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Chng 2
i S Boole & Cc Cng Lun L
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
2
Ni dung
i s Boole
i s chuyn mch
Cc cng lun l
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
3
i s Boole
i s Boole c th gii bit n ln u tin bi
George Boole qua tc phm An Investigation of the
Laws of Thought vo nm 1854
i s Boole 2 phn t: cc hng v bin Boole ch c
mang 2 gi tr 0 hoc 1 ( LOW / HIGH )
Cc bin Boole biu din cho mt khong in p trn
ng dy hoc ti ng nhp/ng xut ca mch
Gi tr 0 hoc 1 c gi l mc lun l (logic level)
Mchlun l
ng nhp ng xut
A
x
F
y
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
4
i s Boole
i s Boole, cng tng t nh cc h i s khc,c xy dng thng qua vic xc nh ngha mt snhng vn c bn sau: Min (domain), l tp hp (set) cc phn t (element) m
trn nh ngha nn h i s
Tp hp cc php ton (operation) thc hin c trnmin
Mt tp hp cc nh (postulate), hay tin (axiom)c cng nhn khng qua chng minh. nh phi mbo tnh nht qun (consistency) v tnh c lp(independence)
Mt tp hp cc h qu (consequence) c gi l nh l(theorem), nh lut (law) hay quy tc (rule)
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
5
nh Huntington
Pht biu bi nh ton hc Anh E.V.Huntington trn
c s h thng ha cc cng trnh ca G. Boole
S dng cc php ton trong lun l mnh
(propositional logic)
1. Tnh ng (closure)
Tn ti min B vi t nht 2 phn t phn bit v 2 php
ton + v sao cho:
Nu x v y l cc phn t thuc B th x + y cng l 1
phn t thuc B (php cng lun l - logical addition)
Nu x v y l cc phn t thuc B th x y cng l 1
phn t thuc B (php nhn lun l - logical multiplication)
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
6
nh Huntington
2. Tnh ng nht (identity)Nu x l mt phn t trong min B th
Tn ti 1 phn t 0 trong B , gi l phn t ng nhtvi php ton + , tha mn tnh cht x + 0 = x
Tn ti 1 phn t 1 trong B , gi l phn t ng nhtvi php ton , tha mn tnh cht x 1 = x
3. Tnh giao hon (commutative)
Giao hon ca php + :x + y = y + x
Giao hon ca php :x y = y x
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
7
nh Huntington
4. Tnh phn phi (distributive)
Php c tnh phn phi trn php +
x (y + z) = (x y) + (x z)
Php + c tnh phn phi trn php
x + (y z) = (x + y) (x + z)
5. B (complementation)
Nu x l 1 phn t trong min B th s tn ti mt phn
t khc gi l x (hay x ), l phn t b ca x tha mn:
x + x = 1 v
x x = 0
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
8
Tnh cht i ngu (Duality)
Quan st cc nh Hungtinton, ta thy chng mang tnhi xng (symmetry) tc l cc nh xut hin theo cp
Mi nh trong 1 cp c th c xy dng t nh cnli bng cch Thay i cc php ton 2 ngi ( + | )
Thay i cc phn t ng nht ( 0 | 1 )
C th suy ra mt kt qu no t cc nh bng cch Hon i php ton + vi php ton
Hon i phn t ng nht 0 vi phn t ng nht 1
iu ny th hin tnh i ngu i s Boole
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
9
Cc nh l c bn (fundamental theorem)
Cc nh l c chng minh t cc nh Huntington v ccnh i ngu theo 2 cch Chng minh bng phn chng (contradiction) Chng minh bng quy np (induction)
nh l 1 (Null Law)
1.a x + 1 = 1 1.b x 0 = 0
nh l 2 (Involution)
(x ) = x
nh l 3 (Idempotency)
3.a x + x = x 3.b x x = x
nh l 4 (Absorption)
4.a x + x y = x 4.b x (x + y) = x
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
10
Cc nh l c bn
nh l 5 (Simplification)
5.a x + x y = x + y
5.b x (x + y ) = x y
nh l 6 (Associative Law)
6.a x + (y + z) = (x + y ) + z = x + y + z
6.b x (y z) = (x y) z = x y z
nh l 7 (Consensus)
7.a x y + x z + y z = x y + x z
7.b (x + y) (x + z) (y + z) = (x + y) (x + z)
nh l 8 (De Morgans Law)
8.a (x + y) = x y
8.b (x y) = x + y
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
11
Ti gin biu thc boole
Y = A(AB + ABC)
= A(AB(1 + C)) distributive
= A(AB(1)) Null Law
= A(AB) identity
= (AA)B Associative Law
= AB Idempotency
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
12
V d
Ti gin
x = ACD + ABCD
z = (A + B)(A+B)
De Morgans
z = ((a+c) . (b+d))
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
13
V d
Ti gin
x = ACD + ABCD
= CD( A + AB )
= CD( A + B ) = ACD + BCD
z = (A + B)(A+B)
= AA + AB + AB + BB = 0 + (A+A)B + B = B
De Morgans
z = ((a+c) . (b+d))
= (a+c) + (b+d) = ac + bd
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
14
Ti gin biu thc bool sau
a)
b)
Ti gin biu thc boole sau s dng nh l DeMorgan
Bi tp
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
15
i s chuyn mch (switching algebra)
i vi i s Boole, min khng b hn ch (khng c gii hnt ra i vi s lng cc phn t trong min)
Gii hn xem xt i s Boole vi 2 phn t ng nht.
i s Boole 2 phn t
Nm 1937, Claude Shannon hin thc i s Boole 2 phn tbng mch in vi cc chuyn mch (switch)
Chuyn mch l thit b c 2 v tr bn: tt (off) hay m (on)
2 v tr ny ph hp biu din cho 0 hay 1
i s Boole 2 phn t cn c gi l i s chuyn mch
Cc phn t ng nht c gi l cc hng chuyn mch (switchingconstant)
Cc bin (variable) biu din cc hng chuyn mch c gi l ccbin chuyn mch (switching variable) tn hiu
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
16
Bng s tht (Truth Table)
Phng tin m t s ph thuc ca ng xut vo mc lun l
(logic level) ti cc ng nhp ca mch
Lit k tt c cc t hp c th ca mc lun l ti cc ng nhp v
kt qu mc lun l tng ng ti ng xut ca mch
S t hp ca bng N-ng nhp: 2N
A B x
0 0 1
0 1 0
1 0 1
1 1 0
A B C x
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1?
A
Bx
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
17
Cc php ton chuyn mch
i s chuyn mch s dng
cc php ton trong lun l
mnh vi tn gi khc
Php ton AND
Php ton 2 ngi tng
ng vi php nhn lun l
Php ton OR
Php ton 2 ngi tng
ng vi php cng lun l
x y x y x + y x
0 0 0 0 1
0 1 0 1 1
1 0 0 1 0
1 1 1 1 0
Php ton NOT Php ton 1 ngi tng
ng vi php b lun l
Bng s tht ccphp chuyn mch
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
18
Cc php ton chuyn mch
Cc php ton chuyn mch c th c hin thc bi mch phncng
Bng s tht c th s dng nh 1 cng c dng xc minhquan h gia cc php ton chuyn mch
S dng bng s tht chng minh nh l De Morgan(x + y) = x y
x y x y x + y (x + y) x y
0 0 1 1 0 1 1
0 1 1 0 1 0 0
1 0 0 1 1 0 0
1 1 0 0 1 0 0
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
19
Biu thc (expression) chuyn mch
Biu thc chuyn mch l mt quan h hu hn cc
hng, bin, biu thc chuyn mch lin kt vi nhau
bi cc php ton AND, OR v NOT
V d
y + 1 , x x + x , z ( x + y )
E = ( x + y z ) ( x + y ) + ( x + y )
literal c s dng m ch bin hay b ca bin
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
20
Biu thc (expression) chuyn mch...
Mt biu thc c th c chuyn thnh nhiu dng tng ngbng cch s dng cc lut Boole
E = (x + y z) (x + y) + (x + y)
E1 = x x + x y + x y z + y y z + x y E3 =x + x y
E2 = x + x (y + y z) + x y E4 =x + y
Ti sao phi chuyn i dng ca cc biu thc ?
Cc thnh phn tha (redundant) trong biu thc literal lp ( x x hay x + x) bin v b ( x x hay x + x) hng (0 hay 1)
Khng hin thc cc thnh phn tha ca biu thc vo mch
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
21
Hm (function) chuyn mch
Hm chuyn mch (switching function) l mt php gn xc nh
v duy nht ca nhng gi tr 0 v 1 cho tt c cc t hp gi tr
ca cc bin thnh phn
Hm c xc nh bi danh sch cc tr hm ti mi t hp gi
tr ca bin (bng s tht)
Tn ti nhiu biu thc biu din cho 1 hm
S lng hm chuyn mch vi n bin l 2 lu tha 2n
x y x y x y E1 = x + x y E2 = x + y
0 0 1 1 1 1 1
0 1 1 0 0 0 0
1 0 0 1 0 1 1
1 1 0 0 0 1 1
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
22
nh l khai trin Shannon
f(x1, x2, , xn) = x1 . f(1, x2, , xn)
+ x1 . f(0, x2, , xn)
f(x1, x2, , xn) = ( x1 + f(0, x2, , xn) )
. ( x1 + f(1, x2, , xn) )
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
23
Cc php ton chuyn mch khc
Php ton NAND
Php ton 2 ngi tng
ng vi (NOT AND)
Php ton NOR
Php ton 2 ngi tng
ng vi (NOT OR)
Php ton Exclusive OR
E = x y = x y + x y
Php ton XNOR (Ex. NOR) E = ( x y ) = x y + x y
Bin NAND NOR Ex. OR XNOR
x y (x . y) (x + y) x y (x y)
0 0 1 1 0 1
0 1 1 0 1 0
1 0 1 0 1 0
1 1 0 0 0 1
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
24
Cng lun l
i s chuyn mch c th thc hin cc cng vic
trong i tht, cn phi c
Thit b vt l thc hin cc php ton chuyn mch
Tn hiu vt l (in p, ) thay th cho cc bin chuyn
mch
Cng (gate) hay cng lun l (logic gate) l tn chung
dng gi cc thit b vt l thc hin cc php ton
chuyn mch vi chnh xc (accuracy) v thi gian
tr (delay) chp nhn c
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
25
Cng lun l
Mi cng c biu din bi 1 biu tng (schematic
symbol) c trng cng vi 1 s chn (pin, terminal)
tng trng cho cc bin chuyn mch
Mt biu thc chuyn mch bt k lun c th c
hin thc trong i tht bng cch kt ni cc cng
lun l li vi nhau
Mch lun l (logic circuit) hay mch chuyn mch
(switching circuit)
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
26
Biu tng ca cc cng lun l
Cng AND
Cng OR
Cng NOT (cng o - inverter)
Cng NAND
Cng NOR
Cng XOR
Cng XNOR
Cc cng nhiu hn 2 ngnhp
x
yx . y
x
yx + y
x
y(x . y)
x x
x
yx y
x
y(x y)
x
y(x + y)
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
27
Dng tng ng
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
28
Nguyn tc Bubble Pushing
y bong bng i ngc (t output) hoc i ti (t ngnhp) thay i tnh cht cng t AND sang OR vngc li.
y bong bng t ng ra sang ng nhp, bong bngxut hin trn tt c ng nhp, v tnh cht cng thayi.
y bong bng trn tt c ng nhp tin v ng xut.Bong bng xut hin trn ng xut v tnh cht cngthay i.
AB
YAB
Y
AB
YAB
Y
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
29
Din dch biu tng cng lun l
Dng tng ng ca cng AND
Ng xut mc cao khi tt c cc ng nhp mc cao
Ng xut mc thp khi mt trong cc ng nhp mc
thp
Mt s cu trc ca cng XOR
E = x y = x y + x y = ( x y + x y )
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
30
Tch cc cao Tch cc thp
Hai trng thi hot ng ca thit b l tch cc(activity) v khng tch cc (inactivity) Xt cc th d i vi in thoi, n, ng c, v.v
Do thi quen, qui c tch cc ng vi lun l 1 cnkhng tch cc ng vi lun l 0
Tch cc cao (active high)tch cc lun l 1 mc in p cao H
Tch cc thp (active low)tch cc lun l 0 mc in p thp L
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
31
Cng OR 7432
Cng NOR 7402
Cng Ex-OR 7486
Mch tch hp
Cng NOT 7404
Cng AND 7408
Cng NAND 7400
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
32
Tp ph bin ca cc php ton
Mt tp cc php ton c gi l ph bin (universal) nu mi
hm chuyn mch u c th c biu din mt cch tng
minh ch bi cc php ton ca tp trn
i vi cc php ton chuyn mch xt, ta c mt s cc tp
ph bin sau
Tp { NOT , AND , OR }
Tp { NOT , AND }
Tp { NOT , OR }
Tp { NAND }
Tp { NOR }
Tp ...
Bt k hm chuyn mch no cng u c th c biu din mt cch
tng minh ch bi cc php ton NOT v AND
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
33
Tnh ph bin ca cng NAND
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
34
Tnh ph bin ca cng NOR
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
35
Xc nh gi tr ng xut mch lun l
S dng biu thc Boole cho ng xut ca mch lun l
Vi A = 0, B = 1, C = 1, D = 1
x = AB C ( A + D )
= 0. 1 . 1 . (0 + 1)
= 1 . 1 . 1 . 1 = 1 . 1 . 1 . 0 = 0
S dng trc tip s mch lun l m khng cn s dng
biu thc Boolean
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
36
Gin xung theo thi gian (Timing Waveform)
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
37
Tng kt
i s Boolean 5 nh Huntington Tnh cht i ngu 8 nh l c bn
i s chuyn mch Thu gn i s Boolean cho min hai phn t {0,1} Cc php ton chuyn mch nh l khai trin Shannon
Cng lun l. Biu din cng lun l. Din dch cng lun l. Cc IC c bn. Tp ph bin php ton. Biu din s dng sng (Timing Waveform)
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
38
Tt c bi tp trong sch Digital System ca Ronal
Tocci
Chng 3 - Logic Gates and Boolean Algebra
Bi tp v nh v c thm
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
39
Bi tp c bn
a. V gin xung cho tn hiu ng ra X ca cng OR
b. Gi s tn hiu A trong hnh trn b ni tt vi t GND (A = 0). V gin
xung cho tn hiu X ca cng OR.
c. Gi s tn hiu A trong hnh trn b ni tt ln ngun +5V VCC (A = 1).
V gin xung cho tn hiu X ca cng OR.
d. Vi cng OR 5 ng nhp, c bao nhiu t hp ng nhp cho php ng xut
mc cao (HIGH or 1)?
A
B
C
1234C
BA
x
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
40
Bi tp c bn
Vit biu thc i s Boole v bng s tht cho ng
xut ca cc mch di y.
12
3 12
3X
12
3
12A
B
C
12
A
B
C
Y
12
3
12
3
12
3
12
C
B
A 12
3 Z
121
2
3
12
3
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
41
Bi tp c bn
Trnh by nguyn l hot ng ca h thng bo ng
di y, bit ci bo ng c kch hot khi tn hiu
iu khin mc cao (HIGH or 1)
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
42
Bi tp c bn
V cc mch lun l tng ng vi cc biu thc i s
Boole sau
= + + +
= ( + )
n gin cc biu thc sau:
= + + + + + +
+ + + +
= + + + + + ( + )
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
43
Bi tp c bn
n gin cc biu thc Boolean sau
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
45
Bi tp c bn
Ti gin cc biu thc sau
a. xyz + xyz + xy
b. (wx)(w+y)(xyz)
c. x(y+wyz) + xy(wz+z)
d. (w+x)(w+x+yz)(w+y)
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
46
Bi tp c bn
Chng minh bng i s cc biu thc sau
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
47
Bi tp c bn
Tm b ca cc biu thc sau y
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
48
Bi tp c bn
a) Xy dng 1 cng NAND 2 ng nhp ch s dng cc
cng NOR 2 ng nhp.
b) Xy dng 1 cng NOR 2 ng nhp ch s dng cc cng
NAND 2 ng nhp.
c) Hin thc biu thc = ch s dng 1 cng NOR
2 ng nhp v 1 cng NAND 2 ng nhp.
d) Hin thc biu thc = ch s dng cc cng
NAND 2 ng nhp.
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
49
Bi tp c bn
a. Bin i cc mch sau y ch s dng cng NAND
b. Bin i mch sau y ch s dng cng NOR
12
12
3
12
3
12
3
A
B
12
X
12
3 12
3X
12
3
12A
B
C
12
B
A
C
X1
2
3
12
3
12
12
3
A
B
C
Y
12
3
12
3
12
3
12
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
50
Bi tp c bn
V k hiu cng lun l thch hp cho cc pht biu sau
y:
Ng xut ch mc cao (HIGH or 1) khi c 3 ng nhp u
mc thp (LOW or 0).
Ng xut ch mc thp khi bt k ng nhp no trong 4
ng nhp mc thp.
Ng xut ch mc thp khi tt c 5 ng nhp u mc
cao.
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
51
Bi tp c bn
Cho s sau
Gi s ci bo ng c kch hot khi tn hiu iu khin
Z mc cao (HIGH or 1). Xc nh cc t hp ng nhp
tch cc h thng bo ng.
Gi s ci bo ng c kch hot khi tn hiu iu khin
Z mc thp (LOW or 0). Hy thay i s mch trn
phn nh r c ch hot ng ca h thng. T xc
nh cc t hp ng nhp tch cc h thng bo ng.
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
52
Bi tp c bn
Xc nh cc t hp ng nhp n LED sng
A
B
C
D
E
NOR
NAND
OR
R1
LED
+5V
NOT
NOT
NOT
AND
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
53
Bi tp m rng
Cho A.B = 0 v A + B = 1, chng minh ng thc sau:
A C + A B + B C = B + C
Cho hm F(A, B, C) c s logic nh hnh v.
a. Xc nh biu thc ca hm F(A, B, C)
b. Chng minh F c th thc hin ch bng 1 cng logic duy
nht.
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
54
Bi tp m rng
Chng minh cc ng thc sau bng i s boole.a. + + = + + +
b. + + = + + +
c. + + = + +
d. = B
e. () =
C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l
55
Bi tp m rng
Mt my bay phn lc p dng h thng kim sot tc quay
(rpm), p lc (pressure) v nhit (temperature) ca ng c s
dng cc sensor, vi chc nng nh sau:
RPM sensor xut 0 ch khi tc < 4800 rpm
P sensor xut 0 ch khi p sut < 220 psi
T sensor xut 0 ch khi nhit < 200F
Hnh bn di m t s hot ng ca n cnh bo. n cnh bo ch
sng khi W mc HIGH (=1)
a. Xc nh iu kin n cnh bo sng.
b. Thit k li mch ch s dng NAND.