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Fundamentals of Logic Design Chap. 2

Boolean Algebra

This chapter in the book includes:

Objectives

Study Guide

2.1 Introduction

2.2 Basic Operation

2.3 Boolean Expression and Truth Table

2.4 Basic Theorem

2.5 Commutative, Associative and Distributive Laws

2.6 Simplification Theorem

2.7 Multiplying Out and Factoring

2.8 DeMorgan’s Laws

Problems

Laws and Theorems of Boolean Algebra

CHAPTER 2

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Fundamentals of Logic Design Chap. 2

Objectives

Topics introduced in this chapter:

• Understand the basic operations and laws of Boolean algebra

• Relate these operations and laws to AND, OR, NOT gates and switches

• Prove these laws using a truth table

• Manipulation of algebraic expression using

- Multiplying out

- Factoring

- Simplifying

- Finding the complement of an expression

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Fundamentals of Logic Design Chap. 2

2.1 Introduction

•Basic mathematics for logic design: Boolean algebra

•Restrict to switching circuits( Two state values 0, 1) – Switching algebra

•Boolean Variable : X, Y, … can only have two state values (0, 1)

•representing True(1) False (0)

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Fundamentals of Logic Design Chap. 2

0X' then 1X if and 1X' then 0X if

01' and 1'0

2.2 Basic Operations

NOT(Inverter)

Gate Symbol

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Fundamentals of Logic Design Chap. 2

111 001 010 000

2.2 Basic Operations

Gate Symbol

Truth Table

AND

진리표

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Fundamentals of Logic Design Chap. 2

111 101 110 000

2.2 Basic Operations

Gate Symbol

Truth Table

OR

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Fundamentals of Logic Design Chap. 2

BAT

BAT

2.2 Basic Operations

AND

OR

Apply to Switch

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Fundamentals of Logic Design Chap. 2

CAB '

2.3 Boolean Expressions and Truth Tables

Logic Expression :

Circuit of logic gates :

연산의 우선순위

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Fundamentals of Logic Design Chap. 2

0000)]'1(1[01)]'01(1[)]'([ BEDCA

'''''' cbbcabacab

BEDCA )]'([

10 literals

2.3 Boolean Expressions and Truth Tables

Logic Expression :

Circuit of logic gates :

Logic Evaluation : A=B=C=1, D=E=0

Literal : a variable or its complement in a logic expression

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Fundamentals of Logic Design Chap. 2

A B A’ F = A’ + B

0 0

0 1

1 0

1 1

1

1

0

0

1

1

0

1

2.3 Boolean Expressions and Truth Tables

2-Input Circuit and Truth Table

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Fundamentals of Logic Design Chap. 2

n variable needs rows n

CBCACAB

2222

)')(('

n times

TABLE 2.1

2.3 Boolean Expressions and Truth Tables

Proof using Truth Table

A B C B’ AB’ AB’+C A+C B’+C (A+C)(B’+C) 0 0 0 1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1

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Fundamentals of Logic Design Chap. 2

2.4 Basic Theorems

0' 1'

)''(

00 11

1 0

XXXX

XX

XXXXXX

XX

XXXX

101'11 ,1 if and ,10'00 ,0 if XX

11)'( EDAB

0)'')('( DABDAB

Operations with 0, 1

Complementary Laws

Idempotent Laws

Involution Laws

Proof

Example

멱등

누승

상보

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Fundamentals of Logic Design Chap. 2

2.4 Basic Theorems with Switch Circuits

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Fundamentals of Logic Design Chap. 2

2.4 Basic Theorems with Switch Circuits

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Fundamentals of Logic Design Chap. 2

2.4 Basic Theorems with Switch Circuits

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Fundamentals of Logic Design Chap. 2

2.5 Commutative, Associative, and Distributive Laws

XYYXYXXY

ZYXZYXZYX

XYZYZXZXY

)()(

)()(

Commutative Laws:

Associative Laws:

Proof of Associate Law for AND

교환

결합

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Fundamentals of Logic Design Chap. 2

Associative Laws for AND and OR

Figure 2-3: Associative Law for AND and OR

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Fundamentals of Logic Design Chap. 2

1 iff 1 ZYXXYZ

XZXYZYX )(

YZXYZXYZYZX

YZXYXZXYZXYXZX

YZYXXZXXZXYZXXZXYX

1)1(

1

)()())((

0 iff 0 ZYXZYX

))(( ZXYXYZX

2.5 Commutative, Associative, and Distributive Laws

Valid only Boolean algebra not for ordinary algebra

AND

OR

Proof

Distributive Laws:

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Fundamentals of Logic Design Chap. 2

2.6 Simplification Theorems

XYXYYYXYXYY 1)()')(('

XXYXXYXXYXX )(

XXYXXYXXYX 1)1(1

YXYXYXYYYX

XYXXXXYX

XYXYXXXYXY

' )'(

)(

)')(( '

Y

X

Y

XY’

Proof

Proof with Switch

Useful Theorems for

Simplification

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Fundamentals of Logic Design Chap. 2

ABBAAF )'(

===

Equivalent Gate Circuits

2.6 Simplification Theorems

Figure 2-4: Equivalent Gate Circuits

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Fundamentals of Logic Design Chap. 2

2.6 Simplification Theorems

Simplify (p. 43-44)

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Fundamentals of Logic Design Chap. 2

2.7 Multiplying Out and Factoring

EFCDBA )(

EDCBA ''

HDEFGABC '

'''' EACECDAB

To obtain a sum-of-product form Multiplying out using distributive laws

BCEBCDA

BCEBCDBCEDA

BCEBCDABCAEADAEDABCA

)1(

))((

Sum of product form:

Still considered to be in

sum of product form:

Not in Sum of product form:

Multiplying out and eliminating redundant terms

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Fundamentals of Logic Design Chap. 2

)'('

))((

EDCAB

FEDCBA

)'')(')('( ECAEDCBA

To obtain a product of sum form all sums are the sum of single variable

2.7 Multiplying Out and Factoring

Product of sum form:

Still considered to be in

product of sum form:

EFDCBA ))(( Not in Product of sum form :

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Fundamentals of Logic Design Chap. 2

2.7 Multiplying Out and Factoring

EXAMPLE 1: Factor A + B'CD. This is of the form X + YZ

where X = A, Y = B', and Z = CD, so

A + B'CD = (X + Y)(X + Z) = (A + B')(A + CD)

A + CD can be factored again using the second distributive law, so

A + B'CD = (A + B')(A + C)(A + D)

EXAMPLE 2: Factor AB' + C'D

Multiplying Out:곱셈전개,Factoring:인수화

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Fundamentals of Logic Design Chap. 2

2.7 Multiplying Out and Factoring

EXAMPLE 3: Factor C'D + C'E' + G'H

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Fundamentals of Logic Design Chap. 2

CD’E

+

AC'

E’

‘

C

E

D' +

A

E’

C’ +

A

B’

D’

EAB’C

D’

E

AB’C

Circuits for SOP and POS form

Sum of product form:

Product of sum form:

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Fundamentals of Logic Design Chap. 2

2.8 DeMorgan’s Laws

'''')'()'( 321321321 XXXXXXXXX

'...''')'...(

''...'')'...(

321321

321321

nn

nn

XXXXXXXX

XXXXXXXX

'')'(

'')'(

YXXY

YXYX

X Y X’ Y’ X + Y ( X + Y )’ X’ Y’ XY ( XY )’ X’ + Y’

0 0

0 1

1 0

1 1

1 1

1 0

0 1

0 0

0

1

1

1

1

0

0

0

1

0

0

0

0

0

0

1

1

1

1

0

1

1

1

0

Proof

DeMorgan’s Laws

DeMorgan’s Laws for n variables

Example

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Fundamentals of Logic Design Chap. 2

2.8 DeMorgan’s Laws

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Fundamentals of Logic Design Chap. 2

A B A’ B A B’ F = A’B+AB’ A’ B’ A B F’ = A’B’ + AB

0 0

0 1

1 0

1 1

0

1

0

0

0

0

1

0

0

1

1

0

1

0

0

0

0

0

0

1

1

0

0

1

CBACABCBACABCAB D )'()'( so ,')'(')''()''(

......)( ......)( XYZZYXZYXXYZ DD

ABBABBABABAA

BABAABBAABBAF

''''''

)')('()''()''()'''('

2.8 DeMorgan’s Laws

Dual: ‘dual’ is formed by replacing AND with OR, OR with AND, 0 with 1, 1 with 0

Inverse of A’B+AB’

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Fundamentals of Logic Design Chap. 2

LAWS AND THEOREMS (a)

Operations with 0 and 1:

1. X + 0 = X 1D. X • 1 = X

2. X +1 = 1 2D. X • 0 = 0

Idempotent laws:

3. X + X = X 3D. X • X = X

Involution law:

4. (X')' = X

Laws of complementarity:

5. X + X' = 1 5D. X • X' = 0

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Fundamentals of Logic Design Chap. 2

LAWS AND THEOREMS (b)

Commutative laws:

6. X + Y = Y + X 6D. XY = YX

Associative laws:

7. (X + Y) + Z = X + (Y + Z) 7D. (XY)Z = X(YZ) = XYZ

= X + Y + Z

Distributive laws:

8. X(Y + Z) = XY + XZ 8D. X + YZ = (X + Y)(X + Z)

Simplification theorems:

9. XY + XY' = X 9D. (X + Y)(X + Y') = X

10. X + XY = X 10D. X(X + Y) = X

11. (X + Y')Y = XY 11D. XY' + Y = X + Y

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Fundamentals of Logic Design Chap. 2

LAWS AND THEOREMS (c)

DeMorgan's laws:

12. (X + Y + Z +...)' = X'Y'Z'... 12D. (XYZ...)' = X' + Y' + Z' +...

Duality:

13. (X + Y + Z +...)D = XYZ... 13D. (XYZ...)D = X + Y + Z +...

Theorem for multiplying out and factoring:

14. (X + Y)(X' + Z) = XZ + X'Y 14D. XY + X'Z = (X + Z)(X' + Y)

Consensus theorem:

15. XY + YZ + X'Z = XY + X'Z 15D. (X + Y)(Y + Z)(X' + Z) = (X + Y)(X' + Z)