Linardis Psifiaka Systymata Eap

7/22/2019 Linardis Psifiaka Systymata Eap

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HPOOPIKH

A'

IANAITH INAPH

E K T A

ATPA 2001


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I

ANAITH INAPH

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KNTANTINO XAATH

K T

A

KPATH KATIKA

T M A

EPAIMO MPATH

EIA BAKAOOY

...

TYPORAMA

/ 2001

ISBN: 9605381958

K : H 21/1

Copyright 2000

& , 26222 : (0610) 314094, 314206 : (0610) 317244

. 2121/1993,

.


5/335

fi

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E ............................................................................................................... 11

1.1 .................................................................................. 13

1.2 ................................................................................ 15

1.3 ....................................................................................................... 17

1.4 ............................................................................................................. 19

1.5 M ................................................................................... 22

1.6 M ............................................... 23

1.7 ........................................ 26

1.8 ............................................................................ 28

1.9 ...................................................................................................... 29

................................................................................................................................... 32

B ................................................................................................................................................ 34

K 2

K

, ,

E ............................................................................................................... 35

2.1 E ......................................................................................................................................... 37

2.2 M ................................................................................................... 38

2.2.1 K ........................................................................................ 39

2.2.2 K BCD ............................................................................................................... 41

2.2.3 K K Gray ..................................... 42

....................................................................................................................... 46

2.3 A ...................................................................................................... 47

2.3.1 ................................................................................................ 48

2.3.2 ...................................................................................................... 50

2.3.3 2 .................. 55

2.4 K ...................................................... 58

2.4.1 K .............................................................................. 60

2.4.2 K ............................................................................... 63


6/335

6 H I A K H X E I A H I

2.4.3 K Hamming ................................................................................................... 63

....................................................................................................................... 69

................................................................................................................................... 72

B ..................................................................................................................... 75

K 3

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E ............................................................................................................... 77

3.1 E ......................................................................................................................................... 82

3.2 Boole ........................................................................................................................... 83

3.2.1 A .................................................................................................... 89

3.2.2 ................................................................................................... 90

3.2.3 B .................................................................................................. 91

....................................................................................................................... 99

3.3 Boole ..................................................................................................... 100

3.3.1 .......................... 103

3.3.2

( ) .............................................................................. 108

.................................................................................................................... 112

3.4 ......................................................................................................... 113

3.4.1 .............................................................................. 115

3.4.2 .............................................................................. 120

.................................................................................................................... 124

3.5 E M .................................................. 125

.................................................................................................................... 128

3.6 K ....................................................................... 129

.................................................................................................................... 137

3.7 ......................................... 137

3.7.1 ..................................................................... 1383.7.2 ............................................................................................................ 139

3.7.3 I .......................................................................................... 142

3.7.4 .............................................................. 144

3.7.5 .................................................................... 145

.................................................................................................................... 148

................................................................................................................................ 149


7/335

B .................................................................................................................. 153

K 4

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4.1 E ..................................................................................................................................... 159

4.2 K .................................................................................................... 159

4.2.1 .......................................................................... 163

4.2.2 MOS CMOS ................................................................ 167

.................................................................................................................... 172

4.3 K ......................................................................... 173

4.3.1 ............................................... 176

4.3.2 E .............................................................................................................. 179

4.3.3 ........................................................ 182

4.3.4 A ................................ 183

.................................................................................................................... 184

4.4 O ................................................................. 185

4.4.1 O ........................................................... 188

4.4.2 ................................................ 192

.................................................................................................................... 1934.5 B .............................................................................. 194

4.5.1 sa0 sa1 ........................................................ 196

.................................................................................................................... 198

4.6 A ................................................................................................ 198

.................................................................................................................... 201

................................................................................................................................ 202

B .................................................................................................................. 203

K 5

A K

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5.1 E ..................................................................................................................................... 207

5.2 M Karnaugh ........................................ 210

5.2.1 X Karnaugh ................................................................................................... 212

7 E P I E X O M E N A


8/335

8 H I A K H X E I A H I

5.2.2 A Karnaugh ....................................................... 216

5.2.3 E 5 .............................. 221

5.2.4 E ........................................ 222

.................................................................................................................... 223

5.3 ....................................................................... 224

.................................................................................................................... 228

5.4 ......................................................... 228

.................................................................................................................... 233

5.5 QuineMcCluskey ................. 234

5.5.1 E ........................................................ 235

5.5.2 ......................................................... 239

5.5.3 M Petrick .............................................................................. 242

5.5.4 H QuineMcCluskey

............................................................................ 245

.................................................................................................................... 248

5.6 M Espresso ................................................................................................................ 249

.................................................................................................................... 252

5.7 Hazards ................................................................................................. 252

.................................................................................................................... 255

................................................................................................................................ 255

B .................................................................................................................. 258

K 6

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6.1 E ..................................................................................................................................... 261

6.2 A .................................................................................................................................... 261

6.2.1 ..................................................................................... 264

6.2.2 A / A ..................................................................................... 2666.2.3 K ................................................. 267

.................................................................................................................... 271

6.3 ..................................................................................................................................... 272

.................................................................................................................... 275

6.4 K .............................................................. 275

.................................................................................................................... 278


9/335

6.5 ............................................................................................................................. 279

.................................................................................................................... 283

................................................................................................................................. 284

B .................................................................................................................. 285

A A A

E A ....................................................... 287

..................................................................................................................................................... 325

9 E P I E X O M E N A


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32/335

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35/335

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39/335

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40/335

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1

2

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4

5

6

7

8

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41/335

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42/335


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43/335

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44/335


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modulo 2, :

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45/335

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2

3

4

5

6

7

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g5 = b5 => 1

g4 = b4 b5 => 0 1 = 1

g3 = b3 b4 => 0 1 = 1

g2 = b2 b3 => 1 1 = 0

g1 = b1 b2 => 0 1 = 1

g0 = b0 b1 => 1 0 = 1

, Gray Bin8421

:

bn = gn (2.2)

bi = gi gi+1 gi+2 gn, 0 i < n (2.2)

O , -

, bn, -

:

bn = gn (2.3)

bi = gi bi+1, 0 i < n. (2.3)


46/335


2.5

H 1101 Gray Bin8421 1001:

b3 = g3 => 1

b2 = g2 b3 => 1 1 = 0

b1 = g1 b2 => 0 0 = 0

b0 = g0 b1 => 1 0 = 1

Afi

2.3

. M Bin8421 Gray:

) 11011, ) 1001010, ) 1111011101110.

. M Gray Bin8421:

) 1010, ) 00010, ) 11000010001.

fi

O : K.

H .

K , ,

(Gray) (BCD).


47/335

2.3 A

A , -

.

,

, -

, . -

, ,

-

( 2).

T , :

T , - .

T

.

T E .

.

,

, .

.

2

.

E -

,

K. Y : ()

, . ( ) ()

( 2). E

. T 2 -. M -

.

,

(), .., -

. -

4 72 . 3 A


48/335


. E "digit" -

"digitus" "".

T , , I-

(700 .X.). O I

, ,

() (

). E, E-

, -

.

, .

2 .X. -

O,

, . H I

. A E -

, Fibonacci (K),

Liber Abaci (1202 .X.) E -

.

, -

,

,

. , -

, .

2.3.1

A ,

,

Y 2.2.1. -

1908, :

1*103+9*102*+0*101+8*100

E 103, 102, 101 100, -

10,

, 0

.

,

:


49/335

n n1 0 . 1 m

:

(2.4)

:

(, >1)

i (0 i < )

i

n+1

m .

n n1 0 -

, 1 2 m .

n (Most Significant Digit),

(MSD), m

(Least Significant Digit), (LSD).

H , , 2.

=2 , =3 , =8 -

, =10 , =16 ..

O >10 -

. 10 15

:

: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14, 15

: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

2.6

O 1ABC

:

163*1+162*(10)+161*(11)+160*(12) => 684410

, , ,

.. 148.510 = 10010100.12 .

N = a bii

i m

n

=-

4 92 . 3 A


50/335

2.3.2 fi

, - 0 1. X

. ,

, bits.

H

n (bits) 2n1,

1*2n1 + 1*2n2 + + 1*21 + 1*20 = 2n1


Afi

2.4

. 1 ;

) 10002, ) 10008, ) 100010, ) 100016.. H 2*101 + 5*100 ( ):

) 250, ) 25, ) 2.5, ) .

. O A01B .

. O A5 . (

16 = 24)

. O 11101100 .

(E 16 = 24).

. O 111011010 .

(E 8=23).

. A (), (),

(E) (T), ,

,

( )

.

.

, b. - b,

:

1. 1234 + 5432 = 6666 4. 23 + 44 + 14 + 32 = 223

2. 41 / 3 = 13 5. 320 / 20 = 12.1

3. 33 / 3 = 11 6. T P (41) = 5


51/335

5 12 . 3 A

Bits

0 0

0 1

1 0

1 1

+

.

0 0

1 0

1 0

1 1

.

.

0 0

1 1

1 0

0 0

*

0

0

0

1

/

0

1

O ,

( 2). 2.7 - .

2.7

N 1011 111:

N = (23*1+22*0+21*1+20*1) + (22*1+21*1+20*1)

= 23*1 +22*(0+1) +21*(1+1) +20*(1+1)

= 23*1 +22*1 +21*2 +20*2

= 23*1 +22*1 +21*(2+1)

= 23*1 +22*(1+1) +21*1

= 23*1 +22*2 +21*1

= 23*(1+1) +21*1

= 23*2 +21

= 24 +21

= 24*1 +23*0 +22*0 +21*1 +20*0

N 100102.

2.2

K , n

d , :

2n1 > 10d1 n = d / log102 d / 0.3

.

.

. 2.2

( bit):


52/335


2.7

, . 2,

()

.

, -

2.8.

2.8

N 1011 111:

N = (23*1+22*0+21*1+20*1) (22*1+21*1+20*1)

= 23

*1 +22

*(01) +21

*(11) +20

*(11)= 23*1 +22*(1) +21*0 +20*0

= 23*1 +22*(212)

= 23*1 +22*(12)

= 23*(11) +22*1

= 23*(0) +22*1

= 23*0 +22*1 +21*0 +20*0

E 01002.

, , , .

2, -

.

, -

-

, . 2,

:

-

:

K 1 1 1 K 1 1 1

1 0 1 1 1 0 1 1

+ 1 1 1 1 1 1

A: 1 0 0 1 0 : 0 0 1 0


53/335

5 32 . 3 A

2.9

O :

: :

1100 111100 1100101 1100 101

1100 1x1100 0011

00000 00110 .

110000 001100 > .

111100 001100 . .

000000 .

:

910 10012 15.2510 1111.012

+ 510 + 01012 + 7.5010 + 111.102

1410 11102 22.7510 10110.112

:

2510 110012 18.7510 10010.112

710 001112 12.5010 1100.102

1810 100102 6.2510 00110.012

:

910 10012 25.510 11001.12

510 1012 6.510 110.124510 10012 127510 1100112

000012 1530510 00000012

1001012 165.7510 110011012

1011012 1100111012

10100101.112


54/335


H :

-

.

2.10

M 1101 :

11012 => 23*1+22*1+21*0+20*1 = 8+4+0+1 => 1310.

H

2. A

Y. H :

) 0 -

, , k=0.

) k 2 Yk k+1 -

. T Yk k.

) k=k+1 () k=0.

2.11

M 4510 :

0 = 45. A 2 :

k=0 => (Y0=1, 1=22), k=1 => (Y1=0, 2=11),

k=2 => (Y2=1, 3=5), k=3 => (Y3=1, 4=2),

:

46 9 101110 1001 2925 27 1011011011012 1101121 5 100110 1012 225 108 110111011012 01101100

001010 09 0100101011012

001010 11011011012

1001 0001010011012

0001 110111012

00111011012

110111012

00010011012


55/335

k=4 => (Y4=0, 5=1), k=5 => (Y5=1, 6=0)

H : 4510

=> 1011012.

5 52 . 3 A

Afi2.5

. T 11010 + 01111 ( ):

) 101001, ) 101010, ) 110101, ) 101000.

. H 110 010 ( ) ( -

):

) 001, ) 010, ) 101, ) 100.

. M 200010 .

2.3.3 2

,

, 0 1. -

bit 0

bit 1, .. 01011

11010 .

0 1

, , - ,

. 11010 -

5 , 6 .

011010.

-

.

2.

2 (2's

Complement) n :

' = 2n

'

. 0 1 -

.


56/335


n bits

2 :

2n1 N < 2n1

,

:

= + (2n B) 2n = A + B' 2n => + '

O 2n, .

H 2.4:

bit, n=3 2n=8. T 2.4 ,

( )

( ).

-

.

2.12

2:

) (1)+(2): 7 6 (+ 2.4),

5, 3. I -

:

(1)+(2) => 6 + 7 = 13 => 13 8 = 5 => 3

) (+1)+(2): 1

6 , -

7, 1. I :

(+1)+(2) => 1 + 6 = 7 => 1

) (1)+(+2): 7 2

,

1, +1. I :

(1)+(+2) => 7 + 2 = 9 => 9 8 = 1 => +1

2.13

A 2 ( 2.4):


57/335

) (1)(2): (2) 8 6 = 2 =>

(+2). K (1)+(+2) =+1.

) (+1)(+2): (+2) 8 2 = 6 => (2).

E (+1)+(2) =1.

T , , -

. H

2 :

. bit , . 01 1 0,

. .

2.14

A :

(+1)(+2). O (+2) :

(+2) => 010 bit 101

1 + 1

110 => (2)

H , -

:

(+1) => 001

( 2) => 101 bit

+ 1 1

111 => (1)

5 72 . 3 A

000

0

+

4

+11

+33

2 +2

0

4

17

35

26

100

001

010

011101

110

111

2.4

2


58/335


2.15

A :

46 17 2 -

8 bit. E 4610 => 001011102 1710 => 000100012:

(+46) => 00101110

( 17) => 11101110

+ 1

00011101 => (+ 29)

Afi

2.6

. T 2 11001000 (

):

) 00110111, ) 00110001, ) 01001000, ) 00111000.

. H 11610 12610 2 -

8 bit.

. O 2.1

;

. O 2.1 -

2;

2.4 K fi

( -

1.2),

,

. M

bits , ,

. -

-

.

H , -

:

T Hamming


59/335

Hamming

,

Hamming

H : -

.

K.

, -

, ( )

. -

.

, '

, : () , -

, ()

bits .

bit .

, -

, . -

-

.

O . -

, ()

.

p , . bit .

p,

, .

,

.

' , n bit ,

5 92 . 4 K


60/335


Hamming

bits

, .. 0110 0101 . -

.

2.4.1 K

n bit

2n bits , .

, () bit

, ,

, .

,

. -

00, 01, 10, 11, 00

01 10,

00 .

, , -

,

, , (, ),

.

. ..

00 11

, , 00 -

01, , -

, .

00 bit (00 => 01)

11 bit (11 => 01).

bit ,

:

1) 000

2) 001 010 100

3) 011 101 110

4) 111


61/335

000, 011, 101 110 (-

), , -

, 2

4, .

1 3

(), ( 2 4). , -

, ,

. , 1 3 .

, , ,

. -

, ,

1 3 bits

.

'

, -

, ( ).

,

.. 1, -

1 . -

, 1 ( 000 -

).

,

, -

:

1) ,

(parity bit),

2) , 1

( ) ( ).

, -

,

.

6 12 . 4 K


62/335


.

.

2.16

:

= 00 001 ( )

= 01 010

= 10 100

= 11 111

, .. 001, -

(101 011 000 => 1) -

1.

, -

.

, ,

, -

.

21 2 () .

,

,

' -

(redundant information).

Afi2.7

.

:

) 100110010, ) 011101010, ) 10111111010001010.

. O ,

. E

, :

) 11110110, ) 00110001, ) 01010101010101010.


63/335

2.4.2 K fi

, ,

.

:

1) 000

2) 001 010 100

3) 011 101 110

4) 111

000 111. -

000 ,

2. , -

111

2 111 ,

. , 000.

, -

, ,

.

, -

,

. -

' . -

,

. , , , -

2.

-

. ,

, - .

, Hamming.

2.4.3 K Hamming

Y 2.4.2,

6 32 . 4 K


64/335


. O

Hamming

, . K -

Hamming,

bits, m1, m2, , mM, ' -

, p1, p2 ,pK, ( Hamming)

+ :

m1m2mMp1p2pK

+ ,

(position or location number),

1 +

. :

,

, -

. ,

,

.

.

A K K HAMMING

Hamming,

, :

1. pi 1, 2, 4, ,

2K1 Hamming -

mi , .. =8 =4

:

p0p1 m0p2 m1 m2 m3p3 m4 m5m6 m7

2. ,

.

2.1 , ,

,

. 2.3 C=c3c2c1c0

( ) .

* () , -

( 0).


65/335

2.2 2.3, Hamming -

c3 , . ' c3= 0 ' c3= 1.

-

c2, c1, c0.

:

6 52 . 4 K

/

0

1

2

3

4

5

6

78

9

10

11

12

c3c

2c

1c

0

0 0 0 0

0 0 0 1

0 0 1 0

0 0 1 1

0 1 0 0

0 1 0 1

0 1 1 0

0 1 1 11 0 0 0

1 0 0 1

1 0 1 0

1 0 1 1

1 1 0 0

.

* ( )

p0

p1

m0

p2

m1

m2

m3

p3

m4

m5

m6

m7

2.3

A

c3= 0 *p0p1m0p2m1m2m3 c3=1 p3m4m5m6m7

c2= 0 *p0p1m0p3m4m5m6 c2=1 p2m1m2m3m7

c1= 0 *p0p2m1p3m4m7 c1=1 p1m0m2m3m5m6

c0= 0 *p1p2m2p3m5m7 c0=1 p0m0m1m3m4m6

3. Hamming, -

, cj=1 pj.

4. A (3) 2.3 -


66/335


:

: pj

' , -

pj, . cj=1,

: '

,

' (cj=1),

:

,

cj=0,

. .

Hamming pj (j=0,,1)

,

cj=1.

X K HAMMING

H Hamming

2.17

2.20:

2.17

K Hamming :

:

1 0 0 1 1 0 1 0

1. H Hamming pj

2j (j=0, 1, ) -

bits :

p0 p1 m0 p2 m1 m2 m3 p3 m4 m5 m6 m7

p0 p1 1 p2 0 0 1 p3 1 0 1 0

2. A 2.3 ,

, cj=1 -

:


67/335

c3=1: p3m4m5m6m7 => p3 1 0 1 0 => p3= 0

c2=1: p2m1m2m3m7 => p2 0 0 1 0 => p2= 1

c1=1: p1m0m2m3m5m6 => p1 1 0 1 0 1 => p1= 1

c0=1: p0m0m1m3m4m6 => p0 1 0 1 1 1 => p0= 0

3. H Hamming pj:

0 1 1 1 0 0 1 0 1 0 1 0

2.18

Hamming :

:

0 1 1 1 1 0 1 0 1 0 1 0

1. , 2.3,

(

):

p3: 0 1 0 1 0 => : c3=0

p2: 1 1 0 1 0 => : c2=1

p1: 1 1 0 1 0 1 => : c1=0

p0: 0 1 1 1 1 1 => : c0=1

2. T bits c2=1

c0=1, , . m1. T -

:

C = c3 c2 c1 c0 = 01012 => 510

5 Hamming. E

5 :

0 1 1 1 0 0 1 0 1 0 1 0

3. H -

:

1 0 0 1 1 0 1 0

:

c3c2c1c0 = 0000

( ).

6 72 . 4 K


68/335


2.19

Z Hamming

01100101 .

T pj 2j (j=0, 1, )

bits

.

p0p1 0 p2 1 1 0 p3 0 1 0 1

A 2.3 , c3=1,

, p3=0,

. O , c2=1

p2=1. c1=1 p1=0

c0=1 p0=1. E, -

Hamming pj :

p0 p1 0 p2 1 1 0 p3 0 1 0 1

Hamming: 1 0 0 1 1 1 0 0 0 1 0 1.

2.20

H 001010101010 Hamming -

. bit. Z .

2.18

2.3. M 2.3

:

) c3=1 => 01010 => => c3=0

) c2=1 => 01010 => => c2=0

) c1=1 => 010101 => => c1=0

) c0=1 => 011111 => => c0=1

:

c3c2c1c0 = 0001

:

0 0 1 0 1 0 1 0 1 0 1 0


69/335

:

1 0 1 0 1 0 1 0 1 0 1 0.

H , :

=> 1 1 0 1 1 0 1 0.

6 92 . 4 K

Afi2.8

. O

Hamming ( ):

) 110111, ) 00110001, ) 01001000, ) 00111000.

. N ,

Hamming ( ):) 000011110111, ) 000011101111, ) 111011111111.

fi2.1

( 2.5). T

,

. H , -

, BCD. T -

2.5. Z :

) A Y 2.2.3 ( 2.1),

, () -

, (.. 2.1

). N -

.

fi

,

.

T

Hamming.

H

, (-

Hamming).


70/335


) ()

, BCD. A K-

2, ,

,

(), BCD.

) bits ,

(K 1).

bits , ,

.

) , -

BCD.

, ,

( 2.6), -

10 +50 K . T

-

. ()

bit.

, :

) (

); E ( bits) ;

M bits -

M

M

0110M

BCD

2.5

H

fi2.2


71/335

7 12 . 4 K

. ,

.

) X ()

, , -

,

(Y 2.4.3).

) A 2.6, -

, -

,

,

. : () ()

(

).

M

M

A

M

M K

2.6


72/335


K

. E

. T

.

O , -

0 1 (), -

bits. -

bits

. K n bits

2

n

. O ,

: () () . H

, , ,

.

-

(),

. E

. T

.

O BCD -

0 9 . -

BCD,

.

O ,

bit. -

, -

. Gray. O -

Gray Bin8421

. H Gray Bin8421, -

, modulo 2.

T ,

, . T -


73/335

, -

E.

H 16 -

A, B, C, D, E, F

10 15.

O ,

i 2 i k

2k.

O , -

, 2.

,

, 1. A,

2.

,

, , ,

0 1 .

T 2 - ,

. A :

2 N' N

, : () bits N ()

.

H

( E 1.2), -

. bits ( ) , ( ), -

, -

(-

).

H Hamming -

:

7 3 Y N O H


74/335


N , Hamming

N.

N , Hamming

2N.

H ( ) -

, -

.

O Hamming , , -

, -

. K Hamming

, - bits . K

-

, ,

bit, ().


75/335

B

Floyd T., Digital Fundamentals, Macmillan, 1990.

2 K

, ( -

) , BCD,

, ASCII Gray. -

.

Glaser A., A History of Binary and Other Nondecimal Numeration,

Southampton PA, Tomash, 1971.

.

Gregg J., Ones and Zeroes Understanding Boolean Algebra, Digital Circuits

and the Logic of Sets, IEEE Press, 1998.

K 0

.

Ifrah G., From One to Zero: a Universal History of Numbers, New York,

Viking, 1985.

.

Kohavi Z., Switching and Finite Automata Theory, McGraw Hill.

1 K -

, -

( ,

Hamming).

Menninger K., Number Words and Number Symbols: A Cultural History of

Numbers, Cambridge, MIT, 1969.

.

Hamming R., Error Detecting and Error Correcting Codes, Bell System Tech.

J., vol 29, pp.147160, April 1950.

B .

7 5B I B I O P A I A


76/335


Rao T., Fujiwara E., ErrorControl Coding for Computer Systems, Prentice

Hall, 1989.

E -

( ).


77/335

Boole

fi

H

-

. H

-

. H

-

, ,

Boole. O

, . , ,

, ,

Boole.

A, -

, -

. T 1938 C. Shannon -

. O -

Boole. H Boole,

( Y-

3.3.2) -

,

(E 1.4).

K

Boole, ( -

) ( ) -

.

K, :

N A.

T Boole.

N Boole.

3


78/335

7 8 K E A A I O 3 : A E B P A B O O L E

T .

T OR, AND, NOT.

N (

) .

N , ( -

), K,

.

T ,

.

X -

(K ).

T ,

.

N

() .

N , ,

/ -

(K-

).

A

Boole

A

A

OR

AND

NOT

K

X Karnaugh

NK

E

M

K


79/335

K 2 ,

. K -

, , -

, -

. E

Boole,

, .

T K, Boole,

, ,

.

,

-

. .. x+x=2x x.x=x2 Boole

x+x=x x.x=x. H , -

E K.

K, -

, K-

. T K ( E

K) : Boole, , -

Boole, , -

, -

, . H

E

Y K. :

) N

Boole (Y 3.2.3).

) N

( Y 3.3.1).

7 9

NAND

NOR

XOR


80/335


) N , ,

(

Y 3.3.2).

) A A -

,

,

(E 3.4).

) N

-

(E-

3.7). , ,

E

.


81/335

8 1

Boole Boole 3.2

O

A 3.2.1

3.2.2

B 3.2.3

I

3.3

O

3.3.1

3.3.2

E

3.4

O

3.4.1

3.4.2

. E M 3.5

K . 3.6

M 3.7.1

3.7.2

I 3.7.3

3.7.4

E 3.7.5

Boole

Efi


82/335


3.1 E

,

: A-

.

A E :

T A-

.

T .

n -

, :

. xi, i=1,, n , . -

,

. y , .

,

. F .

-

, :

. (Combinatorial). ' ,

yt y t, xit

xi t, .:

yt = F(x1t, x2t, ,xnt)

. (Sequential). yt y

t xit xi

t

. , F

Fy Fs, . F={Fy, Fs},

-

:

yt = Fy(x1t, x2t, , xnt, st)

st+1 = Fs(x1t, x2t, , xnt, st)

st t st+1

.

Fx

k

y

xn

x1

3.1


83/335

T , ,

( Fy Fs) ,

st. E,

,

,

:

y = F(x1, x2, ,xn)

t .

, .

, .

-

( relays)

, o . 1938 C. Shannon

A Symbolic Analysis of Relay and Switching Circuits

(1938) -

, o

.

O -

, Boole.

-

G. Boole "" -

. 1854

An Investigation of the Laws of Thought on Which Are Founded the

Mathematical Theories of Logic and Probabilities -

.

Boole, ,

, "-

" . E

Boole -

( ) -

( ).

3.2 Boole

H -

8 33 . 1 E


84/335


, -

. M " "

Shannon (1938) , Boole

. E

Boole, Huntington,

.

T E :

T Boole (, , ).

T .

Boole

.

.

T .

T Boole.

E -

, -

, E. E

,

E. H Boole,

,

, . O ,

, -

A -

.

H Boole , ,

G. Boole,

A , -

.

H Boole , . -

, :

) , . T


85/335

x B : xB. E B, , .. 1, 2, 3

4, : B

= {1, 2, 3, 4}.

) . M

, -

x y B, z, ..

, 3

() 4, 7.

) . A -

, , .

A Boole

( BooleSchroeder). E

, , -

Boole

Huntington (1904).

O BOOLE

Boole , Huntington, :

= < , +, ., , 0, 1 >,

) , ,

) T +, ., ,

:

+ ,

. ,

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) T 0 1 (0,1), - .

) a, b, c

3.1:

8 53 . 2 B O O L E


86/335


, ,

, . a.b ab. ,

a ( -

a), , ,

,

a'.

:

Boole, A0, -

, -

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:

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. B -

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() .

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, . -

a, b B () c (

a b), B. -

, .. -

, .

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a () b -

b () a.

) T A3

a

a

a

0. a, b B ab, B .

1. a+b B, a.b B A2. a+b=b+a, a.b=b.a

A3. a+(b.c)=(a+b).(a+c), a.(b+c)=(a.b)+(a.c)

A4. a+0=a, a.1=a O

A5. a+ =1, a. =0 aa

3.1

A

Boole (Huntington)


87/335

, a.(b+c), -

, a+(b.c). -

.

) T A4 , -

,

B, 0 1 ,

. 0 -

1, .

0 1.

) T A5 a B

, a - , a

1 0. : a, .

.

Boole -

. H Boole, , -

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:

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,

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. a+(b.c)=(a+b).(a+c), .

E ,

.

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-, . a+c = b+c a = b.

) T ( A5)

.

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:

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a

a

8 73 . 2 B O O L E


88/335


) B

) , .

.

O

Boole, -

-

( Boole). E

, .

B , -

Boole. M

, E 3.3. E 3.3

Boole. E, ,

Boole -

:

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I={a, b} , .

A={a} B={b}. I, , A B -

={I,,A,B} - , , 3.2, < , , , > Huntington -

Boole.

+

+ A B I

A B I

A A A I I

B B I B I

I I I I I

.

T

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A A A

B B B

I A B I

x

I

A B

B A

I

x

3.2

A


89/335

3.2.1 A

3.1 A1 A5

. H

,

, ,

, , -

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+ . A3 :

a+(b.c) = (a+b).a+c)

a.(b+c) = (a.b)+(a.c)

H , Boole, -

-

.H (Duality principle) :

(dual) , :

)

) 0 1.

8 93 . 2 B O O L E

Afi

3.1

N , 3.1 ( 3.2), Huntington.

fi3.1

A 3.1:

) N

. (Y: E -

).

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. (Y:

-

).


90/335


Boole :

: Boole ,

. ..:

(a+a).(b'+1) = a, (a.a)+(b'.0) = a

, :

) 1 5

, ,

)

, -

.

T ,

.

Afi

3.2

N :

) F = (a+b').(b+c+0) F = a.b'+b.c.1

) a+(a.b) = a a.(a+b) = a

) (a+b)' = a'.b' (a.b)' = a'+b'

) a+a'.b = a+b a.(a'+b) = a.b

3.2.2

M .. a+(b+c).a+a.b+(b+c).b,

, .

B, .. a, b, c,

, .

E, -

a,b,c / 0, 1 , +, ., . .. a+b.c.(a+c)'

, a.b+(a'+b').(a'+b.c').

H -

Boole.

, -

, : , ,


91/335

. .. a.b+a.c

(a.b)+(a.c). Y ab a.b.

9 13 . 2 B O O L E

Afi3.3

N

a=0, b=1 c=1:

) F = a+bc

) F = (a+b)(a+c)

) F = (a+b)c. N F

F .

3.2.3

A0 A5 , -

E 3.2, ,

Y. T -

.

1

a+a = a,

E:

a = a+0 4

= a+a.a' 5

= (a+a).(a+a') 3

= (a+a).1 5

= a+a 4

a.a = a

E:

a = a.1 4

= a.(a+a') 5

= a.a+a.a' 3

= a.a+0 5

= a.a 4

a+1 = 1,

E:

1 = a+a' 5

= a+(1.a') 4

= (a+1).(a+a') 3

= (a+1).1 5

= a+1 4

a.0 = 0

E:

0 =a.a' 5

= a.(0+a') 4

= a.0+a.a' 3

= a.0+0 5

= a.0 4

2


92/335


4 :

(a+b)+c = a+(b+c)

:

[a+(b+c)][(a+b)+c] = [(a+b)+c][a+(b+c)] 2

{[a+(b+c)](a+b)}+{[a+(b+c)]c} = {[(a+b)+c]a}+{[(a+b)+c](b+c)} 3

[a+(b+c)a]+[ab+(b+c)b]}+{[ac]+[(b+c)c]} =

={[(a+b)a]+[ac]}+{[(a+b)b+bc]+[(a+b)c+c]}{[a]+[ab+(b+bc)]}+{[ac]+[bc+c]} = {[a+ab]+[ac]}+{[(ab+b)+bc]+[c]}3

{[a]+[ab+b]}+{[ac]+[c]} = {[a]+[ac]}+{[b+bc]+[c]} 3

{a+b}+{c} = {a}+{b+c} 3

(a+b)+c = a+(b+c)

-

: (a+b)+c = a+(b+c) = a+b+c.

4

:

(a.b).c =a.(b.c)

:

[a.(b.c)]+[(a.b).c] = [(a.b).c)]+[a.(b.c)] 2

{[a(bc)]+(ab)}{[a(bc)]+c} = {[(ab)c]+a}{[(ab)c]+(bc)} 3

a+(a.b) = a,

E:

a = a.1 4

= a.(b+1) 2

= a.(1+b) 1

= (a.1)+(a.b) 3

= a+(a.b) 4

a.(a+b) = a

E:

a = a+0 4

= a+(b.0) 2

= a+(0.b) 1

= (a+0).(a+b) 3

= a.(a+b) 4

3


93/335

{a[(bc)+b]}{[a+c][(bc)+c]} = {[(ab)+a][a+c]}{[(ab)+b]c} 3

{a[b]}{[a+c][c]} = {[a][a+c]}{[b]c} 3

{ab}{ac+c} = {a+ac}{bc} 3

{ab}{c} = {a}{bc} 3

(ab)c = a(bc)

: (a.b).c = a.(b.c) = a.b.c.

5

a' a:

) a' .

a, a1'=a a2'=a,

.

: a+a1'=1, a+a2'=1 a.a1'=0, a.a2'=0.

E a2' = 1.a2' A4

= (a+a1').a2' A5

= a.a2'+a1'.a2' A3

= 0+a1'.a2' A5

= a.a1'+a1'.a2' A5

= a1'.(a+a2') A3

= a1'

) a' a, . (a')'=a.

a'=(a)', a a'.

6

de Morgan:

) (a+b)' = a'.b', ) (ab)' = a'+b'

T 6) (a'.b') (a+b). A

(a+b) (a'.b') -

A5:

(a+b)+a'.b' = 1 (a+b).(a'.b') = 0

9 33 . 2 B O O L E


94/335


E:

(a+b)+a'.b'= [(a+b)+a'].[(a+b)+b'] (a+b).(a'.b') = a.a'.b'+b.a'.b'

= (1+b).(a+1) = a.a'.b'+a'.b.b'

= 1 = 0

T 6) , a b -

a' b'.

T de Morgan -

.

7

A.

a+a'.b = a+b, a.(a'+b) = a.b

E: E:

a+b = 1.(a+b) A4 a.b = 0+a.b A4

= (a+a').(a+b) A5 = a.a'+a.b A5

= a+a'.b A3 = a(a'+b) A3

T 1 7 3.3.

1. a+a = a a.a = a

2. a+1 = 1 a.0 = 0

3. a+(a.b) = a a.(a+b) = a

4. (a+b)+c = a+(b+c) (a.b).c = a.(b.c)

5. a' (a')' = a

6. (a+b)' = a'.b' (ab)' = a'+b'

7. a+a'. = a+b a.(a'+) = a.b

T 1 7, ,

-

. E K,

K.

3.3

B

Boole


95/335

3.2

: ab+a'c+bc = ab+a'c

A: ab+a'c = (ab+abc)+(a'c+a'bc) 3

= ab+a'c+abc+a'bc A2

= ab+a'c+(a+a')bc A3

= ab+a'c+1.bc A5

= ab+a'c+bc A4

3.3

:

F = (x'+xyz')+(x'+xyz')(x+x'y'z)

A: F = (x'+xyz')+(x'+xyz')(x+x'y'z)

= (x'+xyz').[1+(x+x'y'z)] 3

= x'+xyz' 2

= x'+yz' 7

3.4

: (abcd)' = a'+b'+c'+d'

A: a'+b'+c'+d' = (a'+b')+(c'+d')

= (ab)'+(cd)' 6

= [(ab).(cd)]' 6

= (abcd)'

3.5

: F = xy+wxyz'+x'y

A: F' = (xy+wxyz'+x'y)'= ([xy+wxyz']+[x'y])'

= [xy+wxyz']'.[x'y]' 6

= (xy)'.(wxyz')'.[x'y]' 6

= (x'+y').(w'+x'+y'+z).[x+y'] 6

= (x'+y').[x+y'].(w'+x'+y'+z) A2

9 53 . 2 B O O L E


96/335


= [xx'+x'y'+xy'+y'y'].(w'+x'+y'+z) A3

= [0+x'y'+xy'+y'].(w'+x'+y'+z) A5,1

= [(x'+x+1)y'].(w'+x'+y'+z) 3

= [y'].(w'+x'+y'+z) 2

= (w'y'+x'y'+y'y'+y'z) A3

= (w'+x'+1+z).y' 1,A3

= y' 2

3.6

:

x'y' + x'z + xz' = y'z' + x'z + xz'

A: 3.2 , :

ab+a'c = ab+a'c+bc

-

. -

y'z',

, x'y', :

x'y'+x'z+xz' y'z'+x'z+xz'

[x'y'+xz']+x'z [y'z'+x'z]+xz'

[x'y'+xz'+y'z']+x'z [y'z'+x'z+x'y']+xz'

x'y'+xz'+x'z+y'z' = y'z'+x'z+xz'+x'y'

: O ,

.

. , ,

:

x'y'+x'z+xz' = [x'y'+xz']+x'z

= .

= x'y'+xz'+x'z+y'z'

= y'z'+x'z+xz'+x'y'

= .

= y'z'+x'z+xz'


97/335

3.7

:

xy+x'y'+x'yz = xyz'+x'y'+yz

A: 3.2:

ab+a'c = ab+a'c+bc

3.6:

xy+x'y'+x'yz xyz'+x'y'+yz

[xy+x'yz]+x'y' [xyz'+yz]+x'y'

[xy+x'yz+yz]+x'y' [xyz'+yz+xy]+x'y'

[xy+yz]+x'y' [yz+xy]+x'y'

xy+yz+x'y' = yz+xy+x'y'

3.8

:

E x+y = 0, x = 0 y = 0

A: x+y = 0 x+y = 0

x+y+y' = 0+y' x'+x+y = 0+x'

x+1 = y' 1+y = x' A5

1 = y' 1 = x'

E: x+y = 0 x+y = 0

(x+y).y' = 0.y' (x+y).x' = 0.x'

xy'+yy' = 0 xx'+yx' = 0 2

x.1+0 = 0 0+y.1 = 0 ( y'=1, x'=1)

E: x = 0 y = 0

9 73 . 2 B O O L E


98/335


3.9

:

E xy'+x'y = xz'+x'z, y = z

A: xy'+x'y = xz'+x'z xy'+x'y = xz'+x'z

(xy'+x'y)x = (xz'+x'z)x (xy'+x'y)x' = (xz'+x'z)x'

xy' = xz' x'y = x'z

(xy')' = (xz')'

x'+y = x'+z 6

(x'+y)x = (x'+z)x

xy = xz x'y = x'z

:

xy+x'y = xz+x'z

(x+x')y = (x+x')z

y = z

Afi

3.4

A. N 1 0

B. N : a+ab+ac+ad = a.

. N : (a+b+c+d)' = a'b'c'd'

. N : F = x'+y'+xyz'

E. N : F = a+a'b+a'b'c+a'b'c'd+

T.N F :

F = x'(y'+z')(x+y+z')

Z. N :

(x+y)(x'+y)(x+y')(x'+y') = 0

H. N : E a+b=a+c a'+b=a'+c, b=c

. N : E a+b=a+c ab=ac, b=c

I. N : E ab'+a'b=0, a=b

IA. N Boole,

B , B={1,0,a} (Y: -

a;).

IB. N

:

x'y' + x'z + xz' = y'z' + x'z + xz'


99/335

fi

H Boole ,

, A0 A5

(Huntington), A1 A5 .

H Boole

,

.

Boole ,

, -

, .

I , , -

, .

,, / 0, 1 -

+, ., . K

.

() , 1 7.

9 93 . 2 B O O L E

fi3.2

) X Huntington ( 3.1), -

a+a = a (1, 3.3), -

a+a (Y:

).

) x, y, z

, . d ,

=d. (Y:

. .. =xy+yz, ;

;)


100/335

1 0 0 K E A A I O 3 : A E B P A B O O L E

3.3 Boole

H Boole . A

-

( ). H -

() -

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.

O E :

O , . -

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(A, K)

.

H

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-

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Boole,

E Boole. E -

,

. K -

,

, .

H Boole,

-

.

Boole.


101/335

Boole, :

) , 0 1 . ={0,1},

) +, .

, ={0,1}, -

( 3.4), :

3.4

A

OR AND NOT

+ . '

0 0 0 0 0 1

0 1 1 0 1 0

1 0 1 0

1 1 1 1

O 3.2 -

, ,

Boole. O ( 3.4) -

OR (), AND () NOT () -

Y 3.3.1. OR, AND, NOT, -

,

Boole.

O

, B . T -

-

, ,

,

. -

de Morgan (6, Y 3.2.3).

3.10

N de Morgan :

(a+b)' = a'b'

A: -

1 0 13 . 3 B O O L E


102/335


a b -

:

a b (a+b)' a'b'

0 0 1 1

0 1 0 0

1 0 0 0

1 1 0 0

, .

O , -

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. Y 3.3.1 3.3.2, ,

:

K .

Afi

3.5

a, b, c, d :

) N a+a=a a.a=a

) E ab'+a'b = c, ac'+a'c = b.

) N a, b, c, d, -

:

a'+ab = 0

ab = ac

ab+ac'+cd = c'd

) N :

ab+a'b'+a'bc = abc'+a'b'+bc


103/335

3.3.1 fi

.

, , . , :

1) OR(A ORB) =( ),

2) AND (A AND B) =( ),

3) NOT , ( ), NOT .

A ,

, . (True) (False). H

-

A B 3.5:

3.5

A

A B OR AND A NOT

F F F F F T

F T T F T F

T F T F

T T T T

: () 1,

, () 0, ,

3.5 3.4, .

Boole.

, 0 1

+=OR, .=AND =NOT, -

-

. , ,

, , ..

.

,

(Y 3.2.1), ,

, .

1 0 33 . 3 B O O L E


104/335


3.11

-

, . :

1) , 35

2) , 35

3) , 35

4) , 35.

Z .

A: , ,

, -

(). :

, / , 35

', / ', 35 '

T, , , -

:

1) A AND AND K 3) A' AND AND K

2) A AND ' AND K 4) A' AND ' D K'

H -

OR 1), 2), 3) 4) :

= ( AND AND K) OR (A AND ' AND K ) OR

(A' AND AND K) OR (A' AND ' D K')

M

:

= (..)+(.'.)+('..)+('.'.')

= ..+.'.+'..+''' = ..+.'.+..+'..+''' ( 1)

= ..(+')+(+')..+''' (A A3)

= ..1+1..+''' (A A5)

= .+.+'''

H -

:


105/335

1) 35

2) / 35

3) , 35.

, -

, .

.

, ' =

NOT(), . :

' = (.+.+''')'

- . H

( A 3.6).

M , ,

E T

Y=(E A T B),

A B , ,

A, () B. A -

A, -

() Y. E -

Y ( 3.6),

, A B. E -

Y, , , :

Y = A'.B' + A'.B + A.B

= A'.(B'+B) + A.B = A' + A.B = (A'+A).(A'+B) = A' + B,. (Y=1) (=0) Y=E A T B

:

Y = A' + B (3.1)

H ET, , -

. E

( 3.1), -

1 0 53 . 3 B O O L E

3.6

Y=E A T B

A B Y

F F T

F T T

T F F

T T T


106/335


, . -

3.11 , -

.

3.12

E 3.11 1)

ET:

-

, . :

1) , 35

, 2) , 35,

3) , 35

4) , 35.

Z .

A: A 3.11, -

:

1) (.)' + 2) .'.

3) '.. 4) '.'.'

H :

= [(.)' + ] + [.'.] + ['..] + ['.'.']

= [A'+'+K] + A.'.K + '.. + '.'.'

= A'+'+K.[1 + A.' + '.] +A''.'

= A'+'+K + '.'.'

= A'.[1 + '.'] + ' + K

= A' + ' + K

H :

1) , 2) /, 3) 35.

E = A'+'+K = (A.)'+K :

E A , T K 35.


107/335

1 0 73 . 3 B O O L E

Afi

3.6

A. 3.11, -

. -

,

.

B. Boole

(3.1);

. , B, , E

,

' . H ,

:

) A B ,

) E ,

) A ,

) , E,

) B, A .

Z: () , () -

() -

.

fi3.3

K :

E: { ) E B K, A ,

) A }

T: B K.

M -

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B K A -

. -

. (Y: .

A . T

A; E

1, ;)


108/335


3.3.2 K (

)

O , -

Boole, 1938 C. Shannon

-

, . O,

Boole -

.

, , -

, ' , . -

.

( 3.2) : ,

, ,

() . ,

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x y -

( 3.2, OR) ( 3.2, AND).

y

y

x

x

x

x

K

OR

AND

ANOT

A

K

0

1

3.2

O 3.7 A -

3.2, .

x y A K.


109/335

, ( 3.2

NOT).

3.7

A

x y OR AND x NOT

A A A A A K

A K K A K A

K A K A

K K K K

3.7 (K),

() ,

1 (K=1) (A) 0 (A=0),

3.7 3.4,

OR, AND, NOT. , ( 3.2):

) , -

OR

) ,

AND.

) ,

NOT.

M (K=1, A=0) -

(=OR, =AND) , -

-

. A

3.2 .

O , ,

,

.

3.13

3.3 -

. A, A1

A2. , , A1 A2

1 0 93 . 3 B O O L E


110/335


A ( ), .

: A1=A2=A. , -

, .

. ,

, . ()

() .

B

A2

A1

M

3.3K

A:

) A, B, , -

. T M , , A2

.

A1 B , -

AND, . : A1

AND B. (A1, B)

M, :

= (A1 AND B) AND M.

T M ,

, . AND , A2,

(, ). E:

M = ( AND ) OR A2

A M :

= (A1 AND B) AND ( ( AND ) OR A2).

E A1=A2=A,

:

= (A AND B) AND ( ( AND ) OR A).


111/335

M AND OR

, :

= (A.B).( (.)+A )

= (A.B).(.+A) = (A.B)..+(A.B).A = A.B..+A.B.A

= A.B..+A.B

H ,

, : A1, B, , , A2

A1, B, A2. Y A1=A2=A.

) H , , :

= A.B..+A.B= A.B.(.+1) = A.B.1= A.B,

. , .

E K -

,

, K -

,

( E 3.8).

1 1 13 . 3

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