Monday, December 3, 2018

Boolean Algebra (Basic Laws, POS and SOP expressions)




What is Boolean Algebra?

}  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.

SOP and POS expressions

}  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.

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

}  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.

Basic Rules

Basic Laws


Basic Duality

}  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 +.

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.

De-Morgan’s Law

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;

 

    Replace all values in the expression by its opposite.

    Replace all “+” with “.” and vise versa.




1 comment:

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