1. Digital Logic Design ESGD2201
University of Palestine
Faculty of Engineering and Urban planning
Software Engineering Department
Digital Logic Design ESGD2201
Lecture 5
Boolean Algebra and Logic Simplification
Eng. Mohammed Timraz
Electronics & Communication Engineer
Sunday, 18th June 2008

2. Agenda Boolean Algebra and Logic Simplification
1. Boolean Operations and Expressions.
2. Laws and Rules of Boolean Algebra.
3. DeMorgan’s Theorems.
4. Boolean Analysis of Logic Circuits.
5. Simplification Using Boolean Algebra.
6. Standard Forms of Boolean Expressions.
7. Boolean Expressions and Truth Tables.

3. Boolean Algebra and Logic Simplification
1. Boolean Operations and Expressions.
Boolean algebra is the mathematics of digital systems.
A basic knowledge of Boolean algebra is indispensable to the study and analysis of logic circuits.
Boolean operations and expressions in terms of their relationship to NOT, AND, OR, NAND and NOR gates were introduce.

4. Boolean Algebra and Logic Simplification
1. Boolean Operations and Expressions.
A variable is a symbol used to represent a logical quantity (usually an italic uppercase letter).
The Complement is the inverse of a variable and is indicated by a bar over the variable.
For example:
A is a variable.
Ā is the complement.

5. Boolean Algebra and Logic Simplification
1. Boolean Operations and Expressions.
For example:
If a variable A = 1,
Then Ā = 1. (is the complement).
The complement of variable A is read as “A not” or “A bar”.
Sometimes a prime symbol rather than an overbar is used to denote the complement of a variable,
B’ indicates the complement of B, in this course only the overbar is used.

6. Boolean Algebra and Logic Simplification
Boolean Addition.
The Boolean addition is equivalent to the OR operation and the basic rules as follows:
= 0
= 1
= 1
+ 1 = 1
In Boolean Algebra, a Sum Term is a sum of literals.
In logic circuits, a sum term is produced by an OR operation with no AND operations involved.

7. Boolean Algebra and Logic Simplification
Boolean Addition.
Some examples of sum terms are:
A sum terms is equal to 1 when one or more of the literals in the terms are 1.
A sum terms is equal to 0 if and only if each of the literals is 0.

8. Boolean Algebra and Logic Simplification
Boolean Addition.
Examples:
Determine the values of A, B, C and D which make the sum term Equal to zero.
Solution:
For the sum terms to be 0, each of the literals in the terms must be 0.
There for, A = 0, B = 1 so that , C = 0 , and D = 1 so that
Then =

9. Boolean Algebra and Logic Simplification
Boolean Multiplication.
The Boolean addition is equivalent to the AND operation and the basic rules as follows:
= 0
= 0
= 0
= 1
In Boolean Algebra, a Product Term is a sum of literals.
In logic circuits, a product term is produced by an AND operation with no OR operations involved.

10. Boolean Algebra and Logic Simplification
Boolean Multiplication.
Some examples of product terms are:
A product terms is equal to 1 if each of the literals in the term is 1.
A product terms is equal to 0 when one or more of the literals are 0.

11. Boolean Algebra and Logic Simplification
Boolean Multiplication.
Examples:
Determine the values of A, B, C and D which make the sum term Equal to 1.
Solution:
For the sum terms to be 1, each of the literals in the terms must be 1.
There for, A = 1, B = 0 so that , C = 1 , and D = 0 so that
Then =

12. Boolean Algebra and Logic Simplification
2. Laws and Rules of Boolean Algebra.
The basic laws of Boolean algebra are:
1- Commutative Laws.
2- Associative Laws.
3- Distributive Laws.

13. Boolean Algebra and Logic Simplification
Laws of Boolean Algebra.
1- Commutative Laws.
The commutative law of addition for two variables is written algebraically as:
A + B = B + A

14. Boolean Algebra and Logic Simplification
Laws of Boolean Algebra.
1- Commutative Laws.
Application of commutative law of addition. A B
A + B B A
B + A ≡
A + B = B + A

15. Boolean Algebra and Logic Simplification
Laws of Boolean Algebra.
1- Commutative Laws.
The commutative law of multiplication for two variables is written algebraically as:
AB = BA

16. Boolean Algebra and Logic Simplification
Laws of Boolean Algebra.
1- Commutative Laws.
Application of commutative law of multiplication. A B AB B A BA ≡
AB = BA

17. Boolean Algebra and Logic Simplification
Laws of Boolean Algebra.
2- Associative Laws.
The commutative law of addition for Three variables is written algebraically as:
A + ( B + C ) = ( A + B ) + C

18. Boolean Algebra and Logic Simplification
Laws of Boolean Algebra.
2- Associative Laws.
Application of Associative law of addition. A A B
A + ( B + C ) C
A + ( B + C ) B ≡ C
A + ( B + C ) = ( A + B ) + C

19. Boolean Algebra and Logic Simplification
Laws of Boolean Algebra.
2- Associative Laws.
The commutative law of multiplication for three variables is written algebraically as:
A ( BC ) = ( AB ) C

20. Boolean Algebra and Logic Simplification
Laws of Boolean Algebra.
2- Associative Laws.
Application of Associative law of multiplication. A B C
( AB )C ≡
( BC )
A( BC )
( AB )
A ( BC ) = ( AB ) C

21. Boolean Algebra and Logic Simplification
Laws of Boolean Algebra.
3- Distributive Laws.
The distributive law is written algebraically for three variables as follows:
A ( B + C ) = AB + AC

22. Boolean Algebra and Logic Simplification
Laws of Boolean Algebra.
3- Distributive Laws.
Application of Distributive law. A B C ≡
A ( B + C )
( B + C )
AB + AC
A ( B + C ) = AB + AC

23. Boolean Algebra and Logic Simplification
Laws of Boolean Algebra.
2- Associative Laws.
Application of Associative law of multiplication. A
A( BC ) A
( AB ) B
( AB )C B
( BC ) ≡ C C
A ( BC ) = ( AB ) C

24. Boolean Algebra and Logic Simplification
Rules of Boolean Algebra.
There are 12 basic rules that are useful in manipulating and simplifying Boolean expressions.
Rules 1 through 9 will be viewed in terms of their application to logic gates.
Rules 10 through 12 will be derived in terms of the simpler rules and laws previously discussed.

25. Boolean Algebra and Logic Simplification
Rules of Boolean Algebra.
Table of Basic Rules of Boolean algebra: 1.
A + 0 = A 2.
A + 1 = 1 3.
A . 0 = 0 4.
A . 1 = A

26. Boolean Algebra and Logic Simplification
Rules of Boolean Algebra.
Table of Basic Rules of Boolean algebra: 5.
A + A = A 6.
A + Ā = 1 7.
A . A = A 8.
A . Ā = 0 9.
= A

27. Boolean Algebra and Logic Simplification
Rules of Boolean Algebra.
Table of Basic Rules of Boolean algebra:
10.
A + AB = A
11.
A + ĀB = A + B
12.
(A + B) (A + C)= A + BC