3
3. 3 , 40- 19 (George Boole, 1815-1864). , t, 30- (Claude Elwood
Shannon, 1916-2001 ) .. . - , , , . , , , , , . 3.1 , , , 0 1. . 0
1. 0+0 = 0:0+1 = 1+0 = 11+1 = 1(3-1) :
= 0
= = 0
(3-2) :
= 1
= (3-3) e 0 1 (3-1), (3-2) (3-3). , A, B, C, .a . :
(3-4) . , . , , , , .., .
, . , (3-4), : , , . , 3 , . (3-4) :
, 3-1. , . 3-1 3-4. ( o , , en - . .)ABA+B
000
011
101
111
3-1
3-2
(3-.5)ABAB
000
010
100
111
3-2
, . , . , , , , .., .
, . , 8. 3-3. , ,
. , ABCABAB+C
00000
00101
01000
01101
10000
10101
11011
11111
3-3
. .
01
10
3-4
3.1.1
3.1.2 . . .3.1.3 :
, . . , . 3.1.4 .:
00011
10101
3.1.5 .
3.2
, , , .., , . . . , , , .
(3-6)
. 3-5.ABCA+BA+C(A+B)(A+C)BCA+BC
00000000
00101000
01010000
01111111
10011101
10111101
11011101
11111111
3-5 (3-6)
3.2.1
()
()
()
()3.3 (AND), (OR) (NOT) - , , , (OR), (AND) (NOT).
, , ( . true) ( . false). , .
, ..,. . , .., , , .., ; , .., ; , .., . , , , : . , 3-6.AB
FFF
FTT
TFT
TTT
3-6
, D=A B (D=A AND B), .., . D A B , ... 3-7AB
FFF
FTF
TFF
TTT
3-7
, = (). . ( , = , , = . A
FT
TF
3-8
, , , . F 0, 1 , . , , . , . , . :
, A B", A B" . .
, B", B", .
. . : . : . D , A, B C , , :
3-3: 1 , 0 , D . , A , B , , , D .3.3.1
T () F (), .. A, B C ()T, F, T()F, T, T()T, T, F 3.3.2
, A, B, C D () T, F, T, F (a) T, T, F, F(b) F, T, F, T(c) F, F,
T, T: () F;() T;() T;() T 3.3.3 C : , A: ,B: () C - A B ;- A B .()
C , () C , .
()AB
000
011
101
110
3.4 , 0- 1- , . , .
=1
3-10
, 3-10, , ,
(3-7)
(3-8) (3-7) . (3-8). (3-7) (3-8) (3-7). :
(3-9)
(3-10), , (39), (3-10) . 3-10, - - . .
(3-11)
(3-12) (Augustus De Morgan (1806-1871)). (3-11) 3-11, (3-12)
3-12. , , .
001111
011000
100100
110000
3-11
001111
011011
100111
110000
3-12
3.5 . . . ( 3-10 ), . , , , . :
1()
1() , 3-13. .
0011010100010001
3-13
2()
2()
00001111001100110101010100010001000000110000000100000001
3-14
3-14, .
2() 2() , , . . , , . :
3()
3() . , , , . 3() 3-5. 3() , 3() , . , :
4()
4()
5()
5()
6()
6()
7()
7()(3-13)
,
8()
8()(3-14) 8()! : (I) , (II) . (I) :
(3-15) :
(3-16) (3-15) (3-16) 8() (I). (II), ,
,
(3-17), , , ,
(3-18) (3-17) (3-18) 8() (II). .3.5.1 :
()
()
()
() ()
00110101110001000111011110001000
3.5.2 3.5.1 () , , , , ,
3.5.3
3.6 . . . , , . , 2x8=16 :
9()
9()(3-19) : , ,
9() . 9():
. :
, :
:
10()
10()(3-20) 10()
3()
7()
5() 10() 10(), 10(). . , :
3()
7()
5() :
11()
11()(3-21) 11() :
, :
12()
12()(3-22)
( 12(), , 12() ). 12 () :
9(). 12() :
9().
. , . , , .3.6.1 , , :
()
()
()
()
()
()
()
() ()
8()( )
10()
7()
= 1 4()3.7 ( 3-15) . , a . , . : (normally opened), 3-15(a)
(normally closed), 3-15(). 1 2 . . , , 1 2 , .
3-15 () ()
1 1 2 0 , 1 0 . , , ,
3-16 . () , (), . , 3-16(), , 1 2 . , 3-16(), , 1 2 . , , ,
3-16
() ()
(3-23)
(3-24) , , .., , . , .
3-17
, 3-17
(3-25)
.., .
3-18
3-18
3.7.1
3.7.2
.
3.8 :() () .
(3-26) 4 , . , :
(3-27)
:
(3-28) 5 . - , :
(3-29)
, , . , , . (3-27), (3-29) , (3-27) (3-29) . , , . , ., :
(3-30)
, . . . . , :
, , :
(3-31)
(3-31), , (standard sum of products). .
1. 1 , 1. Z 1 , 1. 1, 0. , 0 . ( / 1, 0.)
3-19 (3-30) . 1 . ! , . 1 ( 1 0 ). 1
00001111001100110101010101001111
3-19
, (standard product of sums), , .
, . , .
(3-32)
. , ..,:
(3-33)
, , . ( !)
(3-34),
(3-35)
, , :
(3-36)
, , . .
1 0. (3-36) 0 / 0. , 0 / 0. ( 0, 1). 0, 1. 3-19 , . 0. : , . ( 0
1 ). 0. 3.8.1() 5 ?
() 01000, 01011, 10001, :
()
() , ,
3.8.2 1 0, 0, 1 0, 1, 1 1, 0, 1, , 0 .3.8.3 .
: 0 000, 010, 100, 110, 111,
3.8.4 () ()
()
()3.9
, , . , 3-19 0. 3, - 4. , 0, 1. : 0, 1. , . i- , i- . , , , . ()
. , , 3-19, :
.3.9.1 () ()
. () , ()
001101011001
:
()
()
()
()
34
35
1
2
1
2
(a)
()
1
2
1
()
2
()
DC
1
D
C
2
DC
1
2
A
B
