Ch4 Boolean Algebra And Logic Simplication1
-
Upload
qundeel -
Category
Technology
-
view
24.996 -
download
10
description
Transcript of Ch4 Boolean Algebra And Logic Simplication1
![Page 1: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/1.jpg)
1
Ch. 4 Boolean Algebra and Logic Simplification
Boolean Operations and Expressions Laws and Rules of Boolean Algebra Boolean Analysis of Logic Circuits Simplification Using Boolean Algebra Standard Forms of Boolean Expressions Truth Table and Karnaugh Map Programmable Logic: PALs and GALs Boolean Expressions with VHDL
![Page 2: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/2.jpg)
2
Information Security Lab.
Introduction
Boolean Algebra• George Boole(English mathematician), 1854
“An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities”
Boolean Algebra {(1,0), Var, (NOT, AND, OR), Thms}
• Mathematical tool to expression and analyze digital (logic) circuits
• Claude Shannon, the first to apply Boole’s work, 1938– “A Symbolic Analysis of Relay and Switching Circuits” at MIT
• This chapter covers Boolean algebra, Boolean expression and its evaluation and simplification, and VHDL program
![Page 3: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/3.jpg)
3
Information Security Lab.
Boolean functions : NOT, AND, OR, exclusive OR(XOR) : odd function exclusive NOR(XNOR) : even function(equivalence)
Basic Functions
![Page 4: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/4.jpg)
4
Information Security Lab.
• ANDZ=X Y or Z=XY Z=1 if and only if X=1 and Y=1, otherwise
Z=0
• ORZ=X + YZ=1 if X=1 or if Y=1, or both X=1and Y=1.
Z=0 if and only if X=0 and Y=0
• NOTZ=X orZ=1 if X=0, Z=0 if X=1
Basic Functions (계속 )
![Page 5: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/5.jpg)
5
Information Security Lab.
Basic Functions (계속 )
![Page 6: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/6.jpg)
6
Information Security Lab.
Boolean Operations and Expressions
•Boolean Addition– Logical OR operation
Ex 4-1) Determine the values of A, B, C, and D that make the sum term A+B’+C+D’
Sol) all literals must be ‘0’ for the sum term to be ‘0’
A+B’+C+D’=0+1’+0+1’=0 A=0, B=1, C=0, and D=1
•Boolean Multiplication– Logical AND operation
Ex 4-2) Determine the values of A, B, C, and D for AB’CD’=1
Sol) all literals must be ‘1’ for the product term to be ‘1’
AB’CD’=10’10’=1 A=1, B=0, C=1, and D=0
![Page 7: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/7.jpg)
7
Information Security Lab.
Basic Identities of Boolean Algebra
The relationship between a single variable X, its complement X, and the binary constants 0 and 1
![Page 8: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/8.jpg)
8
Information Security Lab.
Laws of Boolean Algebra
•Commutative Lawthe order of literals does not matter–A + B = B + A
–A B = B A
![Page 9: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/9.jpg)
9
Information Security Lab.
•Associative Lawthe grouping of literals does not matter–A + (B + C) = (A + B) + C (=A+B+C)
–A(BC) = (AB)C (=ABC)
Laws of Boolean Algebra (계속 )
![Page 10: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/10.jpg)
10
Information Security Lab.
•Distributive Law : A(B + C) = AB + AC
A
B
C
X
YX=Y
Laws of Boolean Algebra (계속 )
![Page 11: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/11.jpg)
11
Information Security Lab.
(A+B)(C+D) = AC + AD + BC + BD
A
BCD
XY X=Y
Laws of Boolean Algebra (계속 )
![Page 12: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/12.jpg)
12
Information Security Lab.
A+0=A
In math if you add 0 you have changed nothing in Boolean Algebra ORing with 0 changes nothing
A
X X=A+0=A
Rules of Boolean Algebra
![Page 13: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/13.jpg)
13
Information Security Lab.
A+1=1
ORing with 1 must give a 1 since if any input is 1 an OR gate will give a 1
A
XX=A+1=1
Rules of Boolean Algebra ( 계속 )
![Page 14: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/14.jpg)
14
Information Security Lab.
A•0=0
In math if 0 is multiplied with anything you get 0. If you AND anything with 0 you get 0
A
XX=A0 = 0
Rules of Boolean Algebra ( 계속 )
![Page 15: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/15.jpg)
15
Information Security Lab.
A•1 =AANDing anything with 1 will yield the anything
A
XX=A1=A
A
Rules of Boolean Algebra ( 계속 )
![Page 16: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/16.jpg)
16
Information Security Lab.
A+A = A
ORing with itself will give the same result
A
A
X
A=A+A =A
Rules of Boolean Algebra ( 계속 )
![Page 17: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/17.jpg)
17
Information Security Lab.
A+A’=1
Either A or A’ must be 1 so A + A’ =1
A
A’
XX=+A’=1
Rules of Boolean Algebra ( 계속 )
![Page 18: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/18.jpg)
18
Information Security Lab.
A•A = A
ANDing with itself will give the same result
A
A
XA=AA=A
Rules of Boolean Algebra ( 계속 )
![Page 19: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/19.jpg)
19
Information Security Lab.
A•A’ =0
In digital Logic 1’ =0 and 0’ =1, so AA’=0 since one of the inputs must be 0.
A
A’
XX=AA’=0
Rules of Boolean Algebra ( 계속 )
![Page 20: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/20.jpg)
20
Information Security Lab.
A = (A’)’If you not something twice you are back to the
beginning
A
XX=(A’)’=A
Rules of Boolean Algebra ( 계속 )
![Page 21: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/21.jpg)
21
Information Security Lab.
A
B
X
A + AB = A
Rules of Boolean Algebra ( 계속 )
![Page 22: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/22.jpg)
22
Information Security Lab.
A + A’B = A + B If A is 1 the output is 1 If A is 0 the output is B
AB
XY X=Y
Rules of Boolean Algebra ( 계속 )
![Page 23: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/23.jpg)
23
Information Security Lab.
A
B
C
X
Y
(A + B)(A + C) = A + BC
Rules of Boolean Algebra ( 계속 )
![Page 24: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/24.jpg)
24
Information Security Lab.
•DeMorgan’s TheoremF(A,A, , + , 1,0) = F(A, A, + , ,0,1)
– (A • B)’ = A’ + B’ and (A + B)’ = A’ • B’– DeMorgan’s theorem will help to simplify digital circuits using NORs and NANDs his theorem states
DeMorgan’s Theorems
![Page 25: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/25.jpg)
25
Information Security Lab.
![Page 26: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/26.jpg)
26
Information Security Lab.
Look at (A +B +C + D)’ = A’ • B’ • C’ • D’
![Page 27: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/27.jpg)
27
Information Security Lab.
Ex 4-3) Apply DeMorgan’s theorems to (XYZ)’ and (X+Y+Z)’
Sol) (XYZ)’=X’+Y’+Z’ and (X+Y+Z)’=X’Y’Z’
Ex 4-5) Apply DeMorgan’s theorems to (a) ((A+B+C)D)’ (b) (ABC+DEF)’ (c) (AB’+C’D+EF)’
Sol) (a) ((A+B+C)D)’= (A+B+C)’+D’=A’B’C’+D’(b) (ABC+DEF)’=(ABC)’(DEF)’=(A’+B’+C’)(D’+E’+F’)(c) (AB’+C’D+EF)’=(AB’)’(C’D)’(EF)’=(A’+B)(C+D’)(E’+F’)
![Page 28: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/28.jpg)
28
Information Security Lab.
Boolean Analysis of Logic Circuits
•Boolean Expression for a Logic Circuit
Figure 4-16 A logic circuit showing the development of the Boolean expression for the output.
![Page 29: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/29.jpg)
29
Information Security Lab.
•Constructing a Truth Table for a Logic Circuit– Convert the expression into the min-terms containing all the input literals
– Get the numbers from the min-terms – Putting ‘1’s in the rows corresponding to the min-terms and ‘0’s in the remains
Ex) A(B+CD)=AB(C+C’) (D+D’) +A(B+B’)CD =ABC(D+D’) +ABC’(D+D’) +ABCD+AB’CD =ABCD+ABCD’+ABC’D+ABC’D’ +ABCD+AB’CD =ABCD+ABCD’+ABC’D+ABC’D’ +AB’CD =m11+m12+m13+m14+m15=(11,12,13,14,15)
![Page 30: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/30.jpg)
30
Information Security Lab.
Truth Table from Logic Circuit
Input OutputA B C D A(B+C
D)0 0 0 0 00 0 0 1 00 0 1 0 00 0 1 1 00 1 0 0 00 1 0 1 00 1 1 0 00 1 1 1 01 0 0 0 01 0 0 1 01 0 1 0 01 0 1 1 11 1 0 0 11 1 0 1 11 1 1 0 11 1 1 1 1
A(B+CD)=m11+m12+m13+m14+m15 =(11,12,13,14,15)
![Page 31: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/31.jpg)
31
Information Security Lab.
Ex 4-8) Using Boolean algebra, simplify this expression
AB+A(B+C)+B(B+C)Sol) AB+AB+AC+BB+BC =B(1+A+A+C)
+AC=B+AC
Simplification Using Boolean Algebra
![Page 32: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/32.jpg)
32
Information Security Lab.
Ex 4-9) Simplify the following Boolean expression(AB’(C+BD)+A’B’)C
Sol) (AB’C+AB’BD+A’B’)C=AB’CC+A’B’C=(A+A’)B’C=B’C
Ex 4-10) Simplify the following Boolean expressionA’BC+AB’C’+A’B’C’+AB’C+ABC
Sol) (A+A’)BC+(A+A’)B’C’+AB’C=BC+B’C’+AB’C =BC+B’(C’+AC)=BC+B’(C’+A)=BC+B’C’+AB’
Ex 4-11) Simplify the following Boolean expression(AB +AC)’+A’B’C
Sol) (AB)’(AC)’+A’B’C=(A’+B’)(A’+C’)+A’B’C=A’+A’B’ +A’C’+B’C+A’B’C =A’(1+B’+C’+B’C)+B’C=A’+B’C’
![Page 33: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/33.jpg)
33
Information Security Lab.
Standard Forms of Boolean Expressions
•The Sum-of-Products(SOP) FormEx) AB+ABC, ABC+CDE+B’CD’
•The Product-of-Sums(POS) FormEx) (A+B)(A+B+C), (A+B+C)(C+D+E)(B’+C+D’)
•Principle of Duality : SOP POS
•Domain of a Boolean ExpressionThe set of variables contained in the expressionEx) A’B+AB’C : the domain is {A, B, C}
![Page 34: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/34.jpg)
34
Information Security Lab.
• Implementation of a SOP ExpressionAND-OR logic
•Conversion of General Expression to SOP FormA(B+CD)=AB +ACD
Ex 4-12) Convert each of the following expressions to SOP form: (a) AB+B(CD+EF) (b) (A+B)(B+C+D)Sol) (a) AB+B(CD+EF)=AB+BCD+BEF (b) (A+B)(B+C+D)=AB+AC+AD+ BB+BC+BD =B(1+A+C+D)+ AC+AD=B+AC+AD
![Page 35: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/35.jpg)
35
Information Security Lab.
Standard SOP Form (Canonical SOP Form)
– For all the missing variables, apply (x+x’)=1 to the AND terms of the expression
– List all the min-terms in forms of the complete set of variables in ascending order
Ex 4-13) Convert the following expression into standard SOP form: AB’C+A’B’+ABC’DSol) domain={A,B,C,D}, AB’C(D’+D)+A’B’(C’+C)(D’+D)+ABC’D =AB’CD’+AB’CD+A’B’C’D’+A’B’C’D+A’B’CD’+A’B’CD+ABC’D =1010+1011+0000+0001+0010+0011+1101 =0+1+2+3+10+11+13 = (0,1,2,3,10,11,13)
![Page 36: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/36.jpg)
36
Information Security Lab.
Product-of-Sums Form
• Implementation of a POS ExpressionOR-AND logic
![Page 37: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/37.jpg)
37
Information Security Lab.
Standard POS Form (Canonical POS Form)
– For all the missing variables, apply (x’x)=0 to the OR terms of the expression
– List all the max-terms in forms of the complete set of variables in ascending order
Ex 4-15) Convert the following expression into standard POS form: (A+B’+C)(B’+C+D’)(A+B’+C’+D)Sol) domain={A,B,C,D}, (A+B’+C)(B’+C+D’)(A+B’+C’+D) =(A+B’+C+D’D)(A’A+B’+C+D’)(A+B’+C’+D) =(A+B’+C+D’)(A+B’+C+D)(A’+B’+C+D’)(A+B’+C+D’)(A+B’+C’+D)=(0100) )(0101)(0110)(1101)= (4,5,6,13)
![Page 38: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/38.jpg)
38
Information Security Lab.
Converting Standard SOP to Standard POS
Step 1. Evaluate each product term in the SOP expression. Determine the binary numbers that represent the product terms
Step 2. Determine all of the binary numbers not included in the evaluation in Step 1
Step 3. Write in equivalent sum term for each binary number Step 2 and expression in POS form
Ex 4-17) Convert the following SOP to POSSol) SOP=
A’B’C’+A’BC’+A’BC+AB’C+ABC=0+2+3+5+7 =(0,2,3,5,7)
POS=(1)(4)(6) = (1, 4, 6) (=(A+B+C’)(A’+B+C)(A’+B’+C))
![Page 39: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/39.jpg)
39
Information Security Lab.
Boolean Expressions and Truth Tables
•Converting SOP Expressions to Truth Table FormatEx 4-18) A’B’C+AB’C’+ABC =(1,4,7)
InputsA B C
OutputX
Product Term
0 0 0 00 0 1 1 A’B’C0 1 0 00 1 1 01 0 0 1 AB’C’1 0 1 01 1 0 01 1 1 1 ABC
![Page 40: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/40.jpg)
40
Information Security Lab.
• Converting POS Expressions to Truth Table Format
Ex 4-19) (A+B+C)(A+B’+C)(A+B’+C’)(A’+B+C’)(A’+B’+C) = (000)(010)(011)(101)(110) = (0,2,3,5,6)
InputsA B C
OutputX Sum Term
0 0 0 0 A+B+C0 0 1 10 1 0 0 A+B’+C0 1 1 0 A+B’+C’1 0 0 11 0 1 0 A’+B+C’1 1 0 0 A’+B’+C1 1 1 1
![Page 41: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/41.jpg)
41
Information Security Lab.
Ex 4-20) Determine standard SOP and POS from the truth tableSol) (a) Standard SOP F=A’BC+AB’C’+ABC’+ABC(b) Standard POS F=(A+B+C)(A+B+C’)(A+B’+C)
(A’+B+C’)
InputsA B C
OutputX
0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 11 0 1 01 1 0 11 1 1 1
![Page 42: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/42.jpg)
42
Information Security Lab.
Boolean Expression
Truth Table
Logic Diagram
![Page 43: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/43.jpg)
43
Information Security Lab.
Karnaugh Map
•Simplification methods–Boolean algebra(algebraic method)–Karnaugh map(map method))–Quine-McCluskey(tabular method)
XY+XY=X(Y+Y)=X
![Page 44: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/44.jpg)
44
Information Security Lab.
![Page 45: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/45.jpg)
45
Information Security Lab.
Three- and Four-input Kanaugh mapsGray code
![Page 46: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/46.jpg)
46
Information Security Lab.
![Page 47: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/47.jpg)
47
Information Security Lab.
![Page 48: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/48.jpg)
48
Information Security Lab.
Gray code sequence generation
![Page 49: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/49.jpg)
49
Information Security Lab.
F(X,Y,Z)=m(0,1,2,6) =(XY+YZ)=X’Y’ + YZ’
![Page 50: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/50.jpg)
50
Information Security Lab.
Example) F(X,Y,Z)=m(2,3,4,5) =XY+XY
0 1 3 2
4 5 7 6
![Page 51: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/51.jpg)
51
Information Security Lab.
Example) F(X,Y,Z)=m(0,2,4,6) = XZ+XZ =Z(X+X)=Z
![Page 52: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/52.jpg)
52
Information Security Lab.
Four-Variable Map16 minterms : m0 ~ m15
Rectangle group – 2-squares(minterms) : 3-literals product term
– 4-squares : 2-literals product term– 8-squares : 1-literals product term– 16-squares : logic 1
![Page 53: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/53.jpg)
53
Information Security Lab.
![Page 54: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/54.jpg)
54
Information Security Lab.
![Page 55: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/55.jpg)
55
Information Security Lab.
F(W, X,Y,Z)=m(0,2,7,8,9,10,11) = WX’ + X’Z’ + W’XYZ
![Page 56: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/56.jpg)
56
Information Security Lab.
Karnaugh Map SOP Minimization
•Mapping a Standard SOP Expression
![Page 57: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/57.jpg)
57
Information Security Lab.
Ex 4-21) Ex 4-22)
![Page 58: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/58.jpg)
58
Information Security Lab.
•Mapping a Nonstandard SOP Expression– Numerical Expression of a Nonstandard Product Term
Ex 4-23) A’+AB’+ABC’A’ AB’ ABC’000 100 110001 101010011
![Page 59: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/59.jpg)
59
Information Security Lab.
Ex 4-24) B’C’+AB’+ABC’+AB’CD’+A’B’C’D+AB’CDB’C’ AB’ ABC’ AB’CD’ A’B’C’D AB’CD0000 1000 1100 1010 0001 10110001 1001 11011000 10101001 1011
![Page 60: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/60.jpg)
60
Information Security Lab.
Karnaugh Map Simplification of SOP Expressions
• Group 2n adjacent cells including the largest possible number of 1s in a rectangle or square shape, 1<=n
• Get the groups containing all 1s on the map for the expression
• Determine the minimum SOP expression form map
![Page 61: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/61.jpg)
61
Information Security Lab.
Ex 4-26) F=B+A’C+AC’D
![Page 62: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/62.jpg)
62
Information Security Lab.
Ex 4-27) (a) AB+BC+A’B’C’ (b) B’+AC+A’C’ (c) A’C’+A’B+AB’D (d) D’+BC’+AB’C
![Page 63: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/63.jpg)
63
Information Security Lab.
Ex 4-28) Minimize the following expressionAB’C+A’BC+A’B’C+A’B’C’+AB’C’
Sol) B’+A’C
![Page 64: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/64.jpg)
64
Information Security Lab.
Ex 4-29) Minimize the following expression
B’C’D’+A’BC’D’+ABC’D’+A’B’CD+AB’CD+A’B’CD’+A’BCD’ +ABCD’+AB’CD’
Sol) D’+B’C
![Page 65: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/65.jpg)
65
Information Security Lab.
Mapping Directly from a Truth Table
![Page 66: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/66.jpg)
66
Information Security Lab.
Don’t Care Conditions
•it really does not matter since they will never occur(its output is either ‘0’ or ‘1’)
•The don’t care terms can be used to advantage on the Karnaugh map
![Page 67: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/67.jpg)
67
Information Security Lab.
Karnaugh Map POS Minimization
•Use the Duality PrincipleF(A,A, , + , 1,0) F*(A,A, + , ,0,1)
SOP POS
![Page 68: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/68.jpg)
68
Information Security Lab.
Ex 4-30) (A’+B’+C+D)(A’+B+C’+D’)(A+B+C’+D) (A’+B’+C’+D’)(A+B+C’+D’)
Sol)
![Page 69: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/69.jpg)
69
Information Security Lab.
Ex 4-31) (A+B+C)(A+B+C’)(A+B’+C)(A+B’+C’)(A’+B’+C)
Sol) (0+0+0)(0+0+1)(0+1+0)(0+1+1)(1+1+0)=A(B’+C) AC+AB’=A(B’+C)
![Page 70: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/70.jpg)
70
Information Security Lab.
Ex 4-32) (B+C+D)(A+B+C’+D)(A’+B+C+D’)(A+B’+C+D)(A’+B’+C+D)
Sol) (B+C+D)=(A’A+B+C+D)=(A’+B+C+D)(A+B+C+D)(1+0+0+0)(0+0+0+0)(0+0+1+0)(1+0+0+1)(0+1+0+0)
(1+1+0+0) F=(C+D)(A’+B+C)(A+B+D)
![Page 71: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/71.jpg)
71
Information Security Lab.
Converting Between POS and SOP Using the K-map
Ex 4-33) (A’+B’+C+D)(A+B’+C+D)(A+B+C+D’)(A+B+C’+D’) (A’+B+C+D’)(A+B+C’+D)
Sol)
![Page 72: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/72.jpg)
72
Information Security Lab.
![Page 73: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/73.jpg)
73
Information Security Lab.
Five/Six –Variable K-Maps
•Five Variable K-Map : {A,B,C,D,E}
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
16 17 19 18
20 21 23 22
28 29 31 30
24 25 27 26
00 01 11 10
00
01
11
10
BCDE A=0
A=1
![Page 74: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/74.jpg)
74
Information Security Lab.
•Six Variable K-Map : {A,B,C,D,E,F}
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
16 17 19 18
20 21 23 22
28 29 31 30
24 25 27 26
00 01 11 10
00
01
11
10
CDEF
00
10 01
11
AB
32 33 35 34
36 37 39 38
44 45 47 46
40 41 43 42
48 49 51 50
52 53 55 54
60 61 62 63
56 57 59 58
![Page 75: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/75.jpg)
75
Information Security Lab.
Ex 4-34)Sol) A’D’E’+B’C’D’+BCD+ACDE
![Page 76: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/76.jpg)
76
Information Security Lab.
Programmable Logic: PALs and GALs
•Basic PAL Operation– Programmable array of AND gates– Fixed OR gate
![Page 77: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/77.jpg)
77
Information Security Lab.
• Implementing a Sum-of-Product Expression
![Page 78: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/78.jpg)
78
Information Security Lab.
![Page 79: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/79.jpg)
79
Information Security Lab.
Ex 4-35) Show how a PAL is programmed for the following function : X=AB’C+A’BC’+A’B’+AC
Sol)
![Page 80: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/80.jpg)
80
Information Security Lab.
PAL Block Diagram
![Page 81: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/81.jpg)
81
Information Security Lab.
PAL Output Combinational Logic
X0=X
X1=X’
![Page 82: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/82.jpg)
82
Information Security Lab.
A Specific PAL
Figure 4-50 Block diagram of the PAL16L8.
![Page 83: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/83.jpg)
83
Information Security Lab.
Basic GAL Operation
• Reprogrammable AND array• Electrically Erasable CMOS(E2CMOS) technology
![Page 84: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/84.jpg)
84
Information Security Lab.
Figure 4-52 GAL implementation of a sum-of-products expression.
![Page 85: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/85.jpg)
85
Information Security Lab.
Ex 4-36) Show how a GAL is programmed for the function: X=A’BC’+A’BC+BC+AB’Sol)
![Page 86: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/86.jpg)
86
Information Security Lab.
The GAL Block Diagram
•OLMCs(Output Logic Macrocells)– OR array and programmable output logic– Typically m and n >= 8
![Page 87: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/87.jpg)
87
Information Security Lab.
![Page 88: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/88.jpg)
88
Information Security Lab.
GAL20V8High Performance E2CMOS PLD
Generic Array Logic™
![Page 89: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/89.jpg)
89
Information Security Lab.
Boolean Expressions with VHDL
•Boolean Algebra in VHDL Programming– VHDL Optimization
Ex 4-37) Write a VHDL grogram for the following function: X=(AC+(BC’)’+D)’+((BC)’)’
-- Program X=(AC+(BC’)’+D)’+((BC)’)’entity alogicft is port(A, B, C, D: in bit; X: out bit);end entity alogicft;architecture expaft of alogicft isbegin X<=not((A and C) or not(B and not C) or D) or not(not B and C);end architecture expaft;
![Page 90: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/90.jpg)
90
Information Security Lab.
-- Program X=(AC+(BC’)’+D)’+((BC)’)’=(A’+C’)(BC’)D’+BC -- =A’BC’D’+BC’D’+BC=(A’+1)BC’D’+BC = BC’D’+BCentity alogicft is port(B, C, D: in bit; X: out bit);end entity alogicft;
architecture expaft of alogicft isbegin X<= (B and not C and not D) or (B and C);end architecture expaft;
![Page 91: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/91.jpg)
91
Information Security Lab.
Levels of Abstractionfor sequential logic circuits
VHDL(1) Behavioral
approach : state diagram or truth table
(2) Data flow approach : Boolean expression or function
(3) Structure approach : logic diagram or schematic describing logic function
![Page 92: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/92.jpg)
92
Information Security Lab.
Digital System Application : 7-Segment LED Driver
Seven-Segment LED driver
![Page 93: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/93.jpg)
93
Information Security Lab.
A B C D
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
g = m(2,3,4,5,6,8,9) =A+BC’+B’C+CD’CD
AB
![Page 94: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/94.jpg)
94
Information Security Lab.
Figure 4-59 Karnaugh map minimization of the segment-a logic expression.
![Page 95: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/95.jpg)
95
Information Security Lab.
Figure 4-60 The minimum logic implementation for segment a of the 7-segment display.
![Page 96: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/96.jpg)
96
Information Security Lab.
-- Program 7-segment driver entity sevensegdrv is port(A, B, C, D: in bit; a,b,c,d,e,f,g: out bit);end entity sevensegdrv;
architecture segment of sevensegdrv isbegin a<= B or D or (A and C) or (not A and not C); --B+D+AC+A’C’
• • • • • • • • •
g<= A or B and C’ or not B and C or C and not D; --A+BC’+B’C+CD’end architecture segment;
VHDL for 7-Segment Driver
![Page 97: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/97.jpg)
97
Information Security Lab.
Summary
• Gate symbols
• Duality PrincipleF(A,A, , + , 1,0) F*(A,A, + , ,0,1)
• DeMorgan’s TheoremF(A,A, , + , 1,0) = F(A, A, + , ,0,1)
![Page 98: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/98.jpg)
98
Information Security Lab.
The relationship between a single variable X, its complement X, and the binary constants 0 and 1
![Page 99: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/99.jpg)
99
Information Security Lab.
• Sum-of-Product(SOP) form Product-of-Sums(POS) form
• Standard(canonical) SOP form Standard POS form
• Universal gates: NAND, NOR• Don’t care conditions• Karnaugh map(3, 4, 5, 6 variables)• PLDs: PAL, GAL• VHDL for logic expressions
![Page 100: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/100.jpg)
100
Information Security Lab.
Boolean Expression
Truth Table
Logic Diagram
VHDL (HDL)
![Page 101: Ch4 Boolean Algebra And Logic Simplication1](https://reader033.fdocuments.net/reader033/viewer/2022061201/547a457cb4af9fe7318b47ff/html5/thumbnails/101.jpg)
End of Ch. 4