1. BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION Part (a)
Chapter 4
BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION
Part (a)

2. Boolean Operations and Expression:
Boolean algebra is the mathematics of digital systems. A basic knowledge of Boolean algebra is indispensable to the study and analysis of logic circuits.
Terms used in Boolean Algebra:
A variable is a symbol (usually an italic uppercase letter) used to represent a logical quantity. Any single variable can have a 1 or 0 value.
The complement is the inverse of a variable and is indicated by a bar over the variable (overbar)

3. Law of Boolean Algebra The basic law of Boolean Algebra:
The commutative laws
The associative laws
The distributive law
Each of the laws is illustrated with two or three variables, but the number of variables is not limited to this.
for addition and multiplication are the same as in ordinary algebra.

4. Commutative Laws of addition
The commutative law of addition for two variables is written as
A + B = B + A
This law states that the order in which the variables are ORed makes no difference.
This law is applied to the OR gate.

5. Commutative law of multiplication
The commutative law of multiplication for two variables is
AB=BA
This law states that the order in which the variables are ANDed make no difference.
This law as applied to the AND gate.

6. Associative Laws
The associative law of addition is written as follows for three variables:
A + (B+C) = (A+B)+C
This law states that when OR more than two variables, the result is the same regardless of the grouping of the variables.

7. Associative law of multiplication
The associative law of multiplication is written as follows for three variables:
A(BC) = (AB)C
This law states that it makes no difference in what order the variables are grouped when ANDing more two variables.

8. Distributive Laws
The distributive law is written for three variables as follows:
A(B+C) = AB + AC
This law states that ORing two or more variables and then ANDing the result with a single variable is equivalent to ANDing the single variable with each of the two or more variables and then ORing the products.
The distributive law also expresses the process of factoring in which the common variable A is factored out of the product terms.

9. Rules of Boolean Algebra

22. Simplify the Boolean Expression using Rules of Boolean Algebra
a) AB(C+B)
b) AB + ABC
Exercise 1

23. DeMorgan’s Theorems DeMorgan’s First Theorem
The complement of a product of variable is equal to the sum of the complements of the variable
AB = A + B
DeMorgan’s Theorems

24. DeMorgan’s Theorems DeMorgan’s Second Theorem
The complement of a sum of variables is equal to the product of the compliments of the variable
A+B = A B
DeMorgan’s Theorems

25. DeMorgan’s Theorems Example
Apply DeMorgan’s Theorem to the expression (A+B+C)D
Let A+B+C = X and D = Y. So its equal to XY = X + Y and can be rewrite as
(A+B+C)D = A + B + C + D
Next, apply DeMorgan’s Theorem to term A + B + C
A + B + C + D = A B C + D
DeMorgan’s Theorems

26. Exercise 2

27. SOP and POS Form
Boolean expressions can be written in the sum-of-products form (SOP) or in the product-of-sums form (POS). These forms can simplify the implementation of combinational logic, particularly with PLDs. In both forms, an overbar cannot extend over more than one variable.
An expression is in SOP form when two or more product terms are summed as in the following examples:
A B C + A B A B C + C D C D + E
An expression is in POS form when two or more sum terms are multiplied as in the following examples:
(A + B)(A + C) (A + B + C)(B + D) (A + B)C

28. Standard expression
Non Standard expression

30. EXAMPLE 1
Convert following Boolean Equation into standard SOP form

31. Binary Representation of SOP
Binary Representation of SOP

32. The Product of Sums Form (POS)
A sum term is a term consisting of the sum (Boolean addition) of literals
(variable or their complements).
When two or more sum terms are multiplied, the resulting expression is a product
of sum (POS).
Example:
The Product of Sums Form (POS)

33. POS Standard Form

35. EXAMPLE 2
Convert following Boolean Equation into standard POS form

36. Binary Representation of POS
Binary Representation of POS

37. Converting Standard SOP to Standard POS

38. Example

39. Converting SOP Expression to Truth Table Format
Converting SOP Expression to Truth Table Format

40. Converting SOP Expression to Truth Table Format

41. Exercise 3