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7/22/2019 Linardis Psifiaka Systymata Eap
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I
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HPOOPIKH
A'
IANAITH INAPH
E K T A
ATPA 2001
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HPOOPIKH
A'
I
ANAITH INAPH
E K T A
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,
.
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fi
K 1
E
, ,
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
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6 H I A K H X E I A H I
2.4.3 K Hamming ................................................................................................... 63
....................................................................................................................... 69
................................................................................................................................... 72
B ..................................................................................................................... 75
K 3
Boole
, ,
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
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B .................................................................................................................. 153
K 4
K
, ,
E ............................................................................................................ 157
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
, ,
E ............................................................................................................ 205
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
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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
K
, ,
E ............................................................................................................ 259
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
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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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E
fi
M Pascal Leibnitz, -
Charles Babbage A M (1832). X
1930,
, , , -
, . O
-
, Colossus A (1943) ENIAC H
(1946).
H (1948) O K
(1958) -
. , ,
-
, , .
. K, :
T A .
T ,
, .
T .
T , , - .
T .
M ,
, ,
.
1
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1 2 K E A A I O 1 : E I A H
H K . A - ,
,
. T
, .
M , , -
, bit, ,
, K.
, , , , ,
K, - , , ,
. -
, ..
0 1.
, 0 1, -
. T, , -
, , -
, .
A
K
Bit
E
A
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1.1
H ,
, . O
,
,
. A , -
, -
.
E :
.
.
T , ,
: A .
A -
.
1.1. "" , .
, -
. , , ,
, , .. .
, -
.
, , -
. H 1.1
, -
, . -
, -
1, 2, 3,
,
,
, -
, ..
,
' , , -
, ..
.
1 31 . 1
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1 4 K E A A I O 1 : E I A H
, , -
. -
,
.
. ( ),
( ),
( -
, ), ..
T , ,
, .. 1.2. a c
, , .. 1.2.,
a, b, c d .
() ()
1
2
3
4
5
6
() ()
tt
a
b
c
b
c
a
d
1.1
() A
()
1.2
() ()
.
Afi
1.1
, ,
:
. . .
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1.2
T
-
.
A E :
N .
T : , , .
, ,
.
O , , - ,
. 1.3
, -
, -
, 0V, 1V, 2V 3V,
"", "", "" "".
N , -
, N . ,
1.3 1.4 -
.
1 51 . 2
1.3:E
1.4
0V
1V
2V
3V
0VA
B
2V
3V
5V
M
, .. "A", "B", .
,
, .. -
,
.
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1 6 K E A A I O 1 : E I A H
1.4 (0V
2V) (3V 5V) , .
, ,
.
-
. ,
, , ,
. E -
E N ,
=E/N.
, ( N)
( ).
( 1.5.), ,
, -
.
.
O
( 1.5.). 1.5. -
1V
1V.
,
, -
-
. 1.5. -
0V 2V "A", 3V 5V -
"B"
2V 3V.
0VA
B
2V
3V
5V
"A" "?"1V 1V +
1V + ???
A
()
M
()
1.5
A
T , ,
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-
. , -
. H ,
,
.
1 71 . 2
Afi1.2
. ,
( ):
) )
) )
. O :
) )
1.3
( E 1.2),
,
. H -
.
E :
E bit
K N (N>5) :
. N
.
. .
.
( ) -
, -
. 1.6.,
{, , , } ,
{A, B},
1, 2 ( 1.6.)
, . ""=A1A2, ""=A1B2, ""=B1A2, ""=B1B2. E
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1 8 K E A A I O 1 : E I A H
-
. , 2K -
K .
T , -
,
. -
"0" "1" -
bit (binary digit). A, 1 bit -
.
t
B
A1
t
()
Z
()
T
{
B
A2{
1.6
-
1
A
B
0 1 0 1 0 1
B A
t5
1 1 0 1 0 0 1
t0 t1 t2 t3 t4 t5 t6 t7
0 1 1 0 0 1 1
1.7
X
, .
1.7
A, B bits,
t0, t1, t2, , t7.
7/22/2019 Linardis Psifiaka Systymata Eap
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1.4
, E 1.1,
-
. ,
.
M :
.
E -
.
,
, .
T
, .. , -
, .. ,
, .
. , ,
.
O A "T A" "-
" : E c e fi z
a , . -
(), (-
1 91 . 3
Afi
1.3
. H bit :
) ) "0" "1"
)
. () ()
. . bits
:
) ) ) ) )
. 246 . bits
;
7/22/2019 Linardis Psifiaka Systymata Eap
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2 0 K E A A I O 1 : E I A H
) ( )
( ), -
(a ). A A "" -
,
" ",
, -
. E
, K 3,
, .
, , -
( ), -
, X, -
"" F -
, Y=F(X)
.
, -
, . -
, ,
, -
. T :
) T .
.
) T .
() -
(),
.
) T . (-
) , -
, .
, -
.., , -
-
,
7/22/2019 Linardis Psifiaka Systymata Eap
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) T , -
( , )
(),
) T ( ),
-
,
) T , '
, ()
, .. , -
, , ..
H "" , .. , -
, -
. H
, -
, . T -
, -
.
-
, - .
2 11 . 4
Afi1.4
" "
, . -
:
1. P .
2. , ,
3. A , -
,
4. A ,
, -
( -
, , , ) ..
-
;
7/22/2019 Linardis Psifiaka Systymata Eap
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2 2 K E A A I O 1 : E I A H
1.5 M
H -
. -
, .
A :
.
N -
.
H , N ,
. H N ( 1.8) N , ..
, N N
. H -
t.
H N -
. X ( 1.9), -
, 1, , N t1, t2,
t3, tN () -
t1+, t2+, t3+, tN+ , - .
N
A
t
1E
1.8
, -
,
, N -
, N -
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, . -
,
, ,
, (
) ,
,
, -
.
2 31 . 5 M
A
1
N
1
N
tk+ tk
t1 t2 t3 tN t
1.9
1.6 M fi
H -
.
A :
-
.
( -
) .
A/D,
, .
-
. 1.10.
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2 4 K E A A I O 1 : E I A H
. E
, .. , -
,
1.10.. H -
.
0 3
T
6 9 12 3 6..
()
..90 3
T
6 9 12 3 6..
()
..9
0 3
0
1
23
T
6 9 12 3 6..
()
..9 0 3
0
1
23
T
6 9 12 3 6..
()
..9
1.10
K
1.11
1.10
M
. T -
, ..
K,
. 1.11. -
. 2, 3, 3, 3, 2, 1, 1, 1, -
. 1.11.
1.10..
E
. T , , -
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. T
. E -
, .. K,
. K' -
,
.
, , -
..
.
-
, M A/D (Analog toDigital Converters). ,
, M D/A (D/A Converters).
O A/D D/A :
) A , .
( ),
28 216 (8 16 bits),
) T , 102
106 /sec.
2 51 . 6 M
Afi1.5
. K ,
:
) )
. ,
. -
. -
. .
;
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2 6 K E A A I O 1 : E I A H
1.7
H (E
1.6)
. , ,
; :
.
:
T -
.
M -
.
T CD
.
T ""
, . -
' ,
, ,
:
A, B, V=A, - R=B I,
, I=A/B Ohm.
T ,
, .. -
, Fourier,
""
.
-
, , :
1. . ,
, .. , -
. -
.
. , ,
7/22/2019 Linardis Psifiaka Systymata Eap
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, ,
,
, K 2.
2. . ,
,
, ( 1.12).
bits .
2 71 . 7
10% 1% 0.1%
A
K
A
1.12
1.13
,
, -
,
, 1.13.
A
,
,
, -
Compact Disc (CD),
.
A
D
D
A
A
A
M M
7/22/2019 Linardis Psifiaka Systymata Eap
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2 8 K E A A I O 1 : E I A H
1.8
H ,
-
, , . A
,
-
, -
.
:
H .
H (C.
Shannon)
H .
H , ,
,
. O ""
O , -
, , - . O "calculators" 19 , .
,
.
H 19 ,
, ()
( , -
) () (,
, ) . M , ,
(, ,
) ,
, , , -
. -
.
M 1938 C. Shannon, , -
7/22/2019 Linardis Psifiaka Systymata Eap
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,
, -
" ". H , -
K 3,
.
H "",
1958
"O K" (IC),
-
. T , , O
K (OK) 1962
(
Moore). T , OK
-
. T
.
O OK
1.1. T
OK -
1.14.A
,
1.1 1.14 . H -
. A -
.
1.9
E 1.8
-
. E , ,
. ,
, ,
,
..
2 91 . 8
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3 0 K E A A I O 1 : E I A H
K. T -
: () '
,
() ' .
-
M E A
: .
T:
A "" .
Y.
(-
).
M T
: E
() ()
.
T:
.
-
(PLA, PROM) .
: T
. ,
, , -
.
T:
M A E (Design Verification)
1.1
K -
-
OK.
7/22/2019 Linardis Psifiaka Systymata Eap
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A. M
"",
A "" .
K 2 -
-
. K, K 3 -
Boole ,
-
.
K 4
,
. K 5
. T, K 6
.
3 11 . 9
E
T
T
K
TTL
PAL, PLA
PROM, PLD
MOS
CMOS 1.14
T
OK
fi1.1
,
K.
(E 1.4) -
() -
.
7/22/2019 Linardis Psifiaka Systymata Eap
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3 2 K E A A I O 1 : E I A H
T . -
.
A .
.
-
.
.
M , .. . O -
.
Bit 0 1 .
K -
. K () 2K-
.
-
-
fi1.2
: ()
( ) 250 , ()
, ()
.
( .. 1.13) :
)
(E 1.6).
) bits (E 1.3).
) (E 1.5).
) T , , -
(E 1.7).
(Y:
.)
7/22/2019 Linardis Psifiaka Systymata Eap
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. -
.
,
. K
.
O
. T
, , , A/D
D/A. Y , .
T A , ,
-
, -
.
T , -
, .
T -
-
. O
.
3 3 Y N O H
7/22/2019 Linardis Psifiaka Systymata Eap
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3 4 K E A A I O 1 : E I A H
B
T
. A E
A. E , , ,
.
Floyd T., Digital Fundamentals, Macmillan, 1990.
1 K , , A-
, -
O K.
11 K
.
Leach D., Malvino A., H, E TZIOA, 1996
1 K
A ,
. O-
K.
11 K -
.
Tokheim R., H, E TZIOA
K
. E -
13 K
.
Hall D., Digital Circuits and Systems, McGrawHill, 1989
K
.
K 10 ,
11 15 -
-
.
7/22/2019 Linardis Psifiaka Systymata Eap
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K
fi
H
,
.
(, BCD,
2), (,
, Gray). A, ,
-
( )
( Hamming) .
K, :
T .
O (Binary, BCD, Gray) .
T
.
N .
T ,
.
N ( ).
T .
( Hamming).
2
K
A
Bit
B
Binary
7/22/2019 Linardis Psifiaka Systymata Eap
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3 6 K E A A I O 2 : Y A I K H K I K O O I H H
T K ,
. H -
, -
(, BCD, Gray). H -
( ) -
( ). H , -
,
(
). H -
.
T K, -
, K
.
BCD
Gray
2
A Hamming
K Hamming
7/22/2019 Linardis Psifiaka Systymata Eap
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2.1 E
H -
.
, :
T .
O
T
.
E (,
, bit) - K.
, ..
,
.
.
(.. -
, ) -
. , -
.
, -
, -
. , -
-
, 0, 1,, 9.
, ,
( E 1.2 1.3) -
"0" "1", bits. E
{0,1}
(binary) bits, ..
011, 110, 000, 1110.
n , . n bits,
3 72 . 1 E
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3 8 K E A A I O 2 : Y A I K H K I K O O I H H
n, 2n. , -
bits,
: < 2.
2.2 M
H ,
-
. H
, -
. A -
(, BCD, Gray).
A :
T -
.
.
T BCD.
T (Gray ) (-
).
T Binary Gray .
Gray .
: ()
()
K. E ()
() , -
Gray.
:
. H , -
, , -
, .. -
( ), (
), (.. I
), .. , -
7/22/2019 Linardis Psifiaka Systymata Eap
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, -
, . , -
.
. -
, -
.
-
, ASCII. , -
7 bits, 128 , :
) , .:
,
,
0 9,
) , . -
( , , stop ..).
-
, .. -
.. , ,
.
: -
.
2.2.1 K
, ,
-
, .. , -
..
:
:
nn1 10
3 92 . 2 M
7/22/2019 Linardis Psifiaka Systymata Eap
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/
0
1
2
3
4
5
6
7
8
9
8 4 2 1
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 1
1 0 0 0
1 0 0 1
2 4 2 1
0 0 0 0
0 0 0 1
0 0 1 0
0 0 1 1
0 1 0 0
1 0 1 1
1 1 0 0
1 1 0 1
1 1 1 0
1 1 1 1
6 4 2 3
0 0 0 0
0 1 0 1
0 0 1 0
1 0 0 1
0 1 0 0
1 0 1 1
0 1 1 0
1 1 0 1
1 0 1 0
1 1 1 1
4 0 K E A A I O 2 : Y A I K H K I K O O I H H
i (i=0, , n)
{0, 1, , K} , :
= nwn + + 1w1 + 0w0
O wi (i=0, , n) -
. i ,
{0, 1, ,
K}. i
{0, 1}, bits.
2.1
, - 0 9,
8, 4, 2, 1. , ,
i bit 0 1, ,
, 0 1. O -
(8 4 2 1)
2.1. (2,
4, 2, 1) (6, 4, 2, 3). -
8 1000 1110 1010, :
8 = 8*1+4*0+2*0+1*0 = 2*1+4*1+2*1+1*0 = 6*1+4*0+2*1+(3)*0.
2.1
K
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4 12 . 2 M
-
. 8, 4, 2, 1, . 23, 22, 21, 20 -
Bin8421.
-
. 2, 4, 2, 1
2.1, 7 1101, 0111.
.
Afi
2.1
A. N {0, 1,
, 15} (8, 4, 2, 1).
B. ,
(6, 4, 2, 3);
. N {8,
7, , 6, 7} (8, 4, 2, 1).
. , -
{0, 2, 4, 6, 8, 10, 12, 14};
2.2.2 K BCD
8, 4, 2, 1 -
BCD (Binary Coded Decimal). 10 10 10
( 2.1, 8 4 2 1).
2.2
H BCD 12345 :
1234510 00010010001101000101BCD
A bits 16 , 0000
1111, BCD . O -
00010010001101000101
3 4 521
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4 2 K E A A I O 2 : Y A I K H K I K O O I H H
1010, 1011, 1100, 1101, 1110 1111
BCD.
E BCD .
A BCD bits
bits .
2.3
O BCD 2469:
0010010001101001
6 942
Afi
2.2
. BCD ;
) 1111, ) 53, ) 1998, ) 2469.
. BCD
;
) 00111001, ) 000100110100,
) 001100100100001, ) 0001001101001100.
2.2.3 K K Gray
. ..
, -
.
Gray. -
, 2.1
: ( 2.1.) Bin8421 (- 2.1 8,4,2,1) ( 2.1.) Gray,
. X 360
, 3 bits . O -
-
. K ,
bit. O
( 2.1) (). T -
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( 2.1), bit
,
. K , -
, 1 0,
. O -
() (
),
(-
).
A , Bin8421 (
2.1) 011 100. T
. H ,
, ,
. , ..
,
, 111 ( 3 ) 000
( 4 ) .
O Gray ( 2.1) -
. 2.1,
bit ( ), -
.
4 32 . 2 M
000
001
010
011
()
Bin8421
100
101
110
111 000
001
011
010
()
Gray
110
111
101
100
2.1
K
Gray , -
(reflection codes). n bit
n1 bit :
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4 4 K E A A I O 2 : Y A I K H K I K O O I H H
) n1 bit
, . -
,
) bit, 0
1 .
2.2
Gray 2 3 bit, bit.
Gray : 1 bit 2 bit 3 bit
0 00 000
1 01 001
11 011
10 010
110
111
101
100
Bin8421 Gray . 2.3
bi (i=0,1,2) Bin8421
gi (i=0,1,2) Gray.
H : gngn1 g1g0 -
Gray (n+1) bits bnbn1 b1b0 Bin8421
. gi Gray
Bin8421 :
gn = bn (2.1.)
gi = bi bi+1, 0 i < n (2.1.)
modulo 2, :
0 0 = 0, 0 1 = 1, 1 0 = 1, 1 1 = 0
2.2
Gray
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/
0
1
2
3
4
5
6
7
Bin8421
b2b1b0
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
Gray
g2g1g0
0 0 0
0 0 1
0 1 1
0 1 0
1 1 0
1 1 1
1 0 1
1 0 0
4 52 . 2 M
2.3
A -
Bin8421 Gray
2.4
H Gray Bin8421 101101, -
111011:
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)
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4 6 K E A A I O 2 : Y A I K H K I K O O I H H
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).
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2.3 A
A , -
.
,
, -
, . -
, ,
-
( 2).
T , :
T , - .
T
.
T E .
.
,
, .
.
2
.
E -
,
K. Y : ()
, . ( ) ()
( 2). E
. T 2 -. M -
.
,
(), .., -
. -
4 72 . 3 A
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4 8 K E A A I O 2 : Y A I K H K I K O O I H H
. 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
.
,
:
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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
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2.3.2 fi
, - 0 1. X
. ,
, bits.
H
n (bits) 2n1,
1*2n1 + 1*2n2 + + 1*21 + 1*20 = 2n1
5 0 K E A A I O 2 : Y A I K H K I K O O I H H
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
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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):
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5 2 K E A A I O 2 : Y A I K H K I K O O I H H
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
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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
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5 4 K E A A I O 2 : Y A I K H K I K O O I H H
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
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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 -
.
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5 6 K E A A I O 2 : Y A I K H K I K O O I H H
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):
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) (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
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5 8 K E A A I O 2 : Y A I K H K I K O O I H H
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
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Hamming
,
Hamming
H : -
.
K.
, -
, ( )
. -
.
, '
, : () , -
, ()
bits .
bit .
, -
, . -
-
.
O . -
, ()
.
p , . bit .
p,
, .
,
.
' , n bit ,
5 92 . 4 K
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6 0 K E A A I O 2 : Y A I K H K I K O O I H H
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
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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
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6 2 K E A A I O 2 : Y A I K H K I K O O I H H
.
.
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.
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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
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6 4 K E A A I O 2 : Y A I K H K I K O O I H H
. 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).
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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 -
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6 6 K E A A I O 2 : Y A I K H K I K O O I H H
:
: 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 -
:
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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
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6 8 K E A A I O 2 : Y A I K H K I K O O I H H
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
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:
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).
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7 0 K E A A I O 2 : Y A I K H K I K O O I H H
) ()
, 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
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7 12 . 4 K
. ,
.
) X ()
, , -
,
(Y 2.4.3).
) A 2.6, -
, -
,
,
. : () ()
(
).
M
M
A
M
M K
2.6
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7 2 K E A A I O 2 : Y A I K H K I K O O I H H
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 -
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, -
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
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7 4 K E A A I O 2 : Y A I K H K I K O O I H H
N , Hamming
N.
N , Hamming
2N.
H ( ) -
, -
.
O Hamming , , -
, -
. K Hamming
, - bits . K
-
, ,
bit, ().
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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
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7 6 K E A A I O 2 : Y A I K H K I K O O I H H
Rao T., Fujiwara E., ErrorControl Coding for Computer Systems, Prentice
Hall, 1989.
E -
( ).
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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
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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
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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
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8 0 K E A A I O 3 : A E B P A B O O L E
) N , ,
(
Y 3.3.2).
) A A -
,
,
(E 3.4).
) N
-
(E-
3.7). , ,
E
.
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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
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8 2 K E A A I O 3 : A E B P A B O O L E
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
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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
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8 4 K E A A I O 3 : A E B P A B O O L E
, -
. M " "
Shannon (1938) , Boole
. E
Boole, Huntington,
.
T E :
T Boole (, , ).
T .
Boole
.
.
T .
T Boole.
E -
, -
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,
E. H Boole,
,
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, -
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.
H Boole , ,
G. Boole,
A , -
.
H Boole , . -
, :
) , . T
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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 +, ., ,
:
+ ,
. ,
.
) T 0 1 (0,1), - .
) a, b, c
3.1:
8 53 . 2 B O O L E
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8 6 K E A A I O 3 : A E B P A B O O L E
, ,
, . a.b ab. ,
a ( -
a), , ,
,
a'.
:
Boole, A0, -
, -
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. A ,
:
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. B -
0 1. B, ,
() .
) A1
, . -
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)
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, a.(b+c), -
, a+(b.c). -
.
) T A4 , -
,
B, 0 1 ,
. 0 -
1, .
0 1.
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, a - , a
1 0. : a, .
.
Boole -
. H Boole, , -
,
:
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.
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. a+(b.c)=(a+b).(a+c), .
E ,
.
) , .., -
-, . a+c = b+c a = b.
) T ( A5)
.
E ,
:
a a 0+
a
a
8 73 . 2 B O O L E
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8 8 K E A A I O 3 : A E B P A B O O L E
) B
) , .
.
O
Boole, -
-
( Boole). E
, .
B , -
Boole. M
, E 3.3. E 3.3
Boole. E, ,
Boole -
:
3.1
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
. A B I
A A A
B B B
I A B I
x
I
A B
B A
I
x
3.2
A
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3.2.1 A
3.1 A1 A5
. H
,
, ,
, , -
. ..
+ . 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 -
).
) N -
. (Y:
-
).
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9 0 K E A A I O 3 : A E B P A B O O L E
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.
, -
, : , ,
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. .. 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
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9 2 K E A A I O 3 : A E B P A B O O L E
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
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{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
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9 4 K E A A I O 3 : A E B P A B O O L E
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
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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
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9 6 K E A A I O 3 : A E B P A B O O L E
= [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'
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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
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9 8 K E A A I O 3 : A E B P A B O O L E
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'
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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, ;
;)
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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 -
() -
. H
.
O E :
O , . -
( A).
O ,
A
.
H ,
.
H ,
(A, K)
.
H
. H -
-
.
Boole,
E Boole. E -
,
. K -
,
, .
H Boole,
-
.
Boole.
7/22/2019 Linardis Psifiaka Systymata Eap
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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
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1 0 2 K E A A I O 3 : A E B P A B O O L E
a b -
:
a b (a+b)' a'b'
0 0 1 1
0 1 0 0
1 0 0 0
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:
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) 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
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3.3.1 fi
.
, , . , :
1) OR(A ORB) =( ),
2) AND (A AND B) =( ),
3) NOT , ( ), NOT .
A ,
, . (True) (False). H
-
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A
A B OR AND A NOT
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: () 1,
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-
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.
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1 0 33 . 3 B O O L E
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1 0 4 K E A A I O 3 : A E B P A B O O L E
3.11
-
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1) , 35
2) , 35
3) , 35
4) , 35.
Z .
A: , ,
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(). :
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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 -
:
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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
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1 0 6 K E A A I O 3 : A E B P A B O O L E
, . -
3.11 , -
.
3.12
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ET:
-
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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.
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1 0 73 . 3 B O O L E
Afi
3.6
A. 3.11, -
. -
,
.
B. Boole
(3.1);
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,
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:
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M -
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A . T
A; E
1, ;)
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1 0 8 K E A A I O 3 : A E B P A B O O L E
3.3.2 K (
)
O , -
Boole, 1938 C. Shannon
-
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.
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.
( 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
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x y A K.
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, ( 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) -
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-
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3.2 .
O , ,
,
.
3.13
3.3 -
. A, A1
A2. , , A1 A2
1 0 93 . 3 B O O L E
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1 1 0 K E A A I O 3 : A E B P A B O O L E
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).
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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