1. ECE 331 – Digital System Design
Boolean Algebra
and
Standard Forms of Boolean Expressions
(Lecture #4)
The slides included herein were taken from the materials accompanying
Fundamentals of Logic Design, 6th Edition, by Roth and Kinney,
and were used with permission from Cengage Learning.

2. Basic Laws and Theorems
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
Spring 2011
ECE Digital System Design

3. Basic Laws and Theorems
Commutative laws:
6. X + Y = Y + X D. 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 D. (X + Y)(X + Y') = X
10. X + XY = X D. X(X + Y) = X
11. (X + Y')Y = XY D. XY' + Y = X + Y
Spring 2011
ECE Digital System Design

4. Basic Laws and Theorems
DeMorgan's laws:
12. (X + Y + Z +...)' = X'Y'Z'... 12D. (XYZ...)' = X' + Y' + Z' +...
Duality:
13. (X + Y + Z +...)D = XYZ D. (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)
Spring 2011
ECE Digital System Design

5. ECE 331 - Digital System Design
Duality (13)
The dual of a Boolean expression can be written by
Replacing AND with OR, and OR with AND
Replacing 0 with 1, and 1 with 0
Leaving literals unchanged
See the Boolean laws and theorems, previously discussed, for examples of Boolean expressions and their duals.
Spring 2011
ECE Digital System Design

6. Distributive Law: Example #1
Distributive law (8): X.(Y + Z) = X.Y + X.Z
Use the distributive law to multiply out the following Boolean expression:
F = (A+B).(C+D).(E+F)
Spring 2011
ECE Digital System Design

7. Distributive Law: Example #2
Distributive law (8D): X + Y.Z = (X+Y).(X+Z)
Use the distributive law to factor the following Boolean expression:
F = A.B + C.D
Spring 2011
ECE Digital System Design

8. Simplification Theorems: Example #1
X.Y + X.Y' = X (X+Y).(X+Y') = X
X + X.Y = X X.(X+Y) = X
(X+Y').Y = X.Y X.Y' + Y = X+Y
Use the simplification theorems to simplify the following Boolean expression:
F = ABC' + AB'C' + A'BC'
Spring 2011
ECE Digital System Design

9. Simplification Theorems: Example #2
X.Y + X.Y' = X (X+Y).(X+Y') = X
X + X.Y = X X.(X+Y) = X
(X+Y').Y = X.Y X.Y' + Y = X+Y
Use the simplification theorems to simplify the following Boolean expression:
F = (A'+B'+C').(A+B'+C').(B'+C)
Spring 2011
ECE Digital System Design

10. Simplification Theorems: Example #3
X.Y + X.Y' = X (X+Y).(X+Y') = X
X + X.Y = X X.(X+Y) = X
(X+Y').Y = X.Y X.Y' + Y = X+Y
Use the simplification theorems to simplify the following Boolean expression:
F = AB'CD'E + ACD + ACF'GH' +ABCD'E +ACDE' + E'H'
(See Programmed Exercise 3.4 on page 75)
Spring 2011
ECE Digital System Design

11. Consensus Theorem: Example #1
(15) X.Y + Y.Z + X'.Z = X.Y + X'.Z
(15D) (X+Y).(Y+Z).(X'+Z) = (X+Y).(X'+Z)
Use the consensus theorem to simplify the following Boolean expression:
F = ABC + BCD + A'CD + B'C'D'
Spring 2011
ECE Digital System Design

12. Consensus Theorem: Example #2
(15) X.Y + Y.Z + X'.Z = X.Y + X'.Z
(15D) (X+Y).(Y+Z).(X'+Z) = (X+Y).(X'+Z)
Use the consensus theorem to simplify the following Boolean expression:
F = (A+C+D')(A+B'+D)(B+C+D)(A+B'+C)
Spring 2011
ECE Digital System Design

13. Consensus Theorem: Example #3
(15) X.Y + Y.Z + X'.Z = X.Y + X'.Z
(15D) (X+Y).(Y+Z).(X'+Z) = (X+Y).(X'+Z)
Use the consensus theorem to simplify the following Boolean expression:
F = AC' + AB'D + A'B'C + A'CD' + B'C'D'
(See Programmed Exercise 3.5 on page 77)
Spring 2011
ECE Digital System Design

14. ECE 331 - Digital System Design
DeMorgan's Law
DeMorgan's Law:
(12) (X + Y + Z + … )' = X'.Y'.Z'...
(12D) (X.Y.Z… )' = X' +Y' + Z' …
Prove (using a truth table): (X+Y)' = X'.Y' X Y
X + Y
(X + Y)' X' Y'
X'.Y' 1
Spring 2011
ECE Digital System Design

15. ECE 331 - Digital System Design
DeMorgan's Law
Graphical representation of DeMorgan's Law x y
(X.Y)'
X' + Y'
(X+Y)'
X'.Y'
Spring 2011
ECE Digital System Design

16. DeMorgan's Law: Example
(12) (X + Y + Z + … )' = X'.Y'.Z'...
(12D) (X.Y.Z… )' = X' +Y' + Z' …
Find the complement of the following Boolean expression using DeMorgan's law:
F = (A + (BC)').((AD)' + C.(B' + D))
Spring 2011
ECE Digital System Design

17. Simplifying Boolean Expressions
Boolean algebra can be used in several ways to simplify a Boolean expression:
Combine terms
Eliminate redundant or consensus terms
Eliminate redundant literals
Add redundant terms to be combined with or allow the elimination of other terms
Spring 2011
ECE Digital System Design

18. Equivalency of Boolean Expressions
Two Boolean expressions are equivalent iff both expressions evaluate to the same value for all combinations of the variables in the expressions.
The equivalency can be proven using
A Truth table
Boolean algebra theorems to manipulate one expression until it is identical to the other.
Boolean algebra theorems to reduce both expressions independently to the same expression.
Spring 2011
ECE Digital System Design

19. Importance of Boolean Algebra
Boolean algebra is used to simplify Boolean expressions.
Simpler expressions leads to simpler logic circuits.
Reduces cost
Reduces area requirements
Reduces power consumption
The objective of the digital circuit designer is to design and realize optimal digital circuits.
Thus, Boolean algebra is an important tool to the digital circuit designer.
Spring 2011
ECE Digital System Design

20. Problem with Boolean Algebra
In general, there is no easy way to determine when a Boolean expression has been simplified to a minimum number of terms or a minimum number of literals.
Karnaugh Maps provide a better mechanism for the simplification of Boolean expressions.
Spring 2011
ECE Digital System Design

21. Circuit Design: Example
For the following Boolean expression:
F(A,B,C) = A.B.C + A'.B.C + A.B'.C + A.B.C'
1. Draw the circuit diagram
2. Simplify using Boolean algebra
3. Draw the simplified circuit diagram
Spring 2011
ECE Digital System Design

22. Standard Forms of Boolean Expressions
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ECE Digital System Design

23. ECE 331 - Digital System Design
Standard Forms
There are two standard forms in which all Boolean expressions can be written:
1. Sum of Products (SOP)
2. Product of Sums (POS)
Spring 2011
ECE Digital System Design

24. ECE 331 - Digital System Design
Sum of Products (SOP)
Product Term
Logical product = AND operation
A product term is the ANDing of literals
Examples: A.B, A'.B.C, A.C', B.C'.D', A.B.C.D
“Sum of”
Logical sum = OR operation
The sum of products is the ORing of product terms.
Spring 2011
ECE Digital System Design

25. ECE 331 - Digital System Design
Sum of Products (SOP)
The distributive laws are used to multiply out a general Boolean expression to obtain the sum of products (SOP) form.
The distributive laws are also used to convert a Boolean expression in POS form to one in SOP form.
A SOP expression is realized using a set of AND gates (one for each product term) driving a single OR gate (for the sum).
Spring 2011
ECE Digital System Design

26. ECE 331 - Digital System Design
Product of Sums (POS)
Sum Term
Logical sum = OR operation
A sum term is the ORing of literals
Examples: A+B, A'+B+C, A+C', B+C'+D'
“Product of”
Logical product = AND operation
The product of sums is the ANDing of sum terms.
Spring 2011
ECE Digital System Design

27. ECE 331 - Digital System Design
Product of Sums (POS)
The distributive laws are used to factor a general Boolean expression to obtain the product of sums (POS) form.
The distributive laws are also used to convert a Boolean expression in SOP form to one in POS form.
A POS expression is realized using a set of OR gates (one for each sum term) driving a single AND gate (for the product).
Spring 2011
ECE Digital System Design

28. ECE 331 - Digital System Design
SOP and POS: Examples
For each of the following Boolean expressions, identify whether it is in SOP or POS form:
1. F(A,B,C) = (A+B).(A'+B'+C').(B+C')
2. F(A,B,C) = A.B.C + B'.C' + A.C' + A'.B.C'
3. F(A,B,C) = A + B.C + B'.C' + A'.B'.C
4. F(A,B,C) = (A'+B'+C).(B+C').(A+C').(B')
5. F(A,B,C) = A.B.C + A'.(B+C) + (A+C').B
6. F(A,B,C) = A + B + C
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ECE Digital System Design

29. ECE 331 - Digital System Design
Questions?
Spring 2011
ECE Digital System Design