnh hc c in ........................................................................................................ 389

Mc lc 6

Chng 55: Cc loi hnh c in c bit .................................................................................... 390

Chng 56: H trc ta , php bin hnh v ng cong ni ting ........................................... 392

Chng 57: Vector ......................................................................................................................... 399

Chng 58: Tam gic T gic..................................................................................................... 404

Chng 59: Hnh trn .................................................................................................................... 411

Chng 60: Hnh hc khng gian .................................................................................................. 414

Chng 61: Cc loi hnh khng gian c bit .............................................................................. 419

Chng 62: Hnh hc s hc .......................................................................................................... 425

Chng 63: Hnh hc vi phn ........................................................................................................ 426

Chng 64: Hnh hc affine ........................................................................................................... 439

Chng 65: Hnh hc fractal v hnh hc hn n ........................................................................ 440

Phn G: Xc sut Thng k ............................................................................................................ 441

Chng 66: L thuyt xc sut ...................................................................................................... 442

Chng 67: Thng k m t v th thng k ............................................................................. 464

Chng 68: L thuyt lm mi ...................................................................................................... 465

Chng 69: L thuyt hng i ..................................................................................................... 466

Chng 70: Ti chnh Thng k bo him .................................................................................. 470

Chng 71: L thuyt tin cy .................................................................................................... 471

Chng 72: L thuyt thng k (tham s v phi tham s) ............................................................. 472

Chng 73: Qu trnh ngu nhin .................................................................................................. 492

Chng 74: Thng k theo kinh nghim ........................................................................................ 504

Chng 75: L thuyt mu ............................................................................................................ 505

Chng 76: L thuyt hi quy v tng quan ............................................................................... 506

Chng 77: Kim nh thng k (tng i v chnh xc)............................................................ 507

Chng 78: D liu phm tr ........................................................................................................ 508

Chng 79: Thng k nhiu bin ................................................................................................... 509

Chng 80: Gii tch chui thi gian ............................................................................................. 511

Chng 81: Gii tch sng cn ....................................................................................................... 512

Chng 82: D bo ........................................................................................................................ 513

Chng 83: Cc bng dng trong l thuyt xc sut - thng k .................................................... 516

Phn H: Bt ng thc ....................................................................................................................... 517

Chng 84: Bt ng thc ............................................................................................................. 518

Phn I: Quy hoch v ti u .............................................................................................................. 560

Mc lc 7

Chng 85: Quy hoch v ti u ................................................................................................... 561

Phn J: L thuyt tr chi .................................................................................................................. 562

Chng 86: L thuyt tr chi ....................................................................................................... 563

Phn K: Ph lc ................................................................................................................................. 564

Chng 87: Bng ch ci v s ..................................................................................................... 565

Chng 88: Cc t vit tt, danh sch k hiu, cc t kh hiu ..................................................... 572

Chng 89: Bng lit k thut ng (theo th t t in) ............................................................... 573

Chng 90: Ngun ti liu tham kho ........................................................................................... 574

Chng 1: Cc con s 9

I. S nguyn t:

1) S nguyn t:

S nguyn t (prime number), l s nguyn dng khc , ch chia ht cho v chnh n.

Danh sch s nguyn t u tin:

- Theo nh l Euclid, th c mt s lng v hn cc s nguyn t.

- Phng on bo co cng trnh nghin cu phng on Goldbach (Goldbach conjecture)

cho thy rng cng trnh tnh ton tt c cc s nguyn t di . Ngha l c

khong s nguyn t, nhng cc s nguyn t khng c lu

tr . c bit l c nhiu cng thc c lng hm m s nguyn t (prime-counting

function) (s lng cc s nguyn t phi di mt gi tr nht nh) nhanh hn so vi tnh

ton cc s nguyn t . iu ny c s dng tnh ton rng c

s nguyn t (khong ) di . Mt tnh ton

khc nhau cho thy c s nguyn t (khong )

di nu gi thuyt Riemann (Riemann hypothesis) l ng.

2) S nguyn t chn (even prime):

- L s nguyn t c dng .

- l s nguyn t chn duy nht. Do , i khi c gi l s nguyn t l nht (the

oddest prime) nh mt s chi ch ch khng ngha ton hc (khng phi l s l).

3) S nguyn t l (odd prime):

- L s nguyn t c dng: .

3 5 7 11 13 17 19 23 29 31

37 41 43 47 53 59 61 67 71 73

4) S nguyn t c tnh cng (additive prime):

L s nguyn t m tng cc ch s ca n cng l s nguyn t.

5) S siu nguyn t (super-prime):

- L s nguyn t c ch s c bn trong chui cc s nguyn t (s nguyn t th

).

Chng 1: Cc con s 10

6) S t nguyn t trong c s (self prime in base ):

- L s nguyn t m khng th c to ra bi bt k mt s nguyn c thm vo tng cc

ch s thp phn ca n.

7) S nguyn t bn tri xn c (left-truncatable prime):

- L s nguyn t m vn gi nguyn tnh nguyn t khi cc ch s thp phn hng u b

loi b lin tc (b b t bn tri).

8) S nguyn t bn phi xn c (right-truncatable prime):

- L s nguyn t m khi cc ch s thp phn cui cng b loi b lin tip.

9) S nguyn t hai pha (two-sided prime):

- Va l s nguyn t bn tri xn c va l s nguyn t bn phi xn c.

- C ng 15 s nguyn t hai mt:

10) S nguyn t an ton (safe prime):

- L s nguyn t , sao cho v ( ) u l s nguyn t.

Chng 1: Cc con s 11

11) S nguyn t Chen (Chen prime):

- c t theo tn nh khoa hc ngi Trung Quc, Jing Run Chen (1933 -1996).

- L s nguyn t, trong l s nguyn t v hoc l mt nguyn t hoc l bn

nguyn t.

12) S nguyn t c lp (isolated prime):

- L s nguyn t sao cho c v u khng phi l s nguyn t.

13) S nguyn t Sophie Germain (Sophie Germain prime):

- c t theo tn nh ton hc ngi Php, Marie-Sophie Germain (1776 1831).

- L s nguyn t , tha v u l s nguyn t.

14) S nguyn t bt quy tc ( ) (( ) irregular prime):

- L s nguyn t sao cho ( ) l mt cp bt quy tc.

- Hin nay ch c s:

15) S nguyn t bt quy tc ( ) (( ) irregular prime):

- L s nguyn t sao cho ( ) l mt cp s bt quy tc.

- Hin nay ch c s: 37.

16) S nguyn t anh em h (cousin prime):

- L cp s ( ) tha c hai u l s nguyn t.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

Chng 1: Cc con s 12

17) S nguyn t sexy (sexy prime):

- L s nguyn t , sao cho ( ) u l nhng nguyn t.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

18) S nguyn t sinh i (prime twins):

- L b hai s nguyn t, tha ( ) u l s nguyn t.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

19) S nguyn t sinh ba (prime triplets):

- L b ba s nguyn t tha ( ) hoc ( ) l s nguyn t.

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

20) S nguyn t sinh t (prime quadruplets):

- L b bn s nguyn t tha ( ) l cc s nguyn t.

( ) ( ) ( ) ( ) ( )

21) S nguyn t Wagstaff (Wagstaff prime):

- c t theo tn nh ton hc ngi M, Samuel Standfield Wagstaff (1945 - ????).

- L s nguyn t c dng l ( ) .

22) S nguyn t Eisenstein khng c phn o (Eisenstein prime without imaginary part):

- c t theo tn nh ton hc ngi c, Ferdinand Gotthold Max Eisenstein (1823

1852).

- S nguyn Eisenstein l thc ti gin (ngha l n c dng: ).

Chng 1: Cc con s 13

23) S nguyn t Gauss (Gaussian prime):

- c t theo tn nh ton hc ngi c, Johann Carl Friedrich Gauss (1777 1855).

- Cc thnh phn chnh ca cc s nguyn Gauss (l s nguyn t c dng ).

24) S nguyn t Pythagoras (Pythagorean prime):

- c t theo tn nh ton hc ngi Hy Lp, Pythagoras ca thnh Samos (khong 570

khong 495).


25) S nguyn t trong lp thng d (prime in residue classes):

- C dng , vi c nh. Cn c gi l s nguyn t ng d modulo ca .

- Ba trng hp c mc ring ca chng: l s nguyn t l , l s nguyn t

Pythagoras , l cc s nguyn t Gauss.

17 41 73 89 97 113 137 193 233 241 3 11 19 43 59 67 83 107 131 139 5 13 29 37 53 61 101 109 149 157 7 23 31 47 71 79 103 127 151 167

11 31 41 61 71 101 131 151 181 191 3 13 23 43 53 73 83 103 113 163 7 17 37 47 67 97 107 127 137 157 19 29 59 79 89 109 139 149 179 199

26) S nguyn t tam gic trung tm (Centered triangular prime):


Chng 1: Cc con s 14

27) S nguyn t hnh vung trung tm (centered square prime):


28) S nguyn t tht gic trung tm (centered heptagonal prime):

- L s nguyn t c dng: ( ) .

29) S nguyn t thp gic trung tm (centered decagonal prime):


30) S nguyn t ngi sao (star prime):


31) S nguyn t Motzkin (Motzkin prime):

- c t theo tn nh ton hc ngi M gc Do Thi, Theodore Samuel Motzkin (1908

1970).

- L s nguyn t m c s cch v dy cung khng giao nhau khc nhau trn mt vng trn

gia im.

32) S nguyn t Carol (Carol prime)

- Do nh ton hc Cletus Emmanuel nghin cu v c t theo tn bn ca ng y, Carol

Chng 1: Cc con s 15

G. Kirnon.


33) S nguyn t c dng (prime of the form ):


34) S nguyn t Markov (Markov prime):

- c t theo tn nh ton hc ngi Nga, Andrey Andreyevich Markov (1856. 1922).

- L s nguyn t , m tn ti s nguyn v sao cho: .

35) S nguyn t dng bc hai nh phn (prime of binary quadratic form):

- L s nguyn t c dng: , vi s nguyn v khng m.

36) S nguyn t bc bn (quartan prime):

- L s nguyn t c dng: , trong .

37) S nguyn t Cuba (Cuban prime):

(1) L s nguyn t , sao cho ( )

( ) cng l s nguyn t.

Chng 1: Cc con s 16

(2) L s nguyn t , sao cho ( )

( ) cng l s nguyn t.

38) S nguyn t nguyn thy (primeval prime).

- L s nguyn t m c nhiu hn hon v ca mt s hoc tt c cc ch s thp phn hn

so vi mt con s nh hn bt k.

39) S nguyn t hnh trn (circular prime):

- L s nguyn t m vn duy tr tnh nguyn t trn bt k hon v theo hnh vng trn ca

nhng ch s .

40) S nguyn t hon v c (permutable prime):

- Bt k hon v no ca cc ch s thp phn l mt s nguyn t.

41) S nguyn t lp n v (repunit prime):

- L s nguyn t ch cha cc ch s thp phn 1.

- Danh sch sau y lit k s lng cc s :

42) S nguyn t xui ngc u ging nhau (palindromic prime):

- L s nguyn t m vn gi nguyn khi c ngc cc ch s thp phn ca chng.

Chng 1: Cc con s 17

43) S nguyn t cnh xui ngc u ging nhau (palindromic wing prime):

- L s nguyn t c dng l ( )

.

44) S emirp (emirp):

- Ch emirp l vit ngc ca t prime.

- S nguyn t tr thnh mt nguyn t khc nhau khi ch s thp phn ca chng b o

ngc.

45) S nguyn t nh din (dihedral prime):

- L s nguyn t m vn duy tr tnh nguyn t khi c ln ngc hoc phn chiu trong mt

mn hnh hin th by on.

46) S nguyn t Smarandache Wellin (SmarandacheWellin prime):

- c t theo tn nh ton hc ngi M gc Rumani, Florentin Smarandache (1954 -

????) v nh ton hc Paul R. Wellin.

- L s nguyn t l nhng chui s ni s nguyn t u tin c vit bng s thp

phn.

47) S nguyn t giai tha (factorial prime):


Chng 1: Cc con s 18

48) S nguyn t Pillai (Pillai prime):

- c t theo tn nh ton hc ngi n , Subbayya Sivasankaranarayana Pillai (1901

1950).

- L s nguyn t , sao cho tn ti sao cho chia ht cho v khng chia ht

cho .

49) S nguyn t Wilson (Wilson prime):

- c t theo tn nh ton hc ngi Anh, John Wilson (1741 1793).

- L s nguyn t m l c ca ( ) .

- Hin nay, ch tm c s:

50) S nguyn t giai tha kp (double factorial prime):

(1) L s nguyn t c dng: .

- Hin ch tm c 7 s. Cc gi tr u tin (theo s ) l:


- Cc gi tr u tin (theo s ) l:

51) S nguyn t Euclid (Euclid prime):

- c t theo tn nh ton hc ngi Hy Lp, Euclid ca thnh Alexandria (khong 300

TCN).


52) S nguyn t primorial (primorial prime):

- L s nguyn t c dng: hoc .

53) S nguyn t nhn (swinging prime):

- L s nguyn t m trong 1 ca mt giai tha nhn .

Chng 1: Cc con s 19

54) S nguyn t Mersenne (Mersenne prime):

- c t theio tn nh ton hc ngi Php, Marin Mersenne (1588 1648).


- Ch mi tm c s nguyn t Mersenne.

55) M nguyn t Mersenne (Mersenne prime exponents):

- L s nguyn t sao cho l s nguyn t.

56) S nguyn t s Thabit (Thabit prime):

- c t theo tn nh ton hc ngi Iraq, Al-bi Thbit ibn Qurra al-arrn (826 -

901).



57) S nguyn t Cullen (Cullen prime):

- c t tn theo nh ton hc ngi Ireland, James Aloysius Cullen (1841 - 1921).


- Hin nay ch mi tm c s:

Chng 1: Cc con s 20

58) S nguyn t Woodall (Woodall prime):

- c t theo tn nh tan hc ngi Anh, Herbert J. Woodall.


59) S nguyn t Proth (Proth prime):

- c t theo tn nh ton hc ngi Php, Franois Proth (1852 1879).

- L s nguyn t c dng: , vi l v .

60) S nguyn t Kynea (Kynea prime):


61) S nguyn t Fermat (Fermat prime):

- c t theo tn nh ton hc ngi Php, Pierre de Fermat (1601 - 1665).


- Ngi ta cho rng ch c 5 s tha mn iu ny.

62) S nguyn t Mersenne kp (double Mersenne prime):

- c t theo tn nh ton hc ngi Php, Marin Mersenne (1588 1648).

- L s nguyn t c dng: , vi l nguyn t.

- Hin nay, ch tm c s.

63) S nguyn t Wieferich (Wieferich prime):

- L s nguyn t sao cho ( ) vi khng phi l mt s m hon ho.

( )

Chng 1: Cc con s 21

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

64) S nguyn t Fermat tng qut c s 10 (generalized Fermat prime base 10):


- Tnh n nay, ch c hai s tha iu ny:

11 101

65) S nguyn t Pierpont (Pierpont prime):

- c t theo tn nh ton hc ngi M gc Connecticut, James P. Pierpont (1866

1938).

- L s nguyn t c dng: vi mt s nguyn .

y cng l lp 1 s nguyn t.

66) S nguyn t Solinas (Solinas prime):

- c t theo tn nh ton hc, Jerome A. Solinas.


67) S nguyn t Leyland (Leyland prime):

- c t theo tn nh ton hc ngi Anh, Paul Leyland.

- L s nguyn t c dng: , vi .

Chng 1: Cc con s 22

68) S nguyn t di (long prime):

- L s nguyn t sao cho, vi mt c s cho trc,

l mt s tun hon. Chng

cn c gi l s nguyn t Reptend y .

- S nguyn t , cho c s10:

69) S nguyn t Fibonacci (Fibonacci prime):

- c t theo tn nh ton hc ngi , Leonardo Pisano Bigollo (khong 1170 khong

1250).

- L s nguyn t trong dy Fibonacci.

70) S nguyn t Lucas (Lucas prime):

- c t theo tn nh ton hc ngi Php, Franois douard Anatole Lucas (1842

1891).

- L s nguyn t trong dy s Lucas.

71) S nguyn t Padovan (Padovan prime):

- c t theo tn kin trc s ngi Anh, Richard Padovan (1935 - ????).

- S nguyn t trong dy Padovan.

72) S nguyn t Pell (Pell prime):

- c t theo tn nh ton hc ngi Anh, John Pell (1611 1685).

- L s nguyn t trong dy s Pell.

Chng 1: Cc con s 23

73) S nguyn t Perrin (Perrin prime):

- c t theo tn R. Perrin.

- S nguyn t trong dy s Perrin.

74) S nguyn t hnh phc (happy prime):

- S hnh phc l s nguyn t.

75) S nguyn t may mn (lucky prime):

- Cc s may mn l s nguyn t.

76) S nguyn t ti thiu (minimal prime):

- L s nguyn t m khng c ngn hn chui con ca cc ch s thp phn to thnh mt

nguyn t.

- C chnh xc s nguyn t ti thiu:

77) S nguyn t c o (unique prime):

- Danh sch cc s nguyn t m di chu k ca vic m rng s thp phn ca l

duy nht (khng c s nguyn t no khc c cng chu k).

78) S nguyn t siu k d (super-singular prime):

- C chnh xc s nguyn t siu k d:

Chng 1: Cc con s 24

79) S nguyn t Wedderburn - Etherington (Wedderburn Etherington prime):

- c t theo tn nh ton hc ngi Scotland, Joseph Henry Maclagan Wedderburn (1882

1948) v nh ton hc ngi Anh, Ivor Malcolm Haddon Etherington (1908 - 1994).

- S Wedderburn - Etherington l s nguyn t.

80) S nguyn t Fortune (Fortunate prime):

- c t theo tn nh ton hc ngi New Zeland, Reo Franklin Fortune (1903 1979).

- Nhng con s Fortune l s nguyn t (chng cng c phng on l s nguyn t

Fortune duy nht).

81) S nguyn t s Genocchi (Genocchi number prime):

- c t tn theo nh ton hc ngi , Angelo Genocchi (1817 1889).

- l con s Genocchi dng duy nht.

82) S nguyn t Gilda (Gildas prime):

- S Gilda l s nguyn t.

- Hin nay, ch mi tm c s.

83) S nguyn t Newman - Shanks Williams (NewmanShanksWilliams prime):

- c t theo tn Morris Newman, Daniel Shanks v Hugh C. Williams.

- L s va nguyn t va l s Newman - Shanks - Williams.

84) S nguyn t Ramanujan (Ramanujan prime):

- c t theo tn nh ton hc ngi n , Srinivasa Ramanujan (1887 1920).

Chng 1: Cc con s 25

- L s nguyn m l con s nh nht cho t nht l s nguyn t t n vi

mi (tt c cc s nguyn l s nguyn t).

85) S nguyn t s Ulam (Ulam prime):

- c t theo tn nh ton hc ngi M gc Ba Lan, Stanislaw Marcin Ulam (1909

1984).

- S Ulam l s nguyn t.

86) S nguyn t Wedderburn - Etherington (Wedderburn Etherington prime):

- c t theo tn nh ton hc ngi Scotland, Joseph Henry Maclagan Wedderburn (1882

1948) v nh ton hc ngi Anh, Ivor Malcolm Haddon Etherington (1908 - 1994).

- S Wedderburn - Etherington l s nguyn t.

87) S nguyn t iu ha (harmonic prime):

- L s nguyn t m khng c cch ( ) v ( ) vi

. Khi l thng Wolstenholme.

88) S nguyn t Higgs bnh phng (Higgs prime for squares):

- L s nguyn t m chia bnh phng ca tch s ca tt c cc s hng trc n.

Chng 1: Cc con s 26

89) S nguyn t khng rng lng (non-generous prime):

- L s nguyn t m nghim nguyn dng nh nht khng phi l mt nghim nguyn

ca .

90) S nguyn t tng-nng-nng (wall-sun-sun prime):

- Mt nguyn t nu chia ht cho s Fibonacci (

) trong k hiu Legendre

c nh ngha l:

(

) {

- Hin khng c s nguyn t no nh vy.

91) S nguyn t Stern (Stern prime):

- c t theo tn nh ton hc ngi c, Moritz Abraham Stern (1807 1894).

- S nguyn t m khng phi l tng hp ca cc s nguyn t nh hn v hai ln bnh

phng ca mt s nguyn khc khng.

- Hin nay, ch tm c s nguyn t Stern:

92) S nguyn t tt (good prime):

- L s nguyn t , sao cho vi , vi l s nguyn t th

.

93) S nguyn t s Bell (Bell number prime):

- c t theo tn nh ton hc ngi Scotland, Eric Temple Bell (1883 1960).

- L s nguyn t bng s lng cc phn hoch ca mt tp c phn t.

94) S nguyn t phn hoch (partition prime):

- S phn hoch l s nguyn t.

Chng 1: Cc con s 27

95) S nguyn t s cototient cao (highly cototient number prime):

- L s nguyn t m mt s cototient nhiu hn bt k s nguyn di n, ngoi tr .

96) S nguyn t bt quy tc (irregular prime).

- L s nguyn t l m phn chia s lp ca trng cyclotomic th .

97) S nguyn t chnh quy (regular prime):

- L s nguyn t m khng phn chia s lp ca trng Cyclotomic th .

98) S nguyn t Mills (Mills prime):

- c t theo tn William H. Mills.

- L s nguyn t c dng: ,vi l Mills lin tc.

- Dng ny l s nguyn t vi l s nguyn dng.

99) S nguyn t tiu hy (annihilating prime):

L s nguyn t sao cho ( ) , trong ( ) l ci bng ca mt chui cc s t nhin.

100) S nguyn t yu (weakly prime):

- S nguyn t m c bt k mt s trong (c s ) ch s ca chng c thay i

thnh mt gi tr bt k khc s lun lun dn n mt s a hp.

Chng 1: Cc con s 28

II. S hon ho:

S hon ho (perfect number), l s nguyn dng m tng tt c cc c s dng khc

chnh n ca n bng chnh n.

V d:

c c dng: { }, ta c . Nn l s hon ho.

c c dng: { }, ta c . Nn l s hon

ho.

Hin ti, tnh n ngy , th gii ch mi tm ra c s hon ho. S hon

ho cui cng c tm ra l vo nm .

Mi s hon ho u c dng: ( ).

Danh sch s u tin:

STT S

ch s Nm tm ra

Ngi tm ra

Hy Lp c i

Hy Lp c i

Hy Lp c i

Hy Lp c i

Leonard Eugene Dickson

Pietro Cataldi

Pietro Cataldi

Leonhard Euler

Ivan Mekheevich Pervushin

R. E. Powers

R. E. Powers

douard Lucas

Raphael M. Robinson

Raphael M. Robinson

Raphael M. Robinson

Raphael M. Robinson

Raphael M. Robinson

Hans Riesel

Hurwitz

Hurwitz

Chng 1: Cc con s 29

Donald B. Gillies

Donald B. Gillies

Donald B. Gillies

Bryant Tuckerman

Landon Curt Noll v Nickel

Landon Curt Noll

Harry Lewis Nelson v David Slowinski

David Slowinski

Colquitt v Welsh

David Slowinski

David Slowinski

David Slowinski v Paul Gage



Armengaud, George Woltman

Spence, George Woltman

Clarkson, George Woltman, Kurowski

Hajratwala, George Woltman, Kurowski

Cameron, George Woltman, Kurowski

Shafer, George Woltman, Kurowski

Findley, George Woltman, Kurowski

Nowak, George Woltman, Kurowski

Curtis Cooper, Boone, George Woltman, Kurowski

Curtis Cooper, Boone, George Woltman, Kurowski

Elvenich, George Woltman, Kurowski

Strindmo, George Woltman, Kurowski

Smith, George Woltman, Kurowski

Danh sch s hon ho u tin:

III. S a gic:

- S a gic (polygonal number) l mt s i din nh s chm hoc s si c sp xp

ca mt a gic u.

Chng 1: Cc con s 30

- Mt s tn thng thng ca n l s tam gic (triangular number), s vung - s chnh

phng (square number), s ng gic (pentagonal number), s lc gic (hexagonal number).

V d:

S cnh

Tn gi

Hnh nh Cng thc

S

tam gic

S vung

(s chnh phng)

S

ng gic

S

lc gic

Cng thc tng qut c th c tnh bng:

( ) ( ) ( )

( )( )

IV. S chnh phng:

S chnh phng (square number), l s m cn bc hai ca n l s nguyn.

Danh sch s chnh phng u tin:

V. Phn s, lin phn s:

- Lin phn s (continued fraction), l phn s m c th vit di dng:

VI. B s Pythagoras:

- B s Pythagoras l b ba cnh ca mt tam gic vung m cc cnh l cc s nguyn

dng.

- Cng thc tng qut, l cng thc Euclid (Euclids formula) sau y:

Vi, , ta c:

Chng 1: Cc con s 31

- Danh sch b s Pythagoras u tin c cho trong bng sau:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

Chng 1: Cc con s 32

VII. B s Heron:

- Tam gic Heron (Heron triangular) l tam gic c di c ba cnh v din tch ca tam

gic l s nguyn.

- Cng thc chnh xc ca tam gic Heron l:

( )

( )

( )( )

( )

( ) ( )( )

Vi,

{

( )

Danh sch b tam gic Heron u tin cho trong bng sau:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

Tam gic Heron kh bng (equable Heronian triangle):

- Hnh kh bng (equable shape) hay hnh hon ho (perfect shape) l hnh c din tch

bng chu vi.

- Ch c tam gic Heron c din tch bng chu vi l:

( ) ( ) ( ) ( ) ( ) Tam gic Heron hu nh u (almost-equilateral Heronian triangle):

- L tam gic Heron m c thm tnh cht cp s c dng ( ).

- Cng thc chnh xc tnh l :

( ) ( )

- Danh sch 10 b tam gic Heron hu nh u u tin cho trong bng sau:

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

VIII. adsf

Chng 2: Cc php chia ht 33

I. Cc du hiu chia ht (divisibility rule):

Cho s nguyn:

Ta c cc du hiu chia ht sau:

{

( ) ( ) ( )

{

{

( ) ( ) ( )

{

{

( )

{

( )

{

II. Bng ng d (table of congruences):

( ) Tha mn v s cc s nguyn dng . ( ) Trng hp c bit ca nh l nh Fermat. ( ) Nghim ca n c gi l s nguyn t Wieferich.

( ) ( ) Mt s l nguyn t khi v ch khi n tha mn biu thc ng d ny (nh l Wilson).

( ) ( ) Nghim ca n c gi l s nguyn t Wilson.

(

) ( ) Tha mn vi mi s nguyn t ln hn (nh l Wolstenholme).

(

) ( ) Nghim ca n c gi l s nguyn t Wolstenholme.


(

) ( ) Phi tha mn bng mt phn v d ti nh l ngc ca nh l Wolstenholme.

(

) ( ) Tha mn vi mi s nguyn t l.

(

) ( ) Nghim ca n gi l s nguyn t Wall Sun Sun.

III. c bi:

c chung ln nht (greatest common divisor), k hiu ( ).

Bi chung nh nht (least common multiple), k hiu ( ).

nh l:

( ) ( )

Thut ton Euclid:

{

( )

( ) ( )

( ) ( )

V d:

( ) ( ) ( ) ( )

( )

IV. Phng trnh Diophantus:

Phng trnh Diophantus (Diophantine equation) c t theo tn ca nh ton hc Hy

Lp c i, Diophantus ca thnh Alexandria (khong (gia 200 v 214) (284 n 298)).

V. ng d:

VI. Thut chia Euclid:

VII. ng d Trung Hoa:

VIII. Chuyn i h c s:


IX. Adshfk

X. adshfk

XI. adsfhk

Chng 3: Ton hc gii tr 36

I. Ngun ti liu:

http://en.wikipedia.org/wiki/List_of_recreational_number_theory_topics

II. Hnh vung ma thut:

- Hnh vung ma thut, ma phng (magic square) c in l mt bng hnh vung kch

thc , c in kn cc s t n theo nguyn tc, tng cc s trn mi hng, v

tng cc s trn mi ct u bng nhau. i khi, i vi ma phng c l, cn c thm

iu kin l tng cc s trn cc ng cho cng bng nhau.

- iu k l l khng tn ti ma phng cp . Nhng cc cp cao hn th li tnh ton c.

Mi ma phng cp l u tn ti.

V d:

- Ta c cng c th tnh v tr th ( ) ca ma phng cp l bng cng thc sau:

((

) ) (( ) )

V d:

III. Ngi sao ma thut:

- Ngi sao ma thut (magic star) l mt a gic hnh sao m tng cc s trn cc ng

Chng 3: Ton hc gii tr 37

thng u bng nhau.

V d:

Hnh su cnh ma thut

(magic hexagram) Hnh by cnh ma thut

(magic heptagram) Hnh tm cnh ma thut

(magic octagram)

IV. fsad

Hkladf

Chng 4: L thuyt s hc i s 38

Shfk

Dsjfl

Dasfjl

dfhkadsf

Chng 5: L thuyt s hc gii tch 39

I. Cc hm s lin quan n l thuyt s hc:

1) Hm c s:

- Hm c s (divisor function), k hiu ( ) l hm s hc c nh ngha l tng ly

tha tt c cc c ca :

( )

- Khi , th hm c s c gi l hm s lng c s (number-of-divisors function).

Lc , k hiu ( ) ( ) ( ) c thay th cho ( ).

- Khi , th hm c s c gi l hm sigma (sigma function) hay hm tng cc c

(sum-of-divisors function). Lc , k hiu l ( ) c thay th cho ( ).

- Tng phn c (aliquot sum), k hiu ( ), c nh ngha l tng tt c cc c ca ,

nhng tr i chnh s .

( ) ( ) ( )

2) S nguyn t cng nhau, totative, hm totient Euler:

- S nguyn dng nguyn t cng nhau (coprime) vi s nguyn dng , nu hai s

khng c cng c no khc :

( )

- Mt s nguyn dng c gi l totative vi s nguyn dng , nu v

nguyn t cng nhau:

V d 1: totative ca l:

- Hm s m tt c cc totative ca c gi l hm totient Euler (Eulers totient

function) hoc hm phi (phi function), k hiu ( ).

Tnh cht:

( ) ( ) ( )

Nu, l s nguyn t v th:

( )

Cng thc tch Euler (Eulers product formula):

( ) (

)

{

V d 2:

Cch 1: totative ca l: . Vy ( ) .

Cch 2: dng khng thc tch Euler: ( ) ( ) (

) (

) .

Tng c s (divisor sum):

( )

3) Hm m s nguyn t:

- Hm nguyn t m (prime-counting function), k hiu ( ), c nh ngha l hm m


s lng tt c cc s nguyn t b hn bng s thc .

Tnh cht:

+ nh l s lng s nguyn t (prime number theorem):

( )

( )

+ Tng ng vi nh l trn, nm 1896, nh ton hc ngi Php, Jacques Salomon

Hadamard ForMem (1865 1963) v nh ton hc ngi B, Charles-Jean tienne Gustave

Nicolas de la Valle Poussin (1866 - 1962) chng minh (mt cch c lp) rng:

( )

( )

4) Phn hoch, hm phn hoch:

- Phn hoch (partial hoc integer partial) ca s nguyn dng l cch vit s ra thnh

tng ca cc s:

V d 1:

- Hm phn hoch (partial function), k hiu ( ), l hm s m s lng phn hoch ca

mt s t nhin .

V d 2:

( )

II. Cc ng thc s hc:

1) ng nht thc Bzout:

- c t theo tn nh ton hc ngi Php, tienne Bzout (1730 - 1783). Bzout chng

minh ng nht thc ny trong a thc. Tuy nhin, gi thuyt ny cho cc s nguyn th c

chng minh bi nh ton hc ngi Php, Claude Gaspard de Bachet Mziriac (1581 - 1638).

- ng nht thc Bzout (Bzouts identity), cn gi l b Bzout (Bzout lemma) l

mt nh l c bn trong l thuyt s hc. Vi l cc s nguyn, khng ng thi bng ,

v l c chung ln nht ca . Khi tn ti s nguyn v sao cho:

Khi :

+ l s nguyn dng nh nht c th vit di dng .

+ Tt c cc s nguyn c dng u l bi s ca . v c gi l h s Bzout

cho ( ), v chng khng phi l duy nht. Mt cp h s Bzout c th c tnh bng cc

thut ton Euclid m rng.

- Ta bit, phng trnh c v s nghim . C th, sau khi mt cp h s

Bzout ( ) c tnh ton (s dng m rng Euclid hoc mt s thut ton khc), tt c

cc cp ( ) khc c th c tm thy bng cch s dng cng thc:

{(

( )

( )) | }


- M rng ra cho nhiu bin, nu:

( )

th tn ti cc s nguyn sao cho:

c cc tnh cht sau:

( ) l s nguyn dng nh nht c dng .

( ) mi s ca dng l bi ca .

( ) l c chung ln nht ca , gha l mi c chung ca u

chia ht cho .

- ng nht thc Bzout khng ch ng trong vnh s nguyn, m cn trong bt k ldeal

(tp con c bit ca vnh quan trng khc (PID). C ngha l, nu l mt PID, v

thuc , v l c chung ln nht ca v th tn ti v trong sao cho

. Bi v, cc Ideal tng ng vi . Trong min tch phn, ng nht thc

Bzout c gi l mt min Bzout.

2) ng nht thc 2 bnh phng ca Brahmagupta Fibonacci:

- c t theo tn nh ton hc v thin vn hc ngi n , Brahmagupta (598 - 668) v

nh ton hc ngi , Leonardo Pisano Bigollo (1170 1250).

- Trong i s, ng nht thc 2 bnh phng ca Brahmagupta Fibonacci

(Brahmagupta Fibonacci two-square identity), cn gi l ng nht thc Fibonacci

(Fibonaccis identity) cho ta kt qu l tch ca hai tng hai bnh phng l tng ca hai bnh

phng. C th l:

( )( ) ( ) ( ) ( ) ( )

V d:

( )( )

- Lin h ti s phc:

Cho l cc s thc, ng nht thc tng ng:

( )( )

( )( ) ( ) ( )

- p dng vi phng trnh Pell:

Brahmagupta p dng tng ca ng trong phng trnh Pell, tc l .

Tng qut hn:

(

)(

) ( ) ( )

ng kt hp b ba ( )( ) (vi ), suy ra b ba mi:

( )

3) ng nht thc 4 bnh phng ca Euler:

ng nht thc 4 bnh phng ca Euler (Eulers four-square identity) l:

(

)(

)

( )

( )

( )

( )

4) ng nht thc 8 bnh phng ca Degen:

- c chng minh nm 1818 bi nh ton hc ngi an Mch, Carl Ferdinand Degen


(1766 1825).

- ng nht thc 8 bnh phng ca Degen (Degens eight-square identity) thit lp v kt

qu ca 2 s, mi s l 1 tng ca 8 bnh phng,c th l:

(

)(

)

( )

( )

( )

( )

( )

( )

( )

( )

5) ng nht thc 16 bnh phng ca Pfister:

Ln u c chng minh bi H.Zassenhaus v W.Eichhorn trong nhng nm 1960 v c

chng minh c lp bi Pfister cng trong khong thi gian . C nhiu bn khc nhau, mt

bn sc tch nht trong s chng l:

vi l:

(

) ( )

(

) ( )

(

) ( )

(

) ( )

(

) ( )

(

) ( )

(

) ( )

(

) ( )

v:

cng vng theo:

M ng thc :


(

)(

)

6) ng nht thc Hermite:

- c t theo tn nh ton hc ngi Php, Charles Hermite (1822 1901).

- ng nht thc Hermite (Hermites identity) pht biu l: vi mi s thc v s nguyn

dng , cho gi tr ca mt tng:

Vi,

l hm sn ca s thc .

III. Cc nh l bnh phng:

1) nh l hai bnh phng ca Fermat (Fermats two-square theorem):

- c chng minh khong nm 1640 bi nh Ton hc ngi Php, Pierre de Fermat (1607

1665).

- nh l ny pht biu l mt s nguyn t c th c vit di dng tng bnh phng

ca s nguyn:

khi v ch khi:

( )

2) nh l ba bnh phng ca Legendre (Legendres three-square theorem):

- c chng minh nm 1798 bi nh Ton hc ngi Php, Adrien-Marie Legendre (1752

1833).

- nh l ny pht biu l mi s t nhin khng c dng ( ) (vi

{ }) u c vit di dng tng bnh phng ca s nguyn:

3) nh l bn bnh phng ca Lagrange (Lagranges four-square theorem):

- c chng minh nm 1770 bi nh Ton hc ngi , Joseph-Louis Lagrange (1736 -

1813).

- nh l ny pht biu l mi s t nhin u c vit di dng tng bnh phng ca

s nguyn:

V d:

4) nh l bn bnh phng ca Jacobi (Jacobis four-square theorem):

- c chng minh nm 1834 bi nh Ton hc ngi c, Carl Gustav Jacob Jacobi (1804

1851).

IV. sdf

adfhk

Chng 6: L thuyt s hc t hp 44

Adsfhk

Adsfhk

adsfhl

Chng 7: L thuyt gii m 45

Sdhfk

Adsfhk

Adsfhk

dsfahk

Chng 8: Logic 47

I. Ngun ti liu: http://en.wikipedia.org/wiki/List_of_mathematical_logic_topics

II. Cc khi nim ton hc:

nh ngha, nh l, tin ,

III. Tin :

http://en.wikipedia.org/wiki/List_of_axioms

IV. Nghch l:

Di y l danh sch cc nghch l, c nhm li theo ch . Cch chia cc nhm l

tng i, mi nghch l c th ph hp vi nhiu ch . V nh ngha ca thut ng

nghch l khc nhau, mt trong s nhng nghch l di y khng c mt s nh ton hc

cng nhn. Danh sch ny thu thp cc nghch l c thu thp t mt s ngun v cc bi

vit ring.

Mc d c coi l nghch l, mt s trong s ny c da trn l lun ngu bin, hoc

phn tch khng y / b li. Nhng nhn chung, cc thut ng ny thng c s dng

m t mt kt qu phn trc gic. Vi kin thc khng chuyn, chng ti ch c th lit k

ra mt vi nghch l tiu biu, v ni dung v l thuyt ca nghch l rt kh hiu v khng

ph bin, nn cn c thi gian cc bn nghin cu su hn v n.

1) mt Logic

Nghc l Catch-22: Mt tnh hung trong mt ngi no ang cn mt ci g m ch

c th c c bng cch khng cn n na

Nghch l v ngi ung ru: Trong qun ru, nu c mt khch ung ru, th tt c mi

ngi trong qun u ung ru.

Nghch l ca s tha k: Cc gi thuyt khng nht qun lun to ra mt kt qu hp l.

Nghch l x s: Nu c mt tm v s x s trng ln, th ta c l do tin mt v x s c

c khng phi l tm v s trng , do xc sut m t c tm v s trng c bit l

rt nh, nhng khng phi l hon ton khng c kh nng trng v c c.

Nghch l (Raven hay Ravens ca Hempel): Quan st mt qu to mu xanh l cy lm tng

kh nng ca tt c cc con qu l en.

2) T tham chiu:

Nhng nghch l c chung mt mu thun pht sinh cng mt tham chiu.

Nghch l ca th co: Mt th co c th co ru, tc, nhng th m nhng ngi n

ng khc khng t lm c. Vy anh ta c th co cho bn thn mnh? (Ph bin ca l

thuyt tp hp nghch l ca Russell.)

Nghch l Berry: Cm t s u tin khng di mi t xut hin t tn cho n

trong chn t.

Lng c su: Nu mt con c su cp mt a tr v ha s tr li a tr nu ngi

cha on chnh xc nhng g con c su s lm. Con c su s lm g nu ngi cha on rng

a tr s khng c tr li?

Nghch l ca Ta n: Mt sinh vin lut ng tr cho c gio ca mnh sau khi thng v

kin u tin ca mnh. Sau , gio vin li yu cu mt sinh vin khc (ngi thua cuc

trong v kin) thanh ton.

Nghch l ca Curry: Nu cu ny l ng, th ng gi Noel tn ti.

Chng 8: Logic 48

Nghch l Epimenides: Mt ngi dn o Crete ni: Tt c dn o Crete l k ni di.

Nghch l ny tng t nh nghch l Liar.

Nghch l ca ngoi l: Nu tn ti mt ngoi l cho mi quy tc, th mi quy tc phi c t

nht mt ngoi l, chnh ngoi l l khng c ngoi l khc. Lun lun c mt ngoi l cho

quy tc, ngoi tr nhng trng hp ngoi l ca quy tc , ngoi l c iu c chp

nhn trong cc quy tc. Trong mt th gii khng c lut l, cn c t nht mt quy tc -

Mt quy tc chng li cc quy tc cn li

Nghch l Grelling-Nelson: T heterological, c ngha l khng p dng i vi bn

thn, vy t heterological c p dng cho bn thn n khng? (Mt nghch l gn vi nghch

l Russell.)

Nghch l Liar (nghch l ca k ni di): Cu ny l sai. y l nghch l t tham chiu

kinh in. Tng t nh cu C th tr li cho cu hi ny khng?, Ti ang ni di, v

Tt c mi th ti ni l mt li ni di.

Nghch l Card: Cc cu theo l ng s tht. Cc tuyn b trc l sai. Mt bin th

ca nghch l liar m khng s dng t tham chiu.

Nghch l Pinocchio: iu g s xy ra nu Pinocchio ni Mi ca ti s c pht trin?

Nghch l Russell: C tn ti tp hp no m cha tt c cc tp m khng cha chnh bn

thn n?

Nghch l Socrates: Ti bit rng ti khng bit g c.

3) ton hc:

Nghch l Cramer: s lng cc im giao nhau ca hai ng cong bc cao c th ln hn

s lng cc im ty cn thit xc nh mt ng cong nh vy.

Nghch l thang my: Thang my c th dng nh ch yu l i theo mt hng, nh l

c c ch to gia ca ta nh v c tho ri trn mi nh v tng hm.


thn n?

4) Thng k:

Nghch l ca chnh xc: Mt m hnh c lng vi mt chnh xc c th mnh hn mt

c lng ln hn vi m hnh v chnh xc cao hn.

Nghch l tnh bn: i vi hu ht mi ngi, bn b ca h c nhiu hn bn m h quen.

Nghch l v nh cn: Tr nh cn c t l t vong cao hn l do ngi m ht thuc. Tr s

sinh c m ht thuc c trng lng s sinh trung bnh thp, nhng tr nh cn sinh ra ht

thuc li c t l t vong thp hn so vi tr nh cn vi nguyn nhn khc. (Mt trng hp

c bit ca nghch l Simpson.)

Hin tng Will Rogers: Cc khi nim ton hc ca trung bnh, cho d nh ngha l trung

bnh hoc trung tm, dn n nghch l r rng kt qu, v d, c th l di chuyn mt mc t

bch khoa ton th cho mt t in s lm tng chiu di nhp trung bnh trn c hai cun

sch.

5) Xc sut:

Nghch l Monty Hall: Bn s chn ca no?

Nghch l hp Bertrand: Mt nghch l xc sut c iu kin c lin quan cht ch vi

Nghch l C b hay cu b Nghch l Bertrand: nh ngha tr khn khc nhau ngu nhin

cho kt qu hon ton khc nhau.

Nghch l ngy sinh nht: C bao nhiu c hi m hai ngi trong mt cn phng c cng

Chng 8: Logic 49

ngy sinh?

Nghch l Borel: Hm mt xc sut c iu kin l khng bt bin theo bin i ta .

Nghch l C b hay cu b: Mt gia nh hai con c t nht mt cu b. Xc sut n c mt

b gi l bao nhiu?

Nghch l sai dng tnh: Mt xt nghim l chnh xc c th cho bn c mt cn bnh,

nhng xc sut bn thc s c n vn c th tr nn rt nh.

Nghch l c vt: Mt cuc nh cc gia hai ngi dng nh c li cho c hai. Tng t

nh bn cht vi nghch l hai phong b.

Nghch l Simpson: Tn ti s lin h trong tiu qun th c th b o ngc trong qun th.

Tn ti hai tp hp s liu h tr ring mt gi thuyt no , nhng, khi tng hp li cng

nhn gi thuyt ngc li.

Vn t ba l bi: Nu rt ngu nhin mt l bi, lm th no bn xc nh mu sc ca

mt bn di?

Vn ca ba t nhn: l mt bin th ca vn Monty Hall.

Nghch l hai phong b: Bn c hai phong b ging nhau, trong mi phong b u cha mt s

tin. Mt phong b cha gp i phong b cn li. Bn c th chn mt phong b v gi s

tin trong . Bn chn mt phong b mt cch ngu nhin nhng trc khi bn m n, bn

c c hi c phong b khc.

6) cng v v cng nh:

Nghch l Burali-Forti ca: Nu s th t hnh thnh mt tp, n s l mt s th t l nh

hn so vi chnh n.

Nghch l Cantor: Khng c s th t ln nht.

Galileos paradox: Though most numbers are not squares, there are no more numbers than

squares.

Nghch l Grand Hotel ca Hilbert: Nu mt khch sn vi v s phng c ngi, vn c

th cha them ngi.


thn n?

Nghch l Skolem: m c tp v hn ca l thuyt tp hp cha tp v hn khng m

c.

Supertasks c th dn n nghch l nh nghch l Ross-Littlewood v nghch l Benardete.

Nghch l Zeno: Bn s khng bao gi chm im B t im A nu bn c o khong cch

bng mt na ca mt na (iu ny cng l mt nghch l vt l.)

7) Hnh hc v topology:

Nghch l Banach-Tarski: Mt qu bng c th b phn hy v tp hp li thnh hai qu bng

c kch thc ging nh bn gc.

Nghch l Banach-Tarski: Ct mt qu bng vo mt s hu hn cc phn, ti lp li cc

ming c c hai qu bng, c hai kch thc bng qu bng u tin.

Nghch l Von Neumann: l mt nghch l tng t trong khng gian hai chiu.

Nghch l tp hp: Mt tp hp c th c phn chia thnh hai tp hp, mi tp u

tng ng vi tp ban u.

Nghch l ng b bin: chu vi ca mt vng t rng l mt nh ngha yu.

Hausdorff nghch l: C tn ti mt tp con C m c ca hnh cu S sao cho S \ C phn

tch (equi decomposable) cng vi hai bn sao ca chnh n..

Chng 8: Logic 50

Tm mnh xp hnh vung: Hai hnh tng t appear to have different areas trong khi hon

tt t cc mnh tng t.

Nghch l Smale: Mt hnh cu c th c ln t trong ra ngoi.

Nghch l Abilene: Mi ngi c th a ra quyt nh khng da trn nhng g h thc s

mun lm, nhng h ngh rng nhng ngi khc mun lm, vi kt qu l tt c mi ngi

quyt nh lm mt ci g m khng ai thc s mun lm, nhng ch h ngh rng tt c

mi ngi khc mun lm.

Nghch l ca Alabama: Tng tng s ch ngi c th thu nh ch ngi ca mt dy gh.

Nghch l ca tiu bang mi: Thm mt tiu bang mi hoc chn biu quyt c th tng s

lng bnh chn ca ngi khc.

Nghch l xanh: Nhng chnh sch c nh gim lng kh thi CO2 trong tng lai c th

dn n pht thi tng trong hin ti.

Cu v c t ca Kavka: Liu c ngi no ung cc cht c khng gy t vong, ch v

mun c c phn thng?

Nghch l Newcomb: Lm th no chi mt trn u vi mt i th ton tr ton thc?

8) Vt l:

Bnh t chy ca Robert Boyle t lm y chnh n (nh hnh trn), nhng ng c chuyn

ng vnh cu khng h tn ti.

Nghch l vng nhit i mt: Mt mu thun gia cc m hnh c tnh nhit nhit i

trong khong thi gian m p, giai on khng b ng bng ca k Creta v Eocen, v nhit

lnh hn bnh thng hin din.

Cc nguyn tc hnh ni ba chiu: S lng thng tin c th c lu tr trong mt khi

lng nht nh khng t l thun vi khi lng nhng t l vi khu vc gii hn khi lng

.

Nghch l bt kh khng p buc: iu g xy ra nu 1 lc khng th ngn cn nhn 1 i

tng bt ng?

9) Thin th hc:

Nghch l thin vn Algol: Trong 1 s cp i tc dng nh h khc nhau v tui, hoc

ngay c khi h suy ngh ln k hoch cng lc.

Nghch l mt tri tr m nht: M nht tr Sun nghch l: cc mu thun r rng gia cc

quan st nc nhng giai on u trong lch s hnh thnh tri t v nhng trin vng ca

thin th cho thy nng lng pht tn ca mt tri tr c c khng lm tan chy

bng trn tri t.

Nghch l GZK: cc tia v tr nng lng cao c quan st thy dng nh vi phm cc

gii hn Greisen-Zatsepin-Kuzmin, y l mt h qu ca thuyt tng i hp.

10) C hc c in:

Nghch l x th: bn trng mc tiu ca mnh, 1 x th buc phi ko nhm trc tip vo

mc tiu, nhng lch sang bn cnh 1 cht.

Nghch l Archimedes: Mt chin hm thit gip ln, vn c th ni trong vi lt nc.

Bnh xe nghch l ca Aristotle: Nhng bnh xe c quay ng tm, s li vt c khong

cch ging nhau ng vi chu vi ca n, thm ch chu vi c khc nhau.

Nghch l ca Carroll: Moment ng lc hc ca 1 cy gy l 0, tuy nhin li ko phi vy.

Nghch l ca DAlembert: dng chy ca mt cht lng khng nht khng sn sinh ra lc

tng hp ln vt rn.

Chng 8: Logic 51

Nghch l ca Denny: B mt c th ca ng vt chn t (chng hn nh rn nc) khng

th gip n y ngi theo chiu ngang.

Nghch l thang my: Thm ch dng nhiu thy trng k o mt cht lng, 1 trong s

s khng ch ra c s thay i ca mt cht lng nu ta thay i p sut kh quyn.

Cha chy t ng Feynman: C cch no 1 bnh phun xoay c khi trong 1 chic xe

tng, v c th ht cht lng xung quanh.

Nghch l Painlev: Nhng vt rn ng lc hc khi tip xc v ma st th mu thun.

Nghch l l tr: Khi khuy 1 cc tr, nhng chic l t tp gia cc, mc d c lc ly tm

y chng ra thnh cc.

11) tr hc:

Nghch l ca Bentley: Trong thuyt v tr ca Newton, lc hp dn c th ko tt c vt cht

vo mt im duy nht.

Nghch l Fermi: t gi thuyt nu nh c mt kh nng nao , tn ti nhiu loi sinh vt

khc trong v tr, vy chng u? Ti sao s hin din ca chng li ko r rng?

Nghch l nhit nng n cht: K t khi v tr khng v hn nh xa, n khng th v

hn thm na.

Nghch l ca Olbers: Ti sao tri m mu en, trong khi c mt ngi sao v tn?

12) in t hc:

Nghch l Faraday: Mt s vi phm r rng ca nh lut Faraday l cm ng in.

13) C hc lng t:

nh l ca Bell: Ti sao o lng lng t ht nhn khng p ng ton hc l thuyt xc

sut?

Th nghim khe i: Vt cht v nng lng c th th hin di dng sng hoc dng ht

ty thuc vo th nghim.

Nghch l Einstein-Podolsky-Rosen: C th lm tng nh hng ca cc h qu ln nhau

trong c hc lng t c hay khng?

Nghch l tuyt chng: Vi bc sng nh c gii hn, tng tit din tn x ca mt mt cu

khng xuyn c bng hai ln din tch mt ct ca n (l gi tr thu c trong c hc c

in).

Nghch l ca Hardy: Lm th no chng ta c th a ra mt suy lun v nhng s vic

trong qu kh, khi m chng ta khng quan st v tha nhn rng hnh ng ang quan st c

thc s nh hng n vic chng ta ang suy lun hay khng?

Nghch l Klein: khi th nng ca mt vt cn c th nng bng vi khi lng bn ph ht

nhn, n s tr nn trong sut.

Bi ton Mott: Quan st v hm sng i xng cu, ta thy n to ra vt tuyn tnh ht.

Nghch l mch lng t LC: cc ngun nng lng c lu tr trn in dung v in cm

bng vi nng lng trng thi c bn ca giao ng lng t.

Lng t thn giao cch cm: hai ngi chi khng th giao tip thc hin nhim v m

nhim v i hi cn phi c giao tip trc tip.

Nghch l con mo ca Schrdinger: mt nghch l lng t-l mo cn sng hay cht

trc khi chng ti nhn thy?

Nguyn l bt nh: th nghim xc nh v tr cn chi phi ng lng, v ngc li.

14) Thuyt tng i:

Nghch l tu v tr ca Bell: lin quan n thuyt tng i.

Chng 8: Logic 52

Nghch l l en thng tin: l en vi phm mt nguyn l ph bin gi ca khoa hc-thng

tin khng th b ph hy.

Nghch l Ehrenfest: trong chuyn ng hc ca mt a cng, quay.

Nghch l ci thang: mt vn tng i c in.

Nghch l tng vn tc ca Mocanu: mt nghch l thuyt tng i hp.

Nghch l ca Supplee: sc ni ca mt i tng mang tnh tng i (chng hn nh mt

vin n) dng nh thay i khi h quy chiu thay i t mt vin n trng thi ngh

thnh vin n ang bay trng thi dng.

Nghch l Trouton-Noble hay n by gc bn phi. C mt m-men xon pht sinh trong

mt h thng tnh khi thay i trng thi khng?

Nghch l sinh i: l thuyt tng i tng qut tin on rng mt ngi thc hin mt

chuyn i du lch s tr nn tr hn anh/ ch em sinh i ca h ang nh.

15) ng nhit hc:

Nghch l Gibbs: trong iu kin kh l tng, entropy c l mt bin s bao qut khng?

Nghch l ca Loschmidt: ti sao c s gia tng khng th trnh khi ca entropy khi cc

nh lut ca vt l l bt bin vi thi gian ngc? S i xng thi gian ngc ca cc nh

lut vt l xut hin mu thun vi nh lut nhit ng lc hc th hai.

Con qu Maxwell: nh lut nhit ng lc hc th hai dng nh vi phm bi mt ca sp

c vn hnh kho lo.

Nghch l Mpemba: trong iu kin nht nh nc nng c th ng nhanh hn so vi nc

lnh, mc d n phi vt qua nhit thp hn trong qu trnh ng bng.

16) Ha hc:

Nghch l Faraday (in ha hc): axid nitric long s n mn thp, trong khi axid nitric m

c th khng.

Nghch l Levinthal: di thi gian cn cho mt chui protein c c trng thi gp

khc nhiu bc, s ngn hn thi gian t d tm tt c cc cu hnh c th.

Nghch l SAR: C ngoi l cho nguyn tc sau: mt thay i nh trong mt phn t gy ra

mt s thay i nh trong din bin ha hc ca n l hon ton thng xuyn.

17) Thi gian:

Nghch l Bootstrap: mt khch du lch c thi hn c th gi cho anh y thng tin m khng

r ngun gc hay khng?

Nghch l Predestination: Mt ngi n ng i ngc thi gian v pht hin nguyn nhn

gy ra mt m chy ni ting. Trong ta nh ni ngn la bt u, ng v tnh chm vo

mt chic n lng du ha v gy ra mt m chy, ngn la ging ht trc truyn

cm hng cho anh ta, mt nm sau khi quay ngc thi gian. Nghch l bootstrap quan h

cht ch vi iu ny, trong , l kt qu ca thi gian i du lch, thng tin hoc cc i

tng xut hin m khng c bt u.

Nghch l bin i theo thi gian: nhng g xy ra khi mt khch du lch c thi hn lm vic

trong qu kh m ngn cn anh ta lm ti ni h u tin?

Nghch l ng ni: bn i ngc thi gian v git cht ng ni ca bn trc khi ng sinh ra

cha hoc m ca bn, iu ny loi tr kh nng bn c th thai, do , bn khng th i

ngc thi gian v git cht ng ni ca bn.

Nghch l ca st nhn Hitler: bn quay ngc thi gian ng lc v git cht mt ngi ni

Chng 8: Logic 53

ting trong lch s trc khi h tr nn ni ting; nhng nu nhng ngi cha bao gi

c ni ting th ng khng th c mc tiu l nhm vo h c.

18) Sinh hc:

Nghch l lm giu: tng ngun thc n c sn trong mt h sinh thi c th dn n s mt

n nh, v thm ch n tuyt chng.

Nghch l Php: Quan st cho thy rng ngi Php mc phi mt t l tng i thp cc

bnh tim mch vnh, mc d c mt ch n tng i giu cht bo bo ha.

Nghch l v ng: s lng ng glycogen ln trong gan khng th c l gii bi s

hp th ng ca n rt nh.

Nghch l ca mu xm: mc d khi lng c bp ca c heo tng i nh, chng vn c

th bi tc cao v kh nng tng tc ln.

Nghch l Ty Ban Nha: cuc tm kim cho thy ngi Hispanics ti Hoa K c xu hng c

sc khe tt hn ng k so vi dn s trung bnh mc d d on tng ch s kinh t x hi

ca h thp.

Nghch l Lombard: Nng ngi ng dy khi bn ang ngi hay ngi xm, gn kheo v c

bn u hot ng cng mt lc, mc d chng i lp vi nhau.

Nghch l Mexico: Tr em Mexico c xu hng c trng lng sinh cao hn d kin so vi

tnh trng kinh t-x hi ca h.

Nghch l ngi thin: bin dao ng nhp tim khi ngi thin ln hn ng k so vi trng

thi trc khi ngi thin v cng nh 3 nhm t th khng thin khc.

Nghch l ca thuc tr su: p dng thuc tr su cho mt dch hi c th lm tng s phong

ph ca cn trng.

Nghch l ca sinh vt ph du: ti sao c rt nhiu loi sinh vt ph du khc nhau, mc d

cuc cnh tranh cc ngun thc n li c xu hng gim vi s lng cc loi?

Nghch l ca Peto: Con ngi b ung th vi tn s cao, trong khi ng vt c v ln hn,

ging nh c voi, th li khng mc phi. Nu ung th l mt s xui ri m tnh cp t

bo, v cc sinh vt ln hn c nhiu t bo hn, v do ung th c kh nng di cn nhiu

hn, ngi ta d on nhng sinh vt ln hn c l cn nhiu nguyn nhn hn na dn

n ung th.

Nghch l Pulsus: Vi mt ng nghe, i khi c th nghe c nhp tim m ko cn chm vo

c tay. Nghch l ny cn c bit vi tn gi Nghch l Xung.

Nghch l Sherman: mt dng d thng ca di truyn trong ca hi chng yu t bt thng

X.

Nghch l thi gian (c sinh vt hc): T tin ca cc loi chim sng vo lc no?

19) Nhn thc:

Nghch l Tritone: Mt o nh ca thnh gic khi chi tun t tng cp giai iu Shepard, i

vi mt s ngi s cm thy bng dn, v trm dn vi mt s khc.

Nghch l Blub: Kha nhn thc ca mt s cc lp trnh vin c kinh nghim, khin h

khng nh gi ng cht lng ca ngn ng lp trnh m h khng bit.

20) Chnh tr:

-Nghch l n nh - khng n nh: Khi hai nc u c v kh ht nhn, t l xy ra mt

cuc chin tranh trc tip gia hai nc gim rt nhiu, nhng kh nng xung t nh hoc

chin tranh gin tip gia hai nc tng ln.

Chng 8: Logic 54

21) Lch s:

-Georg Wilhelm Friedrich Hegel: Chng ta hc hi t lch s iu chng ta khng nn hc t

lch s.

22) Tm l hc:

Nghch l t hp th: Mi quan h mu thun do s t nhn thc c mc cao kt hp vi

mc tm l au kh cao v vi tm l lnh mnh. Ngha l ngi cng hiu bit, s hoc

l au kh nhiu hn hoc l sng vui v hn ngi bnh thng.

23) Kinh t hc:

Nghch l Allais: S thay i ca mt kt qu c th c chia s bi la chn thay th khc

nhau nh hng n s la chn ca ngi dn trong s nhng la chn thay th, mu thun

vi l thuyt tha dng k vng .

Nghch l mi tn thng tin: bn thng tin bn cn a n ra trc khi bn .

-Nghch l Bertrand: Khi hai doanh nghip t n trng thi cn bng Nash th c hai thy

mnh khng c li nhun.

Nghch l ca Braess: B sung thm dung lng cho mt mng li c th lm gim hiu

sut tng th .

Nghch l kinh t nhn khu hc: cc quc gia hoc tiu qun th vi GDP bnh qun u

ngi cao hn, c quan st l c t con hn, mc d dn s giu hn c th h tr tr em

nhiu hn.

Nghch l kinh t nhn khu hc: Cc quc gia hoc tiu qun th vi GDP bnh qun u

ngi cao hn c quan st c t con hn, mc d dn s giu hn c th h tr tr em nhiu

hn.

-Nghch l kim cng - nc (hoc nghch l gi tr): Nc hu ch hn kim cng, nhng

vn r hn kim cng rt nhiu.

-Nghch l Easterlin: i vi cc nc c thu nhp p ng nhu cu c bn, mc

hnh phc khng tng quan vi thu nhp quc dn cho mi ngi.

-Nghch l Gibson: Ti sao li sut v gi tng quan?

-Nghch l Giffen: Tng gi bnh m lm cho ngi ngho n nhiu hn n hn.

-Nghch l Icarus: Mt s doanh nghip mang li s sp ca mnh thng qua nhng thnh

cng ca ring mnh.

- Nghch l Jevons: Tng hiu sut dn n s gia tng ln hn trong nhu cu.

- Nghch l Leontief: Mt s quc gia xut khu cc mt hng thm dng lao ng v hng

ha thm dng vn nhp khu, mu thun vi l thuyt Heckscher-Ohlin.

- Nghch l Lucas: Vn khng chy t cc nc pht trin sang cc nc ang pht trin mc

d thc t l nc ang pht trin c mc thp hn vn cho mi cng nhn, v do li

nhun cao hn vi ngun vn.

- Nghch l Mandeville: Nhng hnh ng c th l xu xa i vi c nhn nhng c th

mang li li ch x hi.

- Nghch l Metzler: Vic p t mc thu nhp khu c th lm gim gi mt hng .

-Nghch l tit kim: Nu tt c mi ngi tit kim c nhiu tin hn trong thi k suy

thoi, th tng cu s gim v s ln lt tng s tit kim thp hn trong dn s.

-Nghch l ca vic suy thoi: Nu tt c mi ngi c gng lm vic trong thi k suy thoi,

mc lng thp hn s lm gim gi, dn n lm pht, dn n tit kim hn na, lm gim

nhu cu v do lm gim vic lm.

Chng 8: Logic 55

-Nghch l nng sut (cn c gi l nghch l ca Solow my tnh): nng sut lao ng c

th i xung, mc d ci tin cng ngh.

-Nghch l Scitovsky: S dng tiu ch Kaldor-Hicks, phn b A c th hiu qu hn vic

phn b B, nhng cng mt lc B c hiu qu hn A.

-Nghch l dch v bo hnh: sa cha thnh cng mt sn phm b li c th lm hi lng

ca ngi tiu dng cao hn trng hp khng c li sn phm xy ra.

-Nghch l ti nguyn: (Li nguyn ti nguyn) cp n nghch l l cc quc gia v vng

lnh th phong ph v ti nguyn thin nhin, c bit l cc ti nguyn khng ti to nh

khong sn v nhin liu, c xu hng tng trng kinh t t hn v kt qu pht trin ti t

hn so vi cc nc c t ti nguyn thin nhin hn.

24) Trit hc:

Paradox of analysis: It seems that no conceptual analysis can both meet the requirement of

correctness and of informativeness.

Buridans bridge: Will Plato throw Socrates into the water or not?

Paradox of fiction: How people can experience strong emotions from purely fictional things?

Fitchs paradox: If all truths are knowable, then all truths must in fact be known.

Paradox of free will: If God knew how we will decide when he created us, how can there be

free will?

Goodmans paradox: Why can induction be used to confirm that things are green, but not to

confirm that things are grue?

Paradox of hedonism: In seeking happiness, one does not find happiness.

Huttons Paradox: If asking oneself Am I dreaming? in a dream proves that one is, what

does it prove in waking life?

Liberal paradox: Minimal Liberty is incompatible with Pareto optimality.

Menos paradox (Learning paradox): A man cannot search either for what he knows or for

what he does not know.

Mere addition paradox: Also known as Parfits paradox: Is a large population living a barely

tolerable life better than a small, happy population?

Moores paradox: Its raining, but I dont believe that it is.

Newcombs paradox: A paradoxical game between two players, one of whom can predict the

actions of the other.

Paradox of nihilism: Several distinct paradoxes share this name.

Omnipotence paradox: Can an omnipotent being create a rock too heavy for itself to lift?

Preface paradox: The author of a book may be justified in believing that all his statements in

the book are correct, at the same time believing that at least one of them is incorrect.

Problem of evil (Epicurean paradox): The existence of evil seems to be incompatible with the

existence of an omnipotent, omniscient, and morally perfect God.

Zenos paradoxes: You will never reach point B from point A as you must always get half-

way there, and half of the half, and half of that half, and so on ... (This is also a paradox of

the infinite).

25) Ch ngha thn b:

Tzimtzum: In Kabbalah, how to reconcile self-awareness of finite Creation with Infinite

Divine source, as an emanated causal chain would seemingly nullify existence. Lurias initial

Chng 8: Logic 56

withdrawal of God in Hasidic panentheism involves simultaneous illusionism of Creation

(Upper Unity) and self-aware existence (Lower Unity), God encompassing logical opposites.

26) C th xem thm:

Logic cng chnh

-T ng tri ngha: mt t c m ha vi ngha i lp.

-Khng c nghch l: Khng ai bao gi y.

-S phi l

-S sut tha th c: Nu mt hnh vi l tha th c, n khng phi l s sut. nh l

Gdel khng y v nh l khng th nh ngha ca Tarski

-B qua tt c cc quy tc: tun theo quy tc ny, iu tt yu l b qua n.

-Vt kh thi: Mt loi o nh quang hc.

-Trang ch ch trng: Nhiu ti liu cha cc trang m trn cc vn bn ghi rng Trang

ny c c tnh trng, do khng c trang no trng.

Giy t chng minh bt hp l: Mt khi ngun ton hc chnh xc dn n mt mu thun

r rng .

-Logic sai lm: Mt quan nim sai lm t l lun khng chnh xc trong tranh lun.

Nghch l o c : Mt tnh hung trong chun mc o c mu thun m khng c

gii quyt r rng .

-Nghch l ca Moravec: suy ngh logic l kh khn i vi con ngi v d dng i vi cc

my tnh, nhng chn mt chic inh c t mt hp inh vt l mt vn cha c gii

quyt.

-Php lut Murphy: Mt lut v lut php: Bt c iu g c th i sai s i sai.

-Nghch l ca quan st vin: Kt qu ca mt s kin hoc th nghim b nh hng bi s

hin din ca cc quan st vin.

-Nghch l sng: Mt khu sng c c im ca c hai sng ngn v sng trng.

-Nghch l ca Anti-Do Thi: Mt cun sch cho rng vic thiu bch hi bn ngoi v i

khng dn n s tan r ca bn sc ca ngi Do Thi, mt l thuyt gy ting vang trong

cng trnh ca Dershowitz v Sartre.

-Chng minh rng ... bng 1

27) Cu :

Nghch l s ma mai ca Stapp: Cc kh nng khc thng lm cho nhn loi hon thnh

nhng k cng l thng.

Nghch l v trng thi: Mt s nghch l lin quan n cc khi nim v tnh trng y t hoc

x hi .

-Cc l thuyt v s hi hc: l thuyt phi l v v l

V. Asdfhk

VI. Asdfhk

adsfhk

Chng 9: Mnh , i s Boole, tp hp 57

I. nh ngha:

1) Mnh :

- Mnh (propostion hay statement) l mt khi nim khng c nh ngha. Ch c 2 thuc

tnh ng (True) hoc Sai (False).

- Cc php ton c bn ca mnh l: php ph nh, php hi, php tuyn, php ko theo,

php tng ng.

2) Tp hp:

- Cha ca l thuyt tp hp l nh ton hc ngi c, Georg Ferdinand Ludwig Philipp

Cantor (1845 1918).

- Tp hp (set, collection) l ni cha cc i tng no . Nhng i tng ny c gi

l phn t (member) ca tp hp.

II. Cc tp hp v mnh quan trng:

1) Cc nghch l: (Xin xem chng 8).

2) Cc tp s:

Tp s nguyn t (prime numbers set), k hiu .

Tp s t nhin (natural numbers set), k hiu .

{ }

(T nhin, loi ngi khng ngh ra s . Trong bng s La M khng h c s ).

Tp s nguyn (intergers set), k hiu .

{ } { } { } { }

Tp s hu t (rational numbers set), k hiu .

{

| }

Tp s v t (irrational numbers set), k hiu .

Tp s thc (real numbers set), k hiu .

Tp s phc (complex numbers set), k hiu .

Trng v hng (scalar field), k hiu .

3) Khong:

- S thc m rng (extended real number) l v .

- Khong (interval) trong tp s thc: (Cc k hiu ny c dng theo tiu chun quc t ISO 31-11).

Tp rng (empty set):

Tp suy bin (degenerate set):

{ } { }

Khong ng (closed interval):

[ ] { }

Khong tri ng, phi m (left-closed, right-open interval):

[ ) [ [ { }

Khong tri m, phi ng (left-open, right-closed interval):


( ] ] ] { }

Khong m (open interval):

( ) ] [ { }

Khong tri ng (left-closed interval):

[ ) [ [ { }

Khong tri m (left-closed interval):

( ) ] [ { }

Khong phi ng (right-closed interval):

( ] ] ] { }

Khong phi m (right-open interval):

( ) ] [ { }

Khong khng b chn c hai u (unbounded at both ends interval):

( ) ] [ { }

- Ngoi ra, nu ta mun miu t cc s t nhin t n th ta dng k hiu, .

V d:

{ } { }

[ ] { }

4) Ct:

Ct Dedekind (Dedekind cut):

c t theo tn nh ton hc ngi c, Julius Wilhelm Richard Dedekind (1831 1916).

III. Cc nh ngha lin quan n mnh :

1) Bng chn tr (truth table):

2) Mnh ng, mnh sai, mnh hng ng, mnh hng sai:

- Mnh ng (true), k hiu hoc .

- Mnh sai (false), k hiu hoc .

- Mnh hng ng (tautology), k hiu , l mnh logic ph thuc vo cc bin m kt

qu ra sau cng lun lun ng.

- Mnh hng sai (contradiction), k hiu , l mnh logic ph thuc vo cc bin m

kt qu ra sau cng lun lun sai.

V d:

Hai ngi n ng ca mt b lc n gp nhau ngoi mt ci lu, ni m hai ngi v ca

h ang chun b sinh con (ngha l vn cha sinh). H ni vi nhau rng:

(i) Ngi th nht ni: Con ca ti l con trai.

(ii) Ngi th hai ni: Con ca ti hoc l con trai hoc l con gi.

y, sau khi sinh, nu l con trai, th pht biu ca ngi th nht l mnh ng, cn

nu l con gi th pht biu ca ngi th nht l mnh sai. Nhng, i vi ngi th hai,

d v anh ta cha sinh th pht biu vn l mt mnh hng ng.

3) Mnh ng hu khp ni, mnh hu nh chc chn:

- Mt mnh gi l ng hu khp ni (almost everywhere a.e) nu mnh ny lun

ng tr ra trn mt tp c o khng (Xin xem chng khng gian o).

- Mt mnh gi l ng hu nh chc chn (almost surely a.s) nu mnh ny c xc

sut ng l (Xin xem chng l thuyt xc sut thng k).


V d:

Gi s, trong mt cuc thi bn cung, nhn vt huyn thoi trong vn hc dn gian Anh, Robin

Hood bn 100 lt tn u trng vo hng tm. Mt anh nng dn gn ni: Robin Hood

(s) bn trng hng tm pht th 101.

Pht biu ny l mt mnh hu nh chc chn v xc sut m Robin Hood bn trng hng

tm (cho n hin ti) l . Tuy nhin n khng phi l mnh hng ng. l l do

m nhng mnh nh vy c thm ch hu nh.

4) Php ph nh:

Ph nh (negation) ca mnh l mt mnh , c l o , k hiu l hoc

hoc hoc hoc ( ) hoc hoc hoc . Mnh o ng khi

sai v sai khi ng.

5) Php tuyn:

Tuyn (disjunction) ca hai mnh l mt mnh , c l hoc , k hiu l

hoc hoc hoc . Mnh sai khi c hai mnh v

cng sai, v ng trong cc trng hp cn li. Ni cch khc, ng khi t nht ng

hoc ng.

Khi c nhiu mnh , ta c th dng cc k hiu sau:

[

Mnh tng hp trn ng khi c t nht mt mnh ng, v ch sai khi tt c u sai.

6) Php hi:

Hi (conjunction) ca hai mnh l mt mnh , c l v , k hiu l

(hay hoc ). Mnh ng khi c hai mnh v cng ng, v sai

trong cc trng hp cn li. Ni cch khc, sai khi t nht sai hoc sai.


Khi c nhiu mnh , ta c th dng cc k hiu sau:

{

Mnh tng hp trn sai khi c t nht mt mnh sai, v ch ng khi tt c u ng.

7) NOR:

- ca hai mnh v , l mt mnh , k hiu, hoc . Mnh

l ng khi tt c mnh l sai, v sai trong cc trng hp cn li:

- Mi tn Peirce (Peirces arrow), k hiu , c t theo tn nh ton hc ngi M,

Charles Sanders Peirce (1839 1914).

8) NAND:

- ca hai mnh l mt mnh , k hiu l hoc hoc ( ) hoc

. Mnh ng khi v ch khi c t nht mt mnh l sai, v sai khi tt c

mnh u ng. Ton t thng c xem nh l tng ng vi .

- Nt bt Sheffer (Sheffer stroke) c k hiu l . N c t theo tn nh logic ngi

M, Henry Maurice Sheffer (1882 1964).

9) Php tuyn loi tr:

Php tuyn loi tr (exclusive disjunction) hay php or loi tr (exclusive or) ca hai mnh

v , k hiu, hoc hoc hoc hoc hoc

. Mnh l ng khi s u vo khc nhau. Tng qut, ca nhiu mnnh

l: Khi tng s mnh l chn/l, nu s mnh ng l l/chn th ca tt c cc

mnh l ng, v sai trong cc trng hp cn li:


10) Mnh c iu kin, iu kin cn, iu kin :

- ko theo (implicate) l mt mnh , k hiu l . Mnh ch sai khi

ng v sai, v ng trong cc trng hp cn li.

- Khi:

Ta c th ni:

suy ra (ko theo) .

l iu kin (sufficient) c .

l iu kin cn (necessary) c .

- Nu ko theo , th ta ni l ko ngc theo (inverse implicate) , k hiu,

hoc ( ) hoc .

11) Mnh song iu kin:

tng ng (equivalent) l mt mnh

Cuoc Vien Chinh Vao Vuong Quoc Toan Hoc

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