1. ECE 331 – Digital System Design
Boolean Algebra
(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. ECE 331 - Digital System Design
Converting POS to SOP
Given an expression in product-of-sums (POS) form, the corresponding sum-of-products (SOP) expression can be obtained by multiplying out, using the two distributive laws:
X(Y + Z) = XY + XZ (3-1)
(X + Y)(X + Z) = X + YZ (3-2)
In addition, the following theorem is very useful for factoring and multiplying out:
(X + Y)(X′ + Z) = XZ + X′Y (3-3)
Fall 2010
ECE Digital System Design

3. Boolean Algebra Exercise:
Convert the following Boolean Expression in POS form to a Boolean Expression in SOP form.
(Hint: Multiply out the expression)
F = (A'+D')(B+C')(C+D+E')
Fall 2010
ECE Digital System Design

4. Boolean Algebra Exercise:
Convert the following Boolean Expression in POS form to a Boolean Expression in SOP form.
(Hint: Multiply out the expression)
F = (A+B+C')(A+B+D)(A+B+E)(A+D'+E)(A'+C)
Fall 2010
ECE Digital System Design

5. ECE 331 - Digital System Design
Boolean Algebra
If we were to multiply out by brute force, we would generate 162 terms, and 158 of these terms would then have to be eliminated to simplify the expression.
Instead, we will use the distributive laws to simplify the process.
Fall 2010
ECE Digital System Design

6. ECE 331 - Digital System Design
Converting SOP to POS
The same theorems that are useful for multiplying out expressions are also useful for factoring.
By repeatedly applying (3-1), (3-2), and (3-3), an expression in sum-of-products (SOP) form can be converted to an expression in product-of-sums (POS) form.
X(Y + Z) = XY + XZ (3-1)
(X + Y)(X + Z) = X + YZ (3-2)
(X + Y)(X′ + Z) = XZ + X′Y (3-3)
Fall 2010
ECE Digital System Design

7. Boolean Algebra Exercise:
Convert the following Boolean Expression in SOP form to a Boolean Expression in POS form.
(Hint: Factor the expression)
F = ACE + BCE + ADE + BDE + ACF + BCF + ADF + BDF
Fall 2010
ECE Digital System Design

8. Boolean Algebra Exercise:
Convert the following Boolean Expression in SOP form to a Boolean Expression in POS form.
(Hint: Factor the expression)
F = AC + A'BD' + A'BE + A'C'DE
Fall 2010
ECE Digital System Design

9. ECE 331 - Digital System Design
Boolean Algebra
Fall 2010
ECE Digital System Design

10. Exclusive-OR and Equivalence
The following theorems of Boolean Algebra apply to Exclusive-OR and Equivalence (Exclusive-NOR):
Fall 2010
ECE Digital System Design

11. Exercise: Simplify the following Boolean Expression.
Boolean Algebra
Exercise:
Simplify the following Boolean Expression.
Fall 2010
ECE Digital System Design

12. Exercise: Simplify the following Boolean Expression.
Boolean Algebra
Exercise:
Simplify the following Boolean Expression.
Fall 2010
ECE Digital System Design

13. Boolean Algebra Exercise: Simplify the following Boolean Expression.
F(A,B,C) = (A xor B)' xor C
Fall 2010
ECE Digital System Design

14. ECE 331 - Digital System Design
Consensus Theorem
The consensus theorem can be stated as follows:
XY + X'Z + YZ = XY + X'Z (3-20)
Dual Form:
(X + Y)(X’ + Z)(Y + Z) = (X + Y)(X’ + Z) (3-21)
Consensus theorem proof:
XY + X'Z + YZ = XY + X'Z + (X + X')YZ
= (XY + XYZ) + (X'Z + X'YZ)
= XY(1 + Z) + X'Z(1 + Y) = XY + X'Z
Fall 2010
ECE Digital System Design

15. Simplifying Boolean Expressions
1. Combining terms. Use the theorem XY + XY′ = X to combine two terms.
2. Eliminating terms. Use the theorem X + XY = X to eliminate redundant terms if possible; then try to apply the consensus theorem (XY + X′Z + YZ = XY + X′Z) to eliminate any consensus terms.
3. Eliminating literals. Use the theorem X + X’Y = X + Y to eliminate redundant literals. Simple factoring may be necessary before the theorem is applied.
4. Adding redundant terms. Redundant terms can be introduced in several ways such as adding xx′, multiplying by (x + x′), adding yz to xy + x′z, or adding xy to x. When possible, the added terms should be chosen so that they will combine with or eliminate other terms.
Fall 2010
ECE Digital System Design

16. ECE 331 - Digital System Design
Boolean Algebra
Example:
Fall 2010
ECE Digital System Design

17. Boolean Algebra
Simplify the following Boolean expression using Boolean algebra:
F(A,B,C,D) = A'BC'D' + A'B'C'D + A'BC'D + ABC'D + A'BCD + ABCD
Fall 2010
ECE Digital System Design

18. Proving Validity of an Equation
Often we will need to determine if an equation is valid for all combinations of values of the variables. Several methods can be used to determine if an equation is valid:
Construct a truth table and evaluate both sides of the equation for all combinations of values of the variables. (This method is rather tedious if the number of variables is large, and it certainly is not very elegant.)
Manipulate one side of the equation by applying various theorems until it is identical with the other side.
Reduce both sides of the equation independently to the same expression.
4. It is permissible to perform the same operation on both sides of the equation provided that the operation is reversible.
Fall 2010
ECE Digital System Design

19. Boolean Algebra Example: Show that
A'BD' + BCD + ABC' + AB'D = BC'D' + AD + A'BC
Solution: Starting with the left side,
Fall 2010
ECE Digital System Design

20. Importance of Boolean Algebra
Boolean Algebra is used to simplify Boolean expressions.
Through application of the Laws and Theorems discussed
Simpler expressions lead to simpler circuit realization, which, generally, reduces cost, area requirements, and power consumption.
The objective of the digital circuit designer is to design and realize optimal digital circuits.
However, in general, there is no easy way to determine when a Boolean expression has been simplified to a minimum number of terms or minimum number of literals.
Fall 2010
ECE Digital System Design

21. Boolean Algebra Exercise: For the following Boolean expression
F(A,B,C) = (A+B+C).(A'+B+C).(A+B'+C).(A+B+C')
1. Using Boolean algebra, simplify the Boolean expression.
2. Derive the Truth table for the simplified expression.
3. From the Truth table, determine an equivalent Boolean
expression.
4. Draw the circuit diagram for the “cheapest” expression.
Fall 2010
ECE Digital System Design

22. ECE 331 - Digital System Design
Questions?
Fall 2010
ECE Digital System Design