Introduction to Boolean algebra, Sum of Products and Product of Sums expressions (SOP and POS expressions), basic laws of Boolean algebra, De-Morgan's law.

1. Chapter: Fundamentals
Lesson: Boolean Algebra
Lecturer: Susantha Herath PGD in IT (MBCS), PGD in Marketing (Uni. Of Kln)
Lecture 04
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2.  Boolean Algebra is the mathematic
representation of operation of logic gates
(and digital circuits).
 Boolean Algebra has variables (input and
output) and operators (AND, OR
Complement).
 Operation of a logic gate (and digital circuit)
can be represented using a Boolean function.
Truth table is used to illustrate the results of
a Boolean function.

3.  There are two types of boolean expressions
◦ Sum of Products (SOP) expressions
◦ Product of Sums (POS) expressions
 SOP expression
◦ A product term is produced when one or more boolean
variables are logically multiplied. It is also called
minterm. When two or more product terms are logically
added a SOP expression is formed.
 POS expression
◦ A sum term is produced when one or more boolean
variables are logically added. This is also called maxtern.
When two or more sum terms are logically multiplied a
POS expression is formed.

4.  In Boolean algebra a variable can have only
either 1 (TRUE) or 0 (FALSE) values. In Boolean
algebra 1 is stands to represent the state of
TRUE, not integer value 1. Also 0 stands to
represent the state of FALSE, not the integer
value 0. Therefore addition and multiplication
works differently than we normally do in
mathematics.
 In Boolean algebra we do only addition and
multiplication.

5. Multiplication (“dot”
used to represent
multiplication).
0.0 = 0
0.1 = 0
1.0 = 0
1.1 = 1
Addition
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1
Negation
0̅ = 1
1̅ = 0

7.  According to basic laws of Boolean algebra
there is an important feature called “Basic
Duality”. It says that every boolean function
has a dual function.
 The duality principle ensures that "if we
exchange every symbol by its dual in a
formula, we get the dual result".
 Everywhere we see 1, change to 0.
 Everywhere we see 0, change to 1.
 Similarly, + to ., and . to +.

8. More examples:
0 . 1 = 0: is a true statement "false and true evaluates
to false“ it’s a basic law.
Now lets replace all values and operations by its
opposite value.
1 + 0 = 1: is the dual of (a): it is a true statement that
"true or false evaluates true.“ it is also a basic law.
Like this, in every formula, if we replace every value
and operation by its opposite, including the result, we
get a valid formula.

9. This is a very useful law in boolean algebra. This allows
to get the complement of a boolean expression. This
represent the basic duality of boolean algebra and
mostly used when we design circuits using NAND and
NOR gates.
(x + y)’ = x’.y’
(x.y)’ = x’ + y’
To get the complement of a boolean expression, do the
following two steps;
1. Replace all values in the expression by its opposite.
2. Replace all “+” with “.” and vise versa.