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/ &
, 2, , 0 1:
n 2n 2 (0_3), 5 (0_31), 8 (0_255)
: : 1910 19/2= 9 1 b0=1 9/2= 4 1 b1=1 4/2= 2 0 b2=0 2/2= 1 0 b3=0 1/2= 0 1 b4=1
2=10011=1910
: . . : 0,375 0,375 x 2 = 0,75, 0, 0,75 b-1=0 0,75 x 2 = 1,5, 1, 0,5 b-2=1 0,5 x 2 = 1,0, 1, 0 b-2=1 2= : ,0112 28,375 : 2= : 11100,0112
(truth table). . 1 0, (true), (false), ( 5V 0V).
OR
H (true) (1), (1)
0 0 0 0 1 1 1 0 1 1 1 1
A B Z
A
B
Z
t
t
Time t
AND
H (1), (1)
A
B
Z
AND
0 0 0 0 1 0 1 0 0 1 1 1
A B Z
A
B
time t
t
t
NOT ()
1
A
Z
NOT
A
0 1
1 0
A
time t
t
NAND ( AND)
(0) (1)
A
B
NAND
Z
0 0 1 0 1 1 1 0 1 1 1 0
A B Z
NOR (NOT OR)
H (1), (0)
0 0 1 0 1 0 1 0 0 1 1 0
A B Z
A
B
Z
time t
t
t
XOR H (1), (1),
0 0 0 0 1 1 1 0 1 1 1 0
A B Z
XNOR (NOT XOR)
H (1) (0), (1)
0 0 1 0 1 0 1 0 0 1 1 1
A B Z
1
A
Z
AND
0 0 0 0 1 0 1 0 0 1 1 1
OR
A
B
Z
0 0 0 0 1 1 1 0 1 1 1 1
NOT
0 11 0
NAND
A
B
Z
0 0 10 1 11 0 11 1 0
NOR
A
B
Z
0 0 10 1 01 0 01 1 0
XOR
0 0 00 1 11 0 11 1 0
XNOR
0 0 10 1 01 0 01 1 1
. : FF, FT, TF, TT (F, T) :
A. B Z A F F T T B F T F T ( ) 0 F F F F 0 0 1 F F F T AND 2 F F T F - - 3 F F T T A 4 F T F F - - 5 F T F T B 6 F T T F OR 7 F T T T OR 8 T F F F NOR 9 T F F T R 10 T F T F Not B 11 T F T T - - 12 T T F F Not A13 T T F T - -14 T T T F NAND15 T T T T 1 1
- : BCD (Binary Coded Decimal) . - 4 bits -
4510 BCD4510=01000101BCD BCD 4bits . 1010011BCD . 4. M [0101][0011]BCD =5310
45
4 50100 0101
Gray - -. , , bit, -.
, , . 1, , 0, , . , , , (parity bit).
{
0 , 1, . 1 , 1, . 010001 1 , 1, : 1 | 010001
. 1 1 . 0 1 10110 1 , 1, : 1 | 10110
Boole George Boole (1815-1884) Augustus De Morgan. Claude Shannon ( MIT) Boole (switching) . Boole .
Boole . . .
:
-
De Morgan
:
:
:
De Morgan AND NOR , OR NAND
De Morgan NAND OR OR :
=
, , . : - ( Boole) ND, OR . NAND, NOR.
( OR) AND , (1 0). 1
Q=ACF+ACG+ACED+ABD+ABEF+ABEG
: , , Boole . :
0 0 0 0 1 1 0 0 0 1 1 1 1 1 0 1 0 1 0 1 00 1 1 1 1 0 01 0 0 1 1 1 11 0 1 1 1 1 11 1 0 1 1 1 11 1 1 1 1 1 1
. ( OR) . :
: F=0. , 0 , 1, . .
0, 1. , , : XOR
XOR : F=1: F=0::
, . . :
:
NAND , NAND DTL & TTL, NAND : :
( P 1)
2. , 0, ( 0, )
3. De Morgan :
4. NAND F=ABC+ABD NAND: :
NAND AND, OR NOT NAND.
R NOR NAND, . NOR NOR NAND. AND, R NOT NOR :
Karnaugh , .
( )
( )
Boole. Karnaugh, .
Karnaugh ABC, Karnaugh :
:
(8) .
( Gray).
:
(4) .
.
- F = (1,2,5,6) : 2 2 . , 3 .
- F = (0,2,4,9,11), :
F = (3,4,5,6,7,8,10,12,14)
. (alarm), 0,4,6,8,9. 0 9 ) ) -.
: :
:
:
:
:
:
. (delay) s. , (hazards), . :
, , . : 1 (hazard).
: . . :
. :
: . .
: . . . Flip Flops flip-flops (1 0), (1) bit. flip-flops, .
flip-flops .
SR-ff Karnaugh, Qn ( ) :
: De Morgan, NAND :
Flp Flop D
flip-flops SR-ff T-ff SR-ff : T-ff ( CLK, T, Qn ) SR-ff Qn+1
S, R Ck, T Qn
:
: CLK=1, CLK,
ff 1.
2. ff
ff ffs :
SISO (Serial In Serial Out). SISO Q1 Q4 (MSB LSB ), . , Q4 MSB Q1 LSB, . : 0 2 2
: LSD ff , ffs ff .
(up-counter) C D QA QB QC QD , (Down Counter) A B C D
MOD-M , S7 S13
1
1 0 1 1 bit ( J-ffs)
, (Reset circuitry). , , .