1. CHAPTER 2 Boolean Algebra This chapter in the book includes:
Objectives
Study Guide
2.1 Introduction
Basic Operation
2.3 Boolean Expression and Truth Table
2.4 Basic Theorem
2.5 Commutative, Associative and Distributive Laws
Simplification Theorem
Multiplying Out and Factoring
DeMorgan’s Laws
Problems
Laws and Theorems of Boolean Algebra
Fundamentals of Logic Design Chap. 2

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
Fundamentals of Logic Design Chap. 2

3. 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)
Fundamentals of Logic Design Chap. 2

4. 2.2 Basic Operations NOT(Inverter) Gate Symbol
Fundamentals of Logic Design Chap. 2

5. 2.2 Basic Operations AND Truth Table Gate Symbol
Fundamentals of Logic Design Chap. 2

6. 2.2 Basic Operations OR Truth Table Gate Symbol
Fundamentals of Logic Design Chap. 2

7. 2.2 Basic Operations Apply to Switch AND OR
Fundamentals of Logic Design Chap. 2

8. 2.3 Boolean Expressions and Truth Tables
Logic Expression :
Circuit of logic gates :
Fundamentals of Logic Design Chap. 2

9. 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
10 literals
Fundamentals of Logic Design Chap. 2

10. 2.3 Boolean Expressions and Truth Tables
2-Input Circuit and Truth Table
A B A’
F = A’ + B
0 0
0 1
1 0
1 1 1
Fundamentals of Logic Design Chap. 2

11. 2.3 Boolean Expressions and Truth Tables
Proof using Truth Table
n times
n variable needs rows
TABLE 2.1
Fundamentals of Logic Design Chap. 2

12. 2.4 Basic Theorems Operations with 0, 1 Idempotent Laws
Involution Laws
Complementary Laws
Proof
Example
Fundamentals of Logic Design Chap. 2

13. 2.4 Basic Theorems with Switch Circuits
Fundamentals of Logic Design Chap. 2

14. 2.4 Basic Theorems with Switch Circuits
Fundamentals of Logic Design Chap. 2

15. 2.4 Basic Theorems with Switch Circuits
Fundamentals of Logic Design Chap. 2

16. 2.5 Commutative, Associative, and Distributive Laws
Commutative Laws:
Associative Laws:
Proof of Associate Law for AND
Fundamentals of Logic Design Chap. 2

17. Associative Laws for AND and OR
Figure 2-3: Associative Law for AND and OR
Fundamentals of Logic Design Chap. 2

18. 2.5 Commutative, Associative, and Distributive LawsOR
Distributive Laws:
Valid only Boolean algebra not for ordinary algebra
Proof
Fundamentals of Logic Design Chap. 2

19. 2.6 Simplification Theorems
Useful Theorems for
Simplification
Proof
Proof with Switch
Fundamentals of Logic Design Chap. 2

20. 2.6 Simplification Theorems
Equivalent Gate Circuits
===
Figure 2-4: Equivalent Gate Circuits
Fundamentals of Logic Design Chap. 2

21. 2.6 Simplification Theorems
Simplify (p )
Fundamentals of Logic Design Chap. 2

22. 2.7 Multiplying Out and Factoring
To obtain a sum-of-product form  Multiplying out using distributive laws
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
Fundamentals of Logic Design Chap. 2

23. 2.7 Multiplying Out and Factoring
To obtain a product of sum form  all sums are the sum of single variable
Product of sum form:
Still considered to be in
product of sum form:
Not in Product of sum form :
Fundamentals of Logic Design Chap. 2

24. 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
Fundamentals of Logic Design Chap. 2

25. 2.7 Multiplying Out and Factoring
EXAMPLE 3: Factor C'D + C'E' + G'H
Fundamentals of Logic Design Chap. 2

26. Circuits for SOP and POS form
Sum of product form:
Product of sum form:
Fundamentals of Logic Design Chap. 2

27. 2.8 DeMorgan’s Laws DeMorgan’s Laws Proof
X Y
X’ Y’
X + Y
( X + Y )’ XY
( XY )’
X’ + Y’
0 0
0 1
1 0
1 1 1
DeMorgan’s Laws for n variables
Example
Fundamentals of Logic Design Chap. 2

28. 2.8 DeMorgan’s Laws
Fundamentals of Logic Design Chap. 2

29. 2.8 DeMorgan’s Laws Inverse of A’B+AB’
0 0
0 1
1 0
1 1 1
Dual: ‘dual’ is formed by replacing AND with OR, OR with AND, 0 with 1, 1 with 0
Fundamentals of Logic Design Chap. 2

30. 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
Fundamentals of Logic Design Chap. 2

31. 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
Fundamentals of Logic Design Chap. 2

32. 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)
Fundamentals of Logic Design Chap. 2