What is Boolean Algebra?
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Boolean Algebra is the
mathematic representation of operation of logic gates (and digital circuits).
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Boolean Algebra has
variables (input and output) and operators (AND, OR Complement).
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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.
SOP and POS expressions
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There are two types of
boolean expressions
◦ Sum of Products (SOP)
expressions
◦ Product of Sums (POS)
expressions
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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.
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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.
You may read more on SOP and POS expressions here https://faculty.etsu.edu/tarnoff/ntes2150/Ch6_v02.pdf
Rules and Laws of Boolean Algebra
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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.
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In Boolean algebra we
do only addition and multiplication.
Basic Rules
Basic Laws
Basic Duality
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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.
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The duality principle
ensures that "if we exchange every symbol by its dual in
a formula, we get the dual result".
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Everywhere we see 1,
change to 0.
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Everywhere we see 0,
change to 1.
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Similarly, + to .,
and . to +.
More examples:
0 . 1 = 0: is a true statement "false and true evaluates to false“
it’s a basic law.
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.
