Logicki Kola i Digitalna Logika

8/17/2019 Logicki Kola i Digitalna Logika

1/63

66..

:

, ;

, ;

; ;

- , - ;

;

;

.

A

C

B

A

Y

A B C

( ) ( )15,5,4,3,011,10,8,2,1 += xm

xm F

AB

AB

A + B

(AB+AB)

A

B

D

14

15

13

2

3

1

6

7

5

A

10

11

9

C

12

B

0 4 8

11

11

1

1

1

X

X

X

( ) ( ) ( )C A B A BC A ++=+

11 =+

=+


2/63

172


3/63

173

6.1.

., , (). , .

XIX -

. . - , . , .. , . - . , - , . , .

6.2.

B . , „1” ( ) „0” ( ), B={1,0}. , 1 0,

. : A,B,C,D,E,… X0,X1, X2, X3, …, , .. , , Y,Y0, Y1, Y2, …, F, F1, F6.

B „+” „·”, , :

.1. , .. A,B,C {B} :

ABBA +=+ , ABBA =

( ) ( ) ( )ACABCBA +=+ , ( ) ( ) ( )CABABCA ++=+


4/63

174

.2. 1 0, 0 A+0=A, 1 A·1=A.

.3. A A + A =1, 0AA =⋅ .

(). , „+”, „·”. , „+” “·” 1 0, „+” „·”, .

() ( : (+) (·) :

( )). (. OR)

, „+”, „ “. (. AND) , „·”, „ “, „&”. . (. NOT) ,

„ “, „“ „~”.

. 6-1, . 6-2 . 6-3.

(+) (·) ( )

0 + 0 = 0 0 · 0 = 0 10 =

0 + 1 = 1 0 · 1 = 0 01=

1 + 0 = 1 1 · 0 = 0

1 + 1 = 1 1 · 1 = 1

. 6-1. . 6-2. . 6-3.

.

0, 1, :

1. , 0 0, (.6-1) 1. 1, 1;

2. , 1, 1, 0, .. 0 0

3. 0, 1, , 1, 0.

- ( ), ( ), ( ). .


5/63

175

, : , (. NAND) - (, ) , (. NOR) (, ). : , .. , (. XOR)

“⊕ “, , .. , (. XNOR) ( ).

. 6-4,. 6-5, . 6-6 .6-7.

)(+ )(⋅ )(⊕ )(⊕

100 =+ 100 =⋅ 000 =⊕ 100 =⊕

010 =+ 110 =⋅ 110 =⊕ 010 =⊕

001 =+ 101 =⋅ 101 =⊕ 001 =⊕

011 =+ 011 =⋅ 011 =⊕ 111 =⊕

. 6-4. . 6-5. . 6-6. . 6-7.

.

:

1. 1 0, 0,

2. 0 1, 1;

3. 0 ,.. 0, 0. , 1;

4. . , ., , 1, 0.

6.3.

, . , . , . . , 0 1. 1, .. = 1, A = 0, : = 0, A = 1.

, . :

A = A (. 6-1)

A0A =+ A1A =⋅ (. 6-2)


6/63

176

11A =+ 00A =⋅ (. 6-3)

AAA =+ AAA =⋅ (. 6-4)

AAA =+ AAA =⋅ (. 6-5)

1AA =+ 0AA =⋅ (. 6-6), ,

:

( ) ( ) CBACBA ++=++ ( ) ( ) CBACBA ⋅⋅=⋅⋅ (. 6-7)

ABBA +=+ ABBA ⋅=⋅ (. 6-8)

( ) CABACBA +=+ ( ) ( )CABACBA +⋅+=⋅+ (. 6-9)

,

, . , , . , , , .

(1) . .

A B C (A+B)·(A+C) (A+B·C)

0 0 0 0 0

0 0 1 0 0

0 1 0 0 0

0 1 1 1 1

1 0 0 1 1

1 0 1 1 1

1 1 0 1 11 1 1 1 1

. 6-8. .

, 23 = 8 . : (A+B) (A+C), :(A+BC), . ,

. 6-8 (A+B)(A+C) (A+BC) A, B C, .


7/63

177

(2) (), , :

( )( )

( ) ( ) BCABCC1ABCACABCACB1A

BCACABABCABACABCABACAACABA

+=++=++=+++=

=+++=+++=+++=++.

- . :

...CBA +++ ...,CBA ⋅⋅= ...CBA ⋅⋅ ...CBA +++= (. 6-10)

:

1. ,

2. .

, :

AABA =+ , ( ) ABAA =+ (. 6-11)

BABAA +=+ , ( ) ABBAA =+ (. 6-12)

ABAAB =+ , ( )( ) ABABA =++ (. 6-13)

( )( )BACACAAB +=+ ,

( )( )BAACCABA +=++ (. 6-14)

CAABBCCAAB +=++ , ( )( )( ) ( )( )CABACBCABA ++=+++ (. 6-15)

CABCBBCAB +=++ , ( )( )( ) ( )CBACBCBBA +=+++ (. 6-16)

(. 6-11), (. 6-12) (. 6-13) .

( ):

( ) ( )[ ] ( )[ ],...C,B,0YA,...C,B,1YA,...C,B,AY ⋅+⋅= (. 6-17)

, . :

. 1.( ) ( ) ( )

( ) ACABCBA

CBBACBCCBACBCBBCACBACABABC

+=+=

=+=++=++=++

. 2.( )( )( ) ( )( ) ( )( )

( )[ ]( ) ( )( ) ( )( ) ACCAB1ACABAACABAB1ACAABBAACABBAB _ BAAACABABA

=++=++=+++=

=+++=+++=+++

. 3. ( ) ( ) ( )[ ] ( )[ ]( ) ( )[ ] ( ) ( ) ( )CBA000ABCBCBAC0B0CB0B0A

C1B1CB1)B1(AACABCBABAC,B,AY

+=++++++=+++++

+++++=++++=


8/63

178

6.4.

, , () . - : - .. ( ), ( ) (, ) ( -). . , .

6.4.1.

. . , . .

n , n , .

i ABC Y

0 000

i AB Y 1 001

0 00 2 010

1 01 3 011

2 10 4 100

3 11 5 1016 110

7 111

. 6-9. .

. 6-10. .

.

N=2 n , N=2 n .

“i” , 0, (N-1), ..(2

n-1). 2, 3 4 ,

. 6-9, . 6-10 . 6-11.


9/63

179

i ABCD Y

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

10 101011 1011

12 1100

13 1101

14 1110

15 1111

. 6-11. .

N F n :

n2

F 2 N = (6-18)

6.4.2.

e . , , . ,

„=“ ( ), . , , , . (, ) . , , (). ( ) ().

, .. (Σ)

(p) ( p). (m) ( , ), , .


10/63

180

, (), ( m). , , (Π) , .. (s) ( s). () ( , ), ,

. , (), ( ).

, :

( ) ( ) ( ).X,X,X,XFF,H,G,DZZ,C,B,AYY 4321=== (m), (M).

: CBACBAABCY ++= : CBCBABCACBAY +++=

: ( )( )( )CABABAY +++= : ( )( )CBACBAY ++++=

: ( )( )HGHGDZ +++= : DGHHGDGHDZ +++= : 432143214321 XXXXXXXXXXXX ++

: 43214321 XXXXXXXX ++++++

, . , , (-). : , (-). , .

, . , ., (, ), , : . .

6.4.2.1.

.. . , (, ) : „m”, „”, i 1j , i 0k . i 1j 1, i 0k 0, j+k=N , N=2 n, 0 N-1: [0,1,2…,2n-1]. : j1i1211i m...mmY +++= ,

: k 0i02i01i M...MMY = .


11/63

181

. ( ) j11211 i,...,i,imY = f (1)=(i 11, i 12 , …, i 1j ) ( )k 00101 i,...,i,iMY ∏= f

(0)=(i 10 , i 10 , …, i 1k ).

,

1, 0.

. 6-12, , .

i ABC Y m i M i

0 0 0 0 m0 CBA ⋅⋅ M0 CBA ++

1 0 0 1 m1 CBA ⋅⋅ M1 CBA ++

2 0 1 0 m2 CBA ⋅⋅ M2 CBA ++

3 0 1 1 m3 CBA ⋅⋅ M3 CBA ++

4 1 0 0 m4 CBA ⋅⋅ M4 CBA ++

5 1 0 1 m5 CBA ⋅⋅ M5 CBA ++

6 1 1 0 m6 CBA ⋅⋅ M6 CBA ++

7 1 1 1 m7 CBA ⋅⋅ M7 CBA ++

. 6-12. .

, , .. i = (0, 1, 2, …, 2 n-1) :

ii mM = (6-19)

. 6-13 A, B, C. : Y=Y(A,B,C), Z=Z(A,B,C), W=W(A,B,C) .

i AB C Y Z W

0 00 0 1 0 1

1 00 1 0 0 1

2 01 0 1 0 0

3 01 1 0 0 1

4 10 0 0 1 0

5 10 1 1 1 0

6 11 0 0 0 1

7 11 1 1 1 1

. 6-13. Y, Z W .


12/63

182

: ( ) ( )( )( )( )CBACBACBACBA6,4,3,1MY ++++++++==∏

: ( ) ( )( ) ( ) ( )ABCCBACBACBA7,5,2,0mY ++== : ( ) ( ) ( ) ( ) ( ) ( )ABCCABBCACBACBA7,6,3,1,0mW ++++==

: ( ) ( )( )( )( )( )CBACBACBACBACBA6,3,2,1,0MZ ++++++++++==∏

6.4.2.2.

, , 0 1.

i ABC D F

0 000 0 x

1 000 1 1

2 001 0 1

3 001 1 x

4 010 0 x

5 010 1 x

6 011 0 0

7 011 1 0

8 100 0 1

9 100 1 0

10 101 0 1

11 101 1 1

12 110 0 0

13 110 1 0

14 111 0 0

15 111 1 x

, ( ) .

. , . , .

. “”(. “don’t care”)

, . „/”, „\”, „-”, „b”, „x” „X”. „x”.

F=F(D,C,B,A) . 6-14.

. 6-14. F(D,C,B,A).

:

( ) ( )15,5,4,3,0x11,10,8,2,1mFxm

+= ( ) ( )15,5,4,3,0x14,13,12,9,7,6MFxM

∏∏= .

6.4.3.

, .

, : () (). ,


13/63

183

1. (, ) 1, 0, . . 6-15 ) )

Y(A,B,C), Z(A,B,C) :( ) ( ) ( ) ( )ABCBCACBA7,3,0mY ++==

( ) ( ) ( ) ( ) ( )CBABCACBACBA4,3,2,1mZ +++==

i ABC Y i ABC Z

0 000 1 0 000 0

1 001 0 1 001 1

2 010 0 2 010 1

3 011 1 3 011 1

4 100 0 4 100 1

5 101 0 5 101 0

6 110 0 6 110 0

7 111 1 7 111 0

) Y(A,B,C). ) Z(A,B,C).

. 6-15. .

0 . 1, 0 . Y Z . 6-15:

( ) ( )( )( )( )( )CBACBACBACBACBA6,5,4,2,1MY ++++++++++==∏

( ) ( )( )( )( )CBACBACBACBA7,6,5,0MZ ++++++++==∏

, .

, , . , , . , . .

. W A,B,C:BAABCCBAW ++= .

, ABC=0 ABC=111.


14/63

184

. 6-16.

i ABC W

0 000 0 =0, B=0, C=0 W=010+000+01=0;

1 001 0 =0, B=0, C=1 W=011+001+01=0;2 010 0 =0, B=1, C=0 W=000+010+00=0;

3 011 0 =0, B=1,C=1 W=001+011+00=0;

4 100 1 =1, B=0, C=0 W=110+100+11=1;

5 101 1 =1, B=0, C=1 W=111+101+11=1;

6 110 0 =1, B=1, C=0 W=100+110+10=0;

7 111 1 =1, B=1, C=1 W=101+111+10=1.

. 6-16. a a ja W(A,B,C).

. () .

. : ( )C,B,AZZ = : CBAY += ,

( )BCBAZ ++= . , .

. 1. ( ) ( ) ( )( )( ) CBABCACBAABCCBACBACBBCAACBACBA

CBBC1CBACBACBB1CCBAC111BACBAY

+++++=++++=

=+++=+++=+=+=

. 2.

( ) ( )( ) ( )( ) ( )( )( )[ ] ( ) ( )( )[ ] ( )( )( )[ ]( )( )[ ]{ } ( )( )( )

( )( )CBACBACBACBACBACBABACBABA

CBACCBABACBA0BABA

CBA0BAACBA0B0CBABCBAZ

++++

++++++=++++++

++=+++++=+++

++=++++=++++=++=

, , . U(X,Y,Z) V=V(X,Y,Z) .

. 3.( ) ( ) ( )

( ) ( )YXZYXYXYZYXZYXYZYXZYXXXYZZZYXZYXYZXXYZZYXZYXZ,Y,XU

+=++=++=

=++++=++++=

. 4.( ) ( )( )( )( ) ( )

( )[ ] ( )( )ZYZXXXZYYYZXZYXZYXZYXZYXZ,Y,XV

++=++

++=++++++++=

e

(), . .


15/63

185

. , , . , , . ,

, . , , . ,.. , . .

. 1.

( ) ( )( )( ) ( )( )

( ) ( )7,6,5,4,2,1m1,2,6,4,5,6,7mCBACBAABCCBACBA

CABABCCBACBACABCCBAABCBACBAA

CCBBACBACB111ACBACBAC,B,AF1

==+++++++=+++++=+++

+++=++=++=

. 2. ( ) ( ) ( )( ) ( )( )CBCBACCCBCBBBA

CBCBACBACBA3,0MC,B,AF2

++=++++=

=+++=++++==∏

. 3.

( ) ( )( ) ( )( )( )( ) ( )( ) ( )( )[ ]( )

( )( ) ( )( )( )( )( )( ) ( )6,3,2,1,0M1,6,2,3,0M

CBACBACBACBACBACBACBA

CBACCBABACBACBAACCBBA

CBACB000ACBACBAC,B,AF3

∏∏ ==

=++++++++++=++++

+++++=+++++++=

=++++++=+++=

. 4. ( ) ( ) ABCBAABC1BAABCCBACBA7,5,4mC,B,AF4 +=+=++==

6.5.

, (, ), , () (), . . 6-17 ), ), ), ), ), ) ), .

, .

A B BA + A B BA ⋅

0 0 0 0 0 0 A A

0 1 1 0 1 0 0 1

1 0 1 1 0 0 1 0

1 1 1 1 1 1) ) ) ()

. 6-17. .


16/63

186

A B BA + A B BA ⋅ A B BA ⊕ A B BA ⊕

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

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

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

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

) ) ) )

. 6-17. .

, 1 A B , .. A=0 B=1 A=1 B=0, A B 0. 1 A B , .. A=0 B=0 A=1 B=1, , 0. , . , ().

Y = BABABA ⋅+⋅=⊕ (6-20)

Y = BABABA ⋅+⋅=⊕ (6-21)

.

, , , . , :

( )AAAAA ⋅=+= (6-22)

BAAB ⋅= (6-23)

( )BABABA ⋅=+=+ (6-24)

:

( )AAAAA +=⋅= (6-25)

( )BABAAB +=⋅= (6-26)BABA +=+ (6-27)


17/63

187

, , . , .. . , .. . .

, : , , , . .

6-1. ?

6-6. ?

6-3. ?

6-4. .

6-5. ?

6-6. : , , (), , , .

6-7. (. 6-12), (. 6-14) (. 6-15) () ; () ; () .

6-8. : () CBBCABA1 +++ ; ()

CDBDBA0 +++ ; () ( )CBBC1 + ; () ( )CACABBCBABCA0 ++++ .

6-9. : () ( )( )CBACBA +++ ; ()

( )( )BACBA +++ ; () ( ) ( )CABBA + ; () ( ) ( ) CCABBA ++ .6-10. .

6-11. Y n . ? ? ? ?

6-12. .

6-13. ? ?

6-14. ? ?

6-15. ?

6-16. ? ?6-17. ?

6-18. ?


18/63

188

i ABC F 1 F 2 F 3

0 000 0 1 0

1 001 1 1 1

2 010 0 1 x3 011 1 x 1

4 100 0 0 x

5 101 1 x x

6 110 1 0 1

7 111 0 0 0

6-19. F 1(A,B,C), F 2 (A,B,C) F 3(A,B,C) . .

F1, F2, F3 6-19.

6-20. , , , ,

: () CBAABC)C,B,A(F1 += ; () ( ) BCCBACBAC,B,AF2 ++= ; ()

( ) ( )( )( )CBACBACBAC,B,AY ++++++= ;() ( ) ( )( )( )CBBACAC,B,AZ +++= .

6-21. ()

( ) 213213211 XXXXXX,X,XF += , () ( ) ( )( )323213212 XXXXXX,X,XF +++= , ()( ) CAABC,B,AY += , () ( ) ( )( )BACAC,B,AZ ++= (1)

; (2) ; (3) .

6-22. : ()

( ) CBACBAC,B,AY += ; () ( ) ( )( )( )CBACBACBAC,B,AZ ++++++= .

6-23. : ()( ) DBACBD,C,B,AY += ; () ( ) CDBDAD,C,B,AY += .

6-24. : ()

( ) ( )( )DBADBD,C,B,AZ +++= ; () ( ) ( )( )DCDBAD,C,B,AZ +++= .

6-25. ?

6-26. .

6-27. , () () .

6.6.

() . , (, ) . . , , .

,

. , .


19/63

189

, , . , é .

, .. . , .

, (Quine McCluskey), . ,

.

6.6.1.

( ) . , . ,

.

. 1.

( )

( )( ) ( )

( ) ( ) ( )CBACBACAABCABACAD1ABCADDBADABCABACADBADBADABCBCA

DBACDCBACADABABCCADBADCBA

CADABABCCADBAABCDCBAD,C,B,AF1

+++=+++=++++

++=+++++=+++

+++=+++++=

=+++++++++=

. 2.( ) ( ) ( ) ( )

CAAB

B1CAC1ABBCAACAABBCCAABC,B,AF2

+=

=+++=+++=++=

F 2 (A,B,C,D) :

. 3.( ) ( ) ( )

( ) ( ) ( ) ( ) CAABB1CAC1ABBCC0ABC0BA

BCC0B0ABCC1B1ABCCAABC,B,AF2

+=+++=+++++=

=+++++=++=

(. 6-15).

6.6.2.

, . , ().


20/63

190

, . ().

, . , . : .

, , Y(A,B) . 6-1.

0

2

3

1 10

11 0

1

0 020

1 31

10

0

Y A

B

BAi

. 6-1.

, . 22 = 4 . , , . .

. :

1. 1 1 . , ..

2. , .. 0 0.

, 6-18 , . 6-2 ) , 1. 6-2 ) , 0.


21/63

191

i A B Y

0 0 0 1

1 0 1 12 1 0 1

3 1 1 0

0

1

0 2

33

2

1

A

B

1

1 0

A

B1

. 6-18. .

) . ) .

. 6-2. .

23 = 8 , . 6-3, .6-4 24 =16 .

C(C)

B(B)

7 5

4620

1 3

A(A)

0

1

00 01 11 10AB

C

B(B)CD

AB

C(C)

A(A)

0 4 12 8

913

1115

51

3 7

1462 10

D(D)

00 01 11 10

00

01

10

11

. 6-3.

.. 6-4.

.

, . ,

. , , , . , , . , , . .


22/63

192

C

A

D

B

1

11

1

1

1

10

1 5

4 12 8

913

15 11

1014

3 7

62

, , ,

, , . , . () .

Y=Y(A,B,C,D) . 6-5.

. 6-5. Y(A,B,C,D) = Σ m(1, 2, 3, 8, 12).

. 6-5 ,DCBAm8 = DCABm12 =

. B , , .. . : ( ) DC A B B DC A DC AB DC B A =+=+ .

, , , B. 1, ( ).

, . , .

, .

, , ( ) . , . ,

, . ( ) , .


23/63

193

. N=2

n ,

n , , ( ) . . 6-6 ), ), )

) .

C

A

D

B

1 1

1 1

1

1 1

1

0 4 12 8

1 5 13 9

3 7 15 11

2 6 14 10

1

1

1

1

11

1 1

X

2

1

4

3X

X

X

81240

1 5 13 9

11157

14 106

3

2

) DBDBF1 += . . 6-6. . ) 4242 XXXXZ += .

. 6-6 ) F 1= F 1 (A,B,C,D), . 6-6 ) Z=Z(X 1, X 2 , X 3, X 4 ), . , . 6-7 ) Y=Y(X 3, X 2 , X 1, X 0 ), . 6-7 ) F 2 = F 2 (D, C, B, A), .

X 2

X3

X1

X01

1

1

1

11

11

81240

1 5 13 9

11157

14 106

3

2

C

B

D

A

1 1 1 1

1111

81240

1 5 13 9

11157

14 106

3

2

) 3232 XXXXY += . ) BAABF2 += .

. 6-7. .

. 6-8 a) ) W=W(X 4, X 3, X 2 , X 1 ) F 3= F 3(A, B, C, D) . .


24/63

194

X3

X

2X

1

X4

1 1 1 1

1 1 1 1

81240

1 5 13 9

11157

14 106

3

2

B

C

A

D

1

1

1

1

1

1

1

181240

1 5 13 9

11157

14 106

3

2

) 1XW = . ) AF3 = .

. 6-8. .

, . , 5 25 = 32 , é . , 5 . , 16 , 16 , , .

6 26 = 64 . .

6.6.2.1.

. : .

. , , .. 1.

. . , . () (r ). a r 2 r (n-r ) r nk ≤ , a n - . .


25/63

195

:

1. , .. .

;2.

, .

. , , , . () , .. . , , .

.

D

A

C

B

1 1

1

1

1

81240

1 5 13 9

11157

14 106

3

2

. 6-9. Y 1.

Y 1=Y 1(A,B,C,D) . 6-9.

p1=m2+m3, p2=m8+m12, p3=m2+m10, p4=m8+m10. : Y 1=p1+p2+p3 Y 1=p1+p2+p4. p1, m3 . p1 () ().

( 1), . p2 , m16.. p3, p4 -.

-, - 1 . , , .

C

A

D

B

1 1 1 1

1 1 1 1

1

181240

1 5 13 9

11157

14 106

3

2

X1

2X

3X

X4

1

1

1

1 1

1

11

81240

1 5 13 9

11157

14 106

3

2

. 6-10. Y 2 = Y 2 (A, B, C, D). . 6-11. j Y 3= Y 3(X 1, X 2 , X 3, X 4 ).


26/63

196

. 6-10 Y 2 =Y(A,B,C,D), , , . . 6-11 j Y 3= Y 3(X 1,X 2 ,X 3,X 4 ), -, -.

. , Z 1= Z 1(A, B, C, D) . 6-12, Z 2 = Z 2 (X 1, X 2 , X 3, X 4 ) . 6-13.

B

C

A

D11

11 1

1

1

1

81240

1 5 13 9

11157

14 106

3

2

X1

2

4X

X

3X

1

1

1

1 1

1

1

10 4 12 8

91351

3 7 15 11

101462

. 6-12. Z 1= Z 1(A, B, C, D) . 6-13. Z 2 = Z 2 (X 1, X 2 , X 3, X 4 )

. , . .

, , :

1. ;

6. , ;

3. , ;

4. 8 ;

5. 4 , , .

, 1- . 1- 1- , . . 1-


27/63

197

. 1- , . 1-. 1- 1-, 1-. 1-

. 1- 1- 1- , .. . 1- 1- 1- .

, , , 1- .

F 1= F 1 (A, B, C, D) .6-14 ), ), )

F 2 = F 2 (A, B, C, D) . 6-15 ), ), ). , ,.e. .

C

D

B

A

1 1 1 1

11

1

1

1

10 4 12 8

91351

3 7 15 11

101462

)

1

11 1D

A

C

B

1 1 1 1

110 4 12 8

91351

3 7 15 11

101462

)

D

B

C

A

1 1 11

1 11

1 1

1

0 4 12 8

91351

3 7 15 11

101462

)

. 6-14. F 1= F 1(A, B, C, D).

F 1 : F 1= D B DC ADC B AC AB D BC A +++++ .

0

1

3

2

D

6

7

5

14

A

10

15

13

11

9

4

B12 8

C

1 1

1 1 1

1

1 1 1

1

)

D

A

111240

1

12 6

3

1

7

5

11

14

13

15

1 1

B

18

C10

1

11

9

)

D

A

111240

1

12 6

3

1

7

5

11

14

13

15

1 1

B

18

C10

1

11

9

)

. 6-15. F 2 = F 2 (A, B, C, D).

F 2 :F 2 = CBADCACBABD +++ .


28/63

198

. o , 0-, 1-. 1,

0-. , , .

. , , . () . () ( 0), . , Z=Z(A, B, C, D) . 6-16 ) ).

0

0

0

00

2

D0

3

1

06 14

0

07

5

15

13

0 4

B12

10

A

11

9

0C

8

)

0D

00

62

0

03

1

07

5

0

1014

0

A

0

015

13

11

9

0 4

B12 8

C

)

. 6-16. Z=Z(A, B, C, D).

Z :

Z= )DC)(DB)(CB)(DCA( +++++ .

6.6.2.2.

. 0 1, . , . , .

, , Y=Y(A,B,C,D), :

( ) ( )15,14,13,12,11,10m9,6,5,2,1mYxm

+= .


29/63

199

Y=1 : m1, m2 , m5 , m2 ,m9, : m10 , m11, m12 , m13, m14, m15 . „ x“ .

. 6-17. „ x” - : 1, , 0, .

D

14

15

13

2

3

1

6

7

5

A

10

11

9

C

12

B0 4 8

1 1

11

x

x

x x

xx

D

14

15

13

2

3

1

6

7

5

A

10

11

9

C

12

B0 4 8

11

11

xx

x x

x xx x

x x

D

14

15

13

2

3

1

6

7

5

A

10

11

9

C

12

B0 4 8

x1 1

x

x

x

11 1

x

x

. 6-17. Y. . 6-18. “x”. . 6-19. “x”.

. 6-18 e „ x” . :

( ) ( ) ( ) DCADCBDCAmmmmmmY 62911 ++=+++++= .

. 6-19 „ x” - m13, m14, m10

1- :

( ) ( ) DCDCmmmmmmmmY 14106213951 +=+++++++= .

e „ x” - m11, m12 , m13 , , „ x” - 0. , . 6-20 ), ) 6-21 ), ), ) ) .

D

14

15

13

2

3

1

6

7

5

A

10

11

9

C

12

B0 4 8

11

11

1

1

1

X

X

X

D

14

15

13

2

3

1

6

7

5

A

10

11

9

C

12

B0 4 8

1X X

X

X

1

1

1

1

X

) C A DC C AY ++=1 . 6-20. ) CBBADBY2 ++=


30/63

200

D

14

15

13

2

3

1

6

7

5

A

10

11

9

C12

B

0 4 8

XX

X X

0 00

00

14

15

13

2

3

1

6

7

5

10

11

9

120 4 8

X X

X

0

0

0

00

X1

2

4X

X

3X

) )ABD)(AC)(AC(Y3 ++++= ) )XXX)(XXX)(XX(Y 431321314 +++++=

A

D

2

3

6

7

1

0

5

4

B

15

14

12

13

10

11

C

9

8

X

X

X 0

0

00

00

0

0

D

14

15

13

2

3

1

6

7

5

A

10

11

9

C

12

B0 4 8

1

1 1

1

11

1

1

1

) )BC)(AC)(BC(Y5 +++= ) DBACBABCBDY6 +++=

. 6-21. .

6.6.2.3. M

. , , : (0-) (7-) , 1- 6- , 6- 5- . , (0-)

(, - ). . 6-22 Y(X 1,X 2 ,X 3,X 4,X 5 )=m(0,3,4,6,7,11,12,14,15,27,28, 30,31).


31/63

201

: Y = 2X 4X 5X + 1X 3X 4X +

1X 4X 5X + 2X 3X 5X + 1X 2X 4X 5X .

14 10

11

913

12 84

5

7

62

3

1

0

15

24 28 20 16

17212925

27 31 19

18223026

23

1

1

1

11

11

1

1

1

1 1

1

X

XX

X

X

1

23

4

5

3X

. 6-22. Y=m(0,3,4,6,7,11,12,14,15,27,28,30,31) .

. 5- 0, 1. , . ,

.

, . , , . (ESPRESSO), (BDD) - .

6-28. ?

6-29. ?

6-30. () :

() ( ) ( )( )( )( )CBCBCBBAC,B,AY ++++= ; () ( ) CBBCCAABC,B,AZ +++= .

6-31. () ; ()

: () ( ) DCBADABCDBCADCBABCDAD,C,B,AF1 ++++= ; ()( ) ( )( )( )( )( )DCBADCBADCBADCBADCBAD,C,B,AF2 +++++++++++++++=


32/63

202

6-32. :

() ( ) ( )0,1,2,4,5MC,B,AY ∏=

() ( ) ( )0,1,4,7MC,B,AY ∏=

() ( ) ( )14,157,8,12,13,0,1,2,4,5,MD,C,B,AY ∏=

() ( ) ( )∏= ,158,10,11,140,2,4,5,6,MD,C,B,AY

() ( ) ( ),148,10,11,120,1,2,3,7,MD,C,B,AY ∏=

() ( ) ( )0,2,6,7mC,B,AY = () ( ) ( )0,1,3,5,6mC,B,AY ( ) ( ) ( ),158,11,12,130,3,4,6,7,mD,C,B,AY = () ( ) ( ),159,10,12,134,5,6,7,8,mD,C,B,AY

=

() ( ) ( ) 157,9,11,13,0,2,4,5,6,mD,C,B,AY = () ( ) ( )158,9,10,11,0,2,3,4,6,mD,C,B,AY =

6-33. :

() ( ) CABAC,B,AF +=

() ( ) CBADABCD,C,B,AF +=

() ( ) CBBDAD,C,B,AF +=

() ( ) CDADBCD,C,B,AF +=

() ( ) ( )( )DCADCBAD,C,B,AF +++++=

() ( ) ( )( )DADCBD,C,B,AF +++=

() ( ) ( )( )DBADCBD,C,B,AF ++++=

6-34. :

() ( ) ( ) ( )6,4,1M7,5,0MC,B,AYxM

∏∏=

() ( ) ( ) ( )15,10,7M14,12,8,6,5,4,3,1MD,C,B,AYxM

∏∏=

() ( ) ( ) ( )15,5,4,3,0M11,10,8,2,1MD,C,B,AYxM

∏∏=

() ( ) ( ) ( )14,10,8M15,11,6,5,4,2,0MD,C,B,AYxM

∏∏=

() ( ) ( ) ( )6,5m7,3,2mC,B,AYxm

+=

() ( ) ( ) ( )10,2,1m15,14,12,8,6,5,4,0mD,C,B,AYxm

+=

() ( ) ( ) ( )15,13,10m12,11,9,5,4,3,2,1mD,C,B,AYxm

+=


33/63

203

6.6.3. -

- (. Quine-McQluckey) a () pX i + p iX = p p + pX i = p X i = X i + X i + .... Y n Y = f(X 1, X 2 , ..., X i , ... X n ), p ( ).

, .

6.6.3.1.

- :

1. (-) ;

2. ();

3. ();

3.1. (-);

3.2. () ;

3.3. () ;

4. ().

- .

.. ( ) ( ) 1, .. 1- , - .

. - é . :

− () - ,−

.. () , .. , () ;

− .. () , .. - , () .

- -.

, - . , , , .


34/63

204

é - - . - () 1- (, ) .

6.6.3.2. -

- . , . , : Y 1= m(0,2,5,6,7,8,10, 12,13,14,15),

Y 2 = m(0,2,3,4,5,6,7,8,9,10,11,12,13) Y 3=Σ m(2,3,7,9,11,13)+Σ x(1,10,15).

1: (-)

0. , :

− , .. 2 n X 1, X 2 , X 3, . . , X n 0 é (2

n – 1) n ,

− n- (),

− Y ( ) 1, 0 (, .. /).

.

. 6-19 0

Y ABC X 1 X 2 X 3 .

0 , ( ) ( ).

0 1 :

0 1,

n (

) 1,

.

i X 1 X 2 X 3

ABCY

0 0 0 0

1 0 0 1

2 0 1 0

3 0 1 1

4 1 0 0

5 1 0 1

6 1 1 0

7 1 1 1

T. 6-19. (0)

.


35/63

205

, , 0, “ “. 0 1. , 1 1- . ,

1, 1-. . 1-, 1-, 1-, . é 1-.

- 1 , , , 1- . , , .

1

0 1- , 1 , .. () .

1, , 2. 1, : 1, 0, , . ( ) 2

”−” (), . , 1 “ “. , , , 2.

2 1 , . ( .)

. Y 1(A,B,C,D)=[f (1) ]= m(0,2,5,6,7,8,10,12,13,14,15) 0, 1 2

. 6-20 ), ) ).

3 2 . , 2 3 , 3 , 2 .

(0,2) (8,10) (0,8) (2,10) 2 (0,2,8,10) (0,8,2,10) 3. 3

(0,2,8,10), 2 : (0,2), (8,10) (0,8), (2,10). , Y 1 - 3, 6.


36/63

206

i BCDY

1 i ABCD Y 1 i ABCD Y 1 i ABCD Y 1

0 0000 1 0 0000 (0,2) 00-0 (0,2,8,10) -0-0 p1

1 0001 0 2 0010 (0,8) -000 (2,6,10,14) --10 p2

2 0010 1 8 1000 (2,6) 0-10 (8,10,12,14) 1--0 p3

3 0011 0 5 0101 (2,10) -010 (5,7,13,15) -1-1 p4

4 0100 0 6 0110 (8,10) 10-0 (6,7,14,15) -11- p5

5 0101 1 10 1010 (8,12) 1-00 (12,13,14,15) 11-- p6

6 0110 1 12 1100 (5,7) 01-1

7 0111 1 7 0111 (5,13) -101

8 1000 1 13 1101 (6,7) 011-

9 1001 0 14 1110 (6,14) -110

10 1010 1 15 1111 (10,14) 1-10

11 1011 0 (12,13) 110-

12 1100 1 (12,14) 11-0

13 1101 1 (7,15) -111

14 1110 1 (13,15) 11-1

15 1111 1 (14,15) 111-

-:

p1 = B D

p2 = C D p3 = A D

p4 = B D

p5 = B C

p6 = A B

) 0 ) 1 ) 2 ) 3

. 6-20. Y 1(A,B,C,D) = m(0,2,5,6,7,8,10,12,13,14,15).

, é è . “ ”.

, . p1, p2 , … . , è . - . . , - 1 , 0 , ( “−”) .

Y 1 -: p1= B D , p2 = C D , p3= A D , p4= B D , p5 = B C p6 = A B . (3) , . 6-20 ).


37/63

207

2: ()

- , 1-,

. , .. ( ). 1- , .. () 1, - .

1.(, 1- .)-

0 2 5 6 7 8 10 12 13 14 15

B D p1

C D p2

A D p3

B D p4

B C p5

A B p6

. 6-21.

Y 1(A,B,C,D) = m(0,2,5,6,7,8,10,12,13,14,15).

1,2, 3, ... . , 0. , , 1- . , 1- “ ”.

Y1(A,B,C,D) = m(0,2,5,6,7,8,10,12,13,14,15) . 6-21.

–M K

- , (). . é ( -).

3: ()

(-) . , -

. è - , , -.


38/63

208

- 3.1: -.

(-) . , “ “ . “ “ (“”), () ( “ ’ ”). K, .. 1- , a ( ) (“ * ”). - () , . “ ” , 1-

- ().

1.-

0 * 2 5 * 6 7 8 10 12 13 14 15

B D p1’

C D p2

A D p3

B D p4’

B C p5

A B p6

. 6-22 ). -

Y 1(A,B,C,D) = m(0,2,5,6,7,8,10,12,13,14,15).

- Y 1(A,B,C,D) = m(0, 2, 5, 6, 7, 8, 10, 12, 13, 14, 15) . 6-22 ). -: p1 = B D p4 = B D , . 6-22 ) . 6-22 ).

, - 1- - ( ).

− () − ()

, .

1- - .-

6 12 14

C D p2

A D p3

B C p5

A B p6

. 6-22 ). Y1 =m(0,2,5,6,7,8,10,12,13,14,15)

-.


39/63

209

(1- ) () ( ) “ ” . , , .. .

( ) . (-) .

, , , .

- 3.2:

( )

, , . .

, . , . 6-23 ), Z 1(A,B,C,D) = m(0,1,2,3,5,6,7,9,13). o , .. – .

. 6-23 ) m2 m

3,

.. 3- 6-. , 6- - p1 p4, 3- p1 p4, p3. m3, m6.

, m1 m5 , m1- , m5 -

.

. 6-23 ) .

- Y 1(A,B,C,D) = m(0,2,5,6,7,8,10,12,13,14,15) - p1 p4

. 6-22 ), .. . 6-24 ).

1-

1- - -

1 2 3 5

-

2 5

A B p1 A B p1

C D p2 C D p2

A D p3 A D p3

A C p4 A C p4

) )

. 6-23.

Z 1(A,B,C,D) = m(0,1,2,3,5,6,7,9,13).


40/63

210

m14 m6 m12 , .. 6 - 16- 14-.

14- , . 6-24 ) . 6-24 ) - . ( m14 , , : m6 m12.)

. 6-24. Y 1(A,B,C,D) == m(0,2,5,6,7,8,10,12,13,14,15).

- 3.3:

( )

, , .

, - . 6-25 ) Z 2 (A,B,C,D) = m(2,3,4,5,10,13,15),

. a p1 p2 , .. p2 p1. , p1 m2 , p2 , m3. ( p2 , m2 m3). , , , -.

1- 1- -

2 3 5 13

-

2 3 5 13

B C D p1 A B C p2

A B C p2 B C D p4

A B C p3

B C D p4

A B D p5

) )

. 6-25. Z 2 = m(2,3,4,5,10,13,15).

1- -

1- -

-

6 12 14

-

6 12

C D p2 C D p2

A D p3 A D p3

B C p5 B C p5

A B p6 A B p6

) )


41/63

211

, . 6-25 ) p4 p3, .. : p4 m5 m13, p3 m5 . p4, p3 . , p4 p5 :

p4 m5 m13, p5 m13. p5 . . 6-25 ) – .

- Y 1(A,B,C,D)= m(0,2,5,6,7,8,10,12,13,14,15).

. 6-24 ) . 6-26 ).

p2 p5 . : p5 p2 - , m6 . . , - p3 p6 : 1 m12.

, , .. (). ( p2 p4) m6 ( p3 p5 ) m12. - , .

Y 1(A,B,C,D) = m(0,2,5,6,7,8,10,12,13,14,15) . 6-26) ), . 6-26 ) - p2 =C D p3= A D .( p2 =C D p6 = A B p3= A D p5 = A B .)

-

-, - . - , - (-). , -, () , -.

1- 1- -

6 12

- 6 12

C D p2 C D p2

A D p3 A D p3

B C p5

A B p6

) )

. 6-26.

- Y 1= m(0,2,5,6,7,8,10,12,13,14,15)


42/63

212

- (“ ” ”), 1, (, 1- ) . (“ * ”). -

-: , , . - - , “ ” .

- , -, , ... . é . , é -

- – , .

. , - - 3.

, , .

Y 1(A,B,C,D) = m(0,2,5,6,7,8,10,12,13,14,15) . 6-26 ) - p2 = C D p3= A D -.

. 6-27 , - p1= B D p4= B D - p2 =C D p3= A D :

Y 1(A,B,C,D) = p1 + p4 + p2 + p3 = B D + B D + C D + A D

- - : Y 1= m(0,2,5,6,7,8,10,12,13,14,15).

4: ()

- - - -, / . -, /

. .

-

6* 12*

C D p2 ’’

A D p3’’

. 6-27.


43/63

213

6.6.3.3.

0, 1, ““ ““ ( “ /“) . 1 0. , , .

, 1

0, (, ) 1, X (, ). , , ( “ * ”). “ * ”. , “ * ” ( x x) . ,

“ * ” 1, x, .. 1- .

, - “ ” " * " 1 x 1- .

, , “ * ”, - “x” “x” () .

-. 1. () , .. x, . -, é .

-

Y 3 =Σ

m(2,3,7,9,11,13) +Σ x(1,10,15).

- Y 3(A,B,C,D) = Σ m (2,3,7,9, 11,13) + Σ x (1,10,15) . 6-28 ),),),). . 6-29, - - p2 , p3 p4.


44/63

214

i ABCD Y 3 i ABCD i ABCD i ABCD

0 0000 0 1* 0001 (1,3) 00-1 (1,3,9,11) -0-1 p1

1* 0001 x 2 0010 (1,9) -001 (2,3,10,11) -01- p2

2 0010 1 3 0011 (2,3) 001- (3,7,11,15) --11 p3

3 0011 1 9 1001 (2,10) -010 ,11,13,15) 1--1 p4

4 0100 0 10* 1010 (3,7) 0-11

5 0101 0 7 0111 (3,11) -011

6 0110 0 11 1011 (9,11) 10-1

7 0111 1 13 1101 (9,13) 1-01

8 1000 0 15* 1111 10,11) 101-

9 1001 1 (7,15) -111

10* 1010 x 11,15) 1-11

11 1011 1 13,15) 11-1

12 1100 0

13 1101 1

-:

p1 = B D

p2 = B C

p3 = C D

p4 = A D

14 1110 0

15* 1111 x

. 6-28. Y 3(A,B,C,D) = Σ m(2,3,7,9,11,13) + Σ x(1,10,15).

, p2 m2 , p3 m7 p4 m13. , p2 , p3 p4 , m3, m9 m11, m2 , m7 m13 - p2 , p3 p4.

M (1- ).-

2 * 3 7 * 9 11 13*

B D p1

B C p2 ’

C D p3’

A D p4’

. 6-29. Y3 = Σm (2,3,7,9,11,13) + Σx (1,10,15).

, .. - p2 , p3 p4

1- .

Y 3 : Y 3 = p2 + p3+p4 = B C +C D + A D .


45/63

215

6-35. -.

) - ) ) .

) F 0 (X 1, X 2 , X 3, X 4 ) = m (1, 2, 3, 7, 8, 9, 10, 11, 14, 15)

) F 1(X 1, X 2 , X 3, X 4 ) = m(1, 2, 3, 5, 7)

) F 2 (X 1, X 2 , X 3, X 4 ) = m(0, 4, 5, 11, 13, 15)

) F 3(X 1, X 2 , X 3, X 4 ) = m(2, 4, 5, 6, 7, 9, 13)

) F 4(X 1, X 2 , X 3, X 4 ) = m(3, 4, 5, 7, 9, 11, 12, 13)

) F 5 (X 1, X 2 , X 3, X 4 ) = m(0, 1, 4, 6, 9, 10, 12, 14)

) F 6 (X 1, X 2 , X 3, X 4 ) = m(3, 4, 5, 7, 8, 10, 11, 14, 15)

) F 7 (X 1, X 2 , X 3, X 4 ) = m(3, 4, 5, 7, 8, 10, 13, 14, 15)

) F 8 (X 1, X 2 , X 3, X 4 ) = m(0, 3, 4, 7, 11, 12, 14, 15)

) F 9(X 1, X 2 , X 3, X 4 ) = m(0, 1, 2, 3, 4, 5, 7, 14, 15)

) F 10 (X 1, X 2 , X 3, X 4 ) = m(3, 4, 5, 6, 7, 9, 13)

) F 11(X 1, X 2 , X 3, X 4 ) = m(3, 4, 5, 7, 9, 11, 12, 13)

) F 12 (X 1, X 2 , X 3, X 4 ) = m(0, 5, 7, 8, 9, 10, 11, 14, 15)

) F 13(X 1, X 2 , X 3, X 4 ) = m(1, 3, 6, 7, 8, 10, 11, 13, 14)

.

6-36. .

) F 14(X 1, X 2 , X 3, X 4 ) = m(0, 4, 5, 6, 8, 12, 14, 15) + x(1, 2, 7, 10)

) F 15 (X 1, X 2 , X 3, X 4 ) = m(0, 5, 8, 12, 15) + x(1, 2, 3, 10, 13, 14)

) F 16 (X 1, X 2 , X 3, X 4 ) = m(1, 2, 3, 4, 5, 9, 11, 12) + x(7, 10, 13, 15)

) F 17 (X 1, X 2 , X 3, X 4 ) = m(0, 4, 5, 6, 8, 12, 14, 15) + x(1, 2, 10)

) F 18 (X 1, X 2 , X 3, X 4 ) = m(2, 4, 6, 10) + x(0, 3, 7, 11, 12, 13, 14, 15)

6-37. (*) ) ) - .

) F 19(A ,B,C,D) = m(1, 5, 7, 8, 11, 15, 17, 20, 22, 24, 26, 30, 31)

) F 20 (A ,B,C,D) = m(0, 2, 3, 4, 6, 10, 13, 14, 18, 22, 23, 27, 31)

) F 21(A ,B,C,D) = m(0, 1, 2, 3, 4, 6, 7, 8, 9, 14, 15, 22, 23, 25, 28, 30, 31)

) F 22 (A ,B,C,D) = m(1, 5, 8, 12, 21, 31) + x (0, 15, 17, 24, 28)

) F 23(A ,B,C,D) = m(0, 2, 8, 10, 11, 16, 18, 24, 26, 30) + x(14, 31)

) F 24(A ,B,C,D) = m(1, 4, 17, 20, 22, 31) + x(0, 3, 6, 19, 29)

) F 25 (A ,B,C,D) = m(1, 5, 8, 12, 21, 31) + x(0, 15, 17, 24, 28)

) F 26 (A ,B,C,D) = m(3, 4, 5, 7, 9, 12, 13, 22, 29, 30) + x(1, 2, 17, 21, 24, 28, 31)


46/63

216

6.7.

, : , , , . , . , , .

, , .

- ( -). , . . .

6.7.1.

, . , .

- -, .. , . . IEEE ( ) IEC ( ). .

, . (ANSI) . , , . , ANSI, .. . , . 6-28 () , (),

. , .


47/63

217

(),

().

ANSI IEC/IEEE

()

A Y

AY =

10

A

1

Y

0

-

()

A

BY

B AY

⋅=

BA Y

0 0 0

0 1 01 0 01 1 1

AY

B

B AY ⋅=

BA Y

0 0

0

0 11 01 1

111

A YB

B AY +=

BA Y

0 0 00 11 01 1 1

11

AY

B

B AY +=

BA Y

0 000 1

1 01 1

1

00

-

AY

B

B AY ⊕=

B A B AY +=

BA Y

0 0 00 11 01 1

1

01

()

-

AY

B

B AY ⊕=

B A ABY +=

BA Y

0 000 1

1 01 1

1

01

()

. 6-28. .


48/63

218

6-28 () () , .. .

,

. (6-22, 6-23 6-24) , , . 6-24 ), ) ) , . (6-25, 6-26 6-27) . 6-25 ), ) ) .

A A

)

A A

)

A

B

AB

( A B )

A B

)

ABA

BB

A

(A+B)

)

A

B

A

B

( A B )

A+B

)

A+BA

B

A+B

(A+B)

)

. 6-24. . . 6-25. .

(), .

6.7.2.

(, . buffer ) . , .. , .

A Y

) )

Y = A

A Y

0 0

1 1

. 6-26. . 6-29.


49/63

219

. 6-26 ) ) : ANSI IEC/IEEE. , Y = A, . 6-29 .

. . , (. driver ).

: - , three-state tri-state buffer : 1 0, Y .. , HiZ, Hi-Z Z. a, ( 0→Y I ), Y ( ∞→Y Z , ∞→Y R ).

(. Enable). 1 (=1), , .. (Y=A). , 0 (=0), (Y=HiZ), ∞→Y R 0→Y I . E 1, =1, .

. 6-27 ), ) - : ANSI IEC/IEEE, . 6-30 .

Y=YA

E

) ANSI. ) IEC/IEEE.

=1

Hi-Z =0

E A Y

0 X Hi-Z

0 01

1 1

. 6-27. . 6-30.

( ) 1


50/63

220

(. 6-28).

R I = 0

E=0

Y

E

A A Y=AAY=HiZ

E=1

Y

Y

. 6-28.

- . , . ,

, .

6.7.3.

, . , - , . ,

.

, . , .. (,) , , .. .

, . , , . , , .

, ., , . è

( ). . , .. a.


51/63

221

. 6-29 ), ) ) e . , ,

( ) CBBAC,B,AY += . , .

, .

, . 6-29 ) ,. 6-29 ) „”, . 6-29 ) „ ”.

C

B

A

) () .

A

C

B

) () .

A

C

B

B

) () .

. 6-29. ( ) CBBAC,B,AY += .

. , . 6-29 ) , .

. 6-29 ) . . 6-29 ) .


52/63

222

. 6-30 . 6-31 .

A

K = 0

Y = 0

)

A

K = 1

Y = A

)

. 6-30.

K = 0

AY = A

)

AY = 1

K = 1 )

. 6-31.

. 6-32 . 6-33 :

BABAY += (6-28)

BAABY += (6-29)

AB

AB

A + B

(AB+AB)

A

B

A B B B A +

0 0 0

0 1 1

1 0 1

1 1 0

. 6-32.

A B

AB

(AB+AB)

A + BB

A

A B B A AB +

0 0 1

0 1 0

1 0 0

1 1 1

. 6-33.

, . 6-32 . 6-33 . . 6-34 ), ) . 6-35 ) ) , .


53/63

223

A

B

C

ABABC ABC

)

B

A AB

C

AB

ABC

)

. 6-34. .

C

A

B A+B+C

A+B

A+B +C

)

A+BA

B

C

A+B

A+B+C

)

. 6-35. .

. 6-36 .

B

A

S

Y

E

Z

E S Y Z

0 B B 0

1 A A

0 B HiZ1

1 A HiZ

. 6-36. .

6-37. ?

S

S

0

1

AY

. 6-37.

6-38. S1 S0 . 6-37

Y. o - .


54/63

224

6-39. . 6-38 S1 S2 , Y . .

S0 1S

A

Y

C

B

A

Y

. 6-38. . 6-39.

6-40. A, B C Y

. 6-39 1.

1Y

AK

YY2Y 3 4

DCB

. 6-40.

6-41. .6-40 0, 1 . ?!

6-42. . 6-41 .

A B

DIR

OE

. 6-41.

OE DIR

00

1

1

. 6-31.

6.7.4. A

. . .

. , .


55/63

225

( ), , . , ,

. - . . 6-42 -

BDCBACBADCAY +++= .

A B CD

Y1

A

A

A

B

B

B

C

C

D

C

D

A B DC

. 6-42. - . (

), , - , , , . - ( )( )( )( )CBDBDCDCAY +++++= . 6-43.

DCBA

2Y

AC

C

D

D

D

B

B

C

. 6-43. - .


56/63

226

, . , .

i A B C Y

0 0 0 0 0

1 0 0 1 0

2 0 1 0 1

3 0 1 1 1

4 1 0 0 1

5 1 0 1 06 1 1 0 1

7 1 1 1 0

, .

, a a e a Z

Z=Z(A,B,C) ( ) ( )

7,5,1,0M6,4,3,2m

∏ ==

. 6-32, . 6-44 ) ). . 6-32. -

( ) ( )7,5,1,06,4,3,2 ∏ == M m Z

7

6

B

1

0

3

2

A

5

4

C

111

1

0

0 0 0

) .

7

6

1

0

3

2

5

4

A

B

C

111

1

0

0 0 0

) .

. 6-44. ( )6,4,3,2m)C,B,A(Z = ( )7,5,1,0M∏= .

, CABAZ += , ( )( )CABAZ ++= .

- . 6-45 ) , - . 6-45 ).


57/63

227

A

C

B

A

Y=AB+AC

A B C

) - .

A B C

C

A

A

B Y=AB+AC

) - .

. 6-45. ( )6,4,3,2m)C,B,A(Z = ( )7,5,1,0M∏= .

, . , .

( ) ( )CABACABAZZ ⋅=+== . . 6-46 . 6-47. . 6-41 ), .

, :

( ) ( ) )CA()BA(CABAZZ +++=+⋅+== . . 6-49, . 6-48.

. 6-41 ) .


58/63

228

Y

A

B

C

A

BA CA

B

Y

≡

YB

A

. 6-47. Z = m(2,3,4,6) . 6-46

A

C

B

A

Y

A B C

YA

B

≡

A

B

Y

. 6-49. Z = ∏ M(0,1,5,7). . 6-48

(6-22) AAA ⋅= . 6-47 )

o . , (6-25) AAA += . 6-47) .

. , - , .

,

. - - , .


59/63

229

6-42. , () ; () .

6-43. - , , () ( ) DDCBBAD,C,B,AY ++= ; () ( ) CDBCBAD,C,B,AF ++= .

6-44. - , ,

() ( ) ( )( )DDBACAD,C,B,AZ +++= ; ()( ) ( )( )DDCBBAD,C,B,AF +++= .

6-45. Y(X1,X2,X3,X4) = 432143121 XXXXXXXXX ++

, (*) .

6-46. Y(X1,X2,X3,X4) = )XXXX)(XXX)(XX( 432132141 ++++++

, (*) 2 .6-47. (*) ( ) ( ) ( )10,8,3m14,13,12,7,5,2,1,0mD,C,B,AY

xm

+=

() () .

6-48. (*) ( ) ( ) ( )4,1M13,11,9,7,6,5,2MD,C,B,AYxM

∏∏=

() () .

6.7.5.

(, ) è . Oaa .

. ( ) (, ) . () . .

è . . ( ) .

, . . ,

. , .


60/63


61/63

231

:

1. , .. , ;

2. ;

3. , - ( ).

6-49. . = 1 0 B=B1B0 , Y=Y 2 Y 1Y 0 . A1, B1, Y 2 .

6-50. .

= 1 0 B=B1B0 , Y=Y 2 Y 1Y 0 . A1, B1, Y 2 .

6-51. (*) . , = 1 0 B=B1B0 . , Y=Y 1Y 0 , : A=B=0, Y=0, A>B, Y=1


62/63

232

,

, , . . , , , – .

.

, . .


63/63

1. . (2007), , . , –.

2. ., . (1993) , III , – .

3. ., ., . (1993), II III, – .

4. Balch, M. (2003). Complete Digital Design, McGraw-Hill Companies, Inc.

5. Floyd, T. (2006). Digital Fundamentals, Ninth Edition, Pearson Prentice Hall .

6. Holdsworth, B., Woods, C. (2003). Digital Logic Design, Fourth Edition, ElsevierPrivate Ltd .

7. Maini, A. (2007). Digital Electronics, John Wiley & Sons Ltd .

8. M. Morris, Mano, Charles Kime (2008). Logic and Computer Design

Fundamentals, Fourth Edition, Prentice Hall .

9. Saha, A., Manna, N. (2007). Digital Principles and Logic Design, Infinity SciencePress, Laxmi Publications Ltd.

10., . (2008). I ( ), , – .

11., ., , . (2008). II ( ), , – .

Logicki Kola i Digitalna Logika

Documents

Transcript of Logicki Kola i Digitalna Logika