Boolean Algebra - mbits-mirafra/digitalDesignCourse GitHub Wiki

What is boolean Algebra?

• Boolean algebra is a mathematical system for manipulating variables that can have one of two values.

     In formal logic, these values are “true” and “false”
     In digital systems, these values are “on”/“off,” “high”/“low,” or “1”/”0”.
     Common Boolean operators include AND, OR, and NOT.
  

Example for common boolean operators:

• The function of Boolean operator can be completely described using a Truth Table

The truth tables of the Boolean operator AND and OR gate are shown below

x	y	xy
0	0	0
0	1	0
1	0	0
1	1	1

x	y	x+y
0	0	0
0	1	1
1	0	1
1	1	1

• The AND operator is also known as the Boolean product “ . ”. The OR operator is the Boolean sum “ + ”.
• The NOT operation is most often designated by an overbar “ ‾ ” or " ' ".

A Boolean function has:

     • At least one Boolean variable,

     • At least one Boolean operator, and

     • At least one input from the set of {0,1}

Why is it used?

Digital circuit designer always like achieve the following goals:

• Cheaper to produce
• Consume less power
• run faster
• smaller circuit

How to do it?

• Computers contain circuits that implement Boolean functions ->Boolean functions can express circuits
• If one can simplify a Boolean function, that express a circuit, then it is possible to archive the above goals

When is it used?

Boolean algebra is used whenever there is a need to express and manipulate logical expressions and binary variables.

1. Digital circuit design:

Essential for building digital devices such as computers, calculators, and other electronic devices.

2. Computer programming:

This is particularly useful in control structures such as if statements, while loops, and for loops.

3. Artificial intelligence:

Boolean algebra is used in artificial intelligence to represent and manipulate logical statements and predicates.

Rules of Boolean Algebra

Sr.no	TOPICS
1	Null Rule
2	Identity Rule
3	Complement Rule
4	Involution Rule

Laws of Boolean Algebra

Sr.no	TOPICS
1	Commutative Law
2	Associative Law
3	Distributive Law
4	Idempotence Law
5	Redundance Law
6	De-Morgans Theorem
7	Consensus Theorem

Precedence Rule

Like arithmetic, Boolean operations follow the rules of precedence (priority):

 NOT operator > AND operator > OR operator

Canonical Form

There are 1+ ways of stating the same Boolean expression. Example:

In order to eliminate the confusion, designers express Boolean express in unified and standardized form, called canonical form. There are two canonical forms for Boolean expressions: sum-of-products and product-of-sums.

• In the sum-of-products form, ANDed variables are ORed together

           Example:  F(x, y, z)=xy + yz + xz
  

• In the product-of-sums form, ORed variables are ANDed together

           Example:  F(x, y, z)=(x+y) * (y+z) * (x+z)