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2-1 Table 2-1 Truth Tables for the Three Basic Logic Operations
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
AND OR NOT
X Y Z X Y X
0 0 0 0 0 0 0 10 1 0 0 1 1 1 01 0 0 1 0 11 1 1 1 1 1
Z Z
2-2
••X
NOT gate orinverter
XX
YZ = X + Y
(a) Graphic symbols
OR gate
X
YZ = X • Y
AND gate
X 0 0 1 1
Y 0 1 0 1
X • Y 0 0 0 1
0 1 1 1
1 1 0 0
X + Y
(AND)
(NOT)
(OR)
(b) Timing diagram
X
Figure 2-1 Digital Logic Gates
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-3
ABC
F = A + B + C + D + E + F
(b) Six-input OR gate
ABC
F = ABC
(a) Three-input AND gateDEF
Figure 2-2 Gates with More than Two Inputs
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-4 Table 2-2 Truth Table for the Function F = X + Y'Z
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
X Y Z F
00001111
00110011
01010101
01001111
2-5 Figure 2-3 Logic Circuit Diagram for F = X + Y'Z
X
•Y
Z
F
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-6
TABLE 2-3Basic Identities of Boolean Algebra
1. 2.3. 4.5. 6.7. 8.9.
10. 11. Commutative12. 13. Associative14. 15. Distributive16. 17. DeMorgan’s
X 0 X X 1 X
X 1 1 X 0 0X X X X X X
X X 1 X X 0X X
X Y Y X XY YX
X Y Z( ) X Y( ) Z X YZ( ) XY( )ZX Y Z( ) XY XZ X YZ X Y( ) X Z( )X Y X Y X Y X Y
Table 2-3 Basic Identities of Boolean Algebra
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-7 Table 2-4 Truth Tables to Verify DeMorgan's Theorem
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
A) X Y B X Y
0011
0101
0111
1000
0011
0101
1100
1010
1000
X Y+ X Y X Y?
2-8 Table 2-5 Truth Table for Boolean Function
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
X Y Z F
00001111
00110011
01010101
00110101
2-9
••
••
X
Y
Z
(a) F = XYZ + XYZ + XZ
(b) F = XY + XZ
••X
Y
F
F
Z
•
••
•
•
Figure 2-4 Implementation of Boolean Function with Gates
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-10
TABLE 2-6Minterms for Three Variables
X Y ZProductTerm Symbol m0 m1 m2 m3 m4 m5 m6 m7
00001111
00110011
01010101
m0m1m2m3m4m5m6m7
10000000
01000000
00100000
00010000
00001000
00000100
00000010
00000001
X Y ZX YZXYZXYZXY ZXYZXYZXYZ
Table 2-6 Minterms for Three Variables
2-11
TABLE 2-7Maxterms for Three Variables
X Y Z Sum Term Symbol M0 M1 M2 M3 M4 M5 M6 M7
00001111
00110011
01010101
M0M1M2M3M4M5M6M7
01111111
10111111
11011111
11101111
11110111
11111011
11111101
11111110
X Y ZX Y ZX Y ZX Y ZX Y ZX Y ZX Y ZX Y Z
Table 2-7 Maxterms for Three Variables
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-12 Table 2-8 Boolean Function of Three Variables
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
(a) X Y Z F F (b) X Y Z E
00001111
00110011
01010101
10100101
01011010
00001111
00110011
01010101
11101100
2-16
m0 m1
m2 m3
(a) (b)
0
1
0 1Y
X
X Y X Y
X YX Y
Figure 2-8 Two-Variable Map
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-9 Representation of Functions in the Map2-17
0
1
(a) XY
1
0 1Y
X
0
1
(b) X + Y
1
0 1Y
X
11
1
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-18
(a)
m0 m1 m3 m2
m4 m5 m7 m6
YZX
0
00 01
X Y Z X Y Z X Y Z X Y Z
X Y Z X Y Z X Y Z X Y Z
11 10
Y
1X
Z
(b)
Figure 2-10 Three-Variable Map
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-11 Map for Example 2-32-19
0
1
Y
Z
X 1
00 01 11 10
1
1
YZX XY
XY
1
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-20
0
1
Y
Z
X
(a)
0
4
00 01 11 10
5
2
YZX
1 3
7 6
XZ
XZ
0
6 4
13
57
2
(b)
Figure 2-12 Three-Variable Map: Flat and on a Cylinder to Show Adjacent Squares
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-13 Product Terms Using Four Minterms2-21
0
1
Y
Z
X
(a)
1
1
00 01 11 10
1
YZX
1
0
1
Y
Z
X
(b)
1
00 01 11 10
1
YZX
1 1
1 1
Y
X
Z
Z
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-14 Maps for Example 2-42-22
0
1
Y
Z
X
(b) F2(X, Y, Z) = m(0, 2, 4, 5, 6) = Z + XY
00 01 11 10
1
YZX
1
1
1 1
0
1
Y
Z
X 1
00 01 11 10
1
YZX
1 1
(a) F1(X, Y, Z) = m(3, 4, 6, 7) = YZ + XZ
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-15 F(X, Y, Z) = SUM m(1, 3, 4, 5, 6)2-22
0
1
Y
Z
X 1
00 01 11 10YZ
X
11
1 1
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-16 F(X, Y, Z) = SUM m (1, 2, 3, 5, 7)2-24
0
1
Y
Z
X
00 01 11 10
1
YZX
1
1
1
1
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-25
(a) (b)
00
01
00 01YZ
WX
m0 m1
m4 m5
m3 m2
m7 m6
m12 m13
m8 m9
m15 m14
m11 m10
Y
Z
W
11 10
11
10
X
Figure 2-17 Four-Variable Map
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-26
(a)
00
01
00 01YZ
WX
Y
Z
W
11 10
11
10
X
0 1 3 2
4 5 7 6
8 9 1011
12 13 1415
XZ
02
8 10
1 3911
X Z
(b)
Figure 2-18 Four-Variable Map: Flat and on a Torus to Show Adjacencies
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-19 Map for Example 2-5: F = Y' + W'Z' + XZ'2-27
00
01
00 01YZ
WXY
Z
W
11 10
11
10
X
1 1 1
1 1 1
1 1
1 1 1
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-20 Map for Example 2-6: F = B'D' + B'C + A'CD'2-28
00
01
00 01CD
ABC
A
11 10
11
10
B
1 1 1
1
1 1 1
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-21 Prime Implicants for Example 2-7: A'D, BD', and A'B2-29
00
01
00 01CD
ABC
A
11 10
11
10
B
1
1
1
11 1
1 1
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-22 Simplification with Prime Implicants in Example 2-82-30
00
01
00 01CD
ABC
D
A
11 10
11
10
B
1
1
1
1 1
1
1
00
01
00 01CD
ABC
DA
11 10
11
10
B
1
1
1
1 1
1
1
(a) Plotting the minterms (b) Essential prime implicants
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-23 Map for Example 2-92-31
1 2
3
00
01
00 01CD
ABC
A
11 10
11
10
B
1
1
1
1
1
1
11
1
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-24 Map for Example 2-10: F = (A' + B')(C' + D')(B' + D)2-32
00
01
00 01CD
ABC
A
11 10
11
10
B
1 01 1
1 00 0
0 00 0
1 01 1
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-25 Example with Don't Care Conditions2-33
00
01
00 01CD
ABC
D
A
11 10
11
10
B
X 00
01
00 01CD
ABC
D
A
11 10
11
10
B
1 1 X X 1 1 X
0 X 1 0 0
X
1 0
0 0 1 0 0 0 1 0
0 0 1 0 0 0 1 0
(b) F = CD + AD(a) F = CD + A B
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-34
Graphics Symbols
Name Distinctiveshape
Rectangularshape
Algebraicequation
Truthtable
X
YF F
X
Y
AND F = XY
X Y F0 0 00 1 01 0 01 1 1
&
X
YF F
X
Y
OR F = X + Y
X Y F0 0 00 1 11 0 11 1 1
≥1
••X F •• FXNOT(inverter) F = X
X F0 11 0
1
X F FXBuffer F = XX F0 01 1
1
••X
YF F
X
Y
NAND F = X • Y
X Y F0 0 10 1 11 0 11 1 0
&
••X
YF •• F
X
Y
NOR F = X + Y
X Y F0 0 10 1 01 0 01 1 0
≥1
X
YF F
X
Y
Exclusive–OR(XOR)
F = XY + XY= X Y
X Y F0 0 00 1 11 0 11 1 0
=1
••X
YF •• F
X
Y
Exclusive–NOR(XNOR)
F = XY + XY= X Y
X Y F0 0 10 1 01 0 01 1 1
=1
••
Figure 2-26 Digital Logic Gates
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-36
(a) AND - NOT
Y ••X
ZXYZ
••X X
(b) NOT – OR
YX
Z
•• XX
(c) NOT
••••••X + Y + Z = XYZ
Figure 2-28 Alternative Graphics Symbols for NAND and NOT Gates
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-29 Three Ways to Implement F = AB + CD2-37
A
B
C
D
F•
••
(c)
(b)
A
B
C
D
F•
•
••
A
B
C
D
F
(a)
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-13 Figure 2-5 Sum-of-Products Implementation
Y
XY
XY
FZ
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-14
A
B
C
D
E
(a) AB + C (D + E ) (b) AB + CD + CE
A
B
C
D
C
E
Figure 2-6 Three-Level and Two-Level Implementation
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
2-15 Figure 2-7 Product-of-Sums Implementation
X
Z
XYZ
YF
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-30 Solution to Example 2-122-38
0
1
Y
Z
X 1
00 01 11 10
1
1
YZX
11
1F = XY + XY + Z
(a)
•
•
•
•••
Z
X
Y
X
YF
(b)
•
•
Z
X
Y
X
YF•
(c)
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-31 Implementing F = A(CD + B) + BC'2-39
C
D
B
A
B
C
F
(a) AND – OR gates
C
D
B
A
B
C
F
(b) NAND gates
•••
•
•••
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-32 Implementing F = (AB' + A'B) E (C + D')2-40
A
B
A
B
CD
E
F
(a) AND – OR gates
A
B
A
B
CD
E
F
(b) NAND gates
X
••
•
••
••
•
•
••
•
2-42
Y ••
(a) OR – NOT
XX + Y + Z
ZY
(b) NOT – AND
XX Y Z = X + Y + Z
Z
••••••
Figure 2-34 Two Graphic Symbols for NOR Gate
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-35 Implementing F = (A + B)(C + D)E with NOR Gates2-43
•
•
E
A
B
C
DF•••
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-36 Implementing F = (AB' + A'B) E (C + D') with NOR Gates2-44
AB
CD
E
FY
•
••
•
• ••
••
AB
••
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-37 Exclusive-OR Constructed with NAND Gates2-45
••
••
••
•X
Y
F = X Y
•
•
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-39 Mutiple-Input Odd Functions2-48
XY
ZP
XYZ
C
P
(a) P = X Y Z (b) C = X Y Z P
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-40 Propagation Delay for an Inverter2-49
•IN OUT
IN
OUT tPHL tPLH
tpd = max (tPHL, tPLH)
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
Figure 2-42 Signal Assignment and Logic Polarity2-51
(a) Positive logic
Signalvalue
Logicvalue
H 1
L 0
(b) Negative logic
Signalvalue
Logicvalue
H 0
L 1
M. Morris Mano & Charles R. KimeLOGIC AND COMPUTER DESIGN FUNDAMENTALS
© 2001 Prentice Hall, Inc.
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