Basic Properties Of Boolean Algebra

A Set of element in the Boolean System indicated by {B=a, b, c,…} has two binary operation AND (.), OR (+) and one unary operator NOT (´). The basic properties of the system are:

1.Both the operations are commutative,
* a+b=b+a
* a.b=b.a

2.Identity elements are there in both the operations,
* a+0=a
* a.1=a

3.Each operation is distributive with respect to the other.
* a+(b.c)=(a+b).(a+c)
* a.(b+c)=(a.b)+(a.c)

4.There exists a´ for each a∈b such that
* a+a´=1
* a.a´=0 

Example: Given the set {0,1} of two elements, where the elements have been denoted by the symbols 0 and 1 as is customary and they have no relation with the numbers 0 and 1 used in arithmetic. Let the two binary operations be denoted by + known as logical addition and (.) known as logical multiplication which have no relation to the operations of addition and multiplication used in arithmetic. In tables 1 and 2 are given the logical sum and logical products, i.e., the results of the above operations on the elements of the set.

 Proved that the set (0,1) with the operations defined in the tables is Boolean.

Solution: Both the operations are Boolean because of the following properties….

In view of these properties of the set {0, 1} and the definition of (+) and (.) as given by the tables 1 and 2, we conclude that it is Boolean

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Concept and Structure Of Boolean Algebra : Chapter 3

Boolean Algebra is a two valued algebra, applied earlier to statements and sets which were either true or false and now to switches which are either closed or open, i.e., ON or Off respectively. George Boole developed this branch of mathematics in his book. “An Investigation of the Laws Of Thought” now known as symbolic logic. This provided the basic logic for operations on binary numbers (1 and 0). Since modern business machines are based on binary system, the symbolic logic of George Boole was found extremely useful and is being considered as the base of Modern Mathematics.

Structure:

• Basic Properties
• Derived Properties
• Boolean Functions
• Canonical Form
• Electrical Switching System
• Boolean Multiplication
• Boolean Addition
• Circuits with Composite Operations

Objectives:
After studying this chapter, you should be able to understand

• Basic, Derived Properties Of Boolean Functions
• Boolean Multiplication and Addition
• Electrical Switching System and Circuits with Composite Operations

 

In the 19th century Symbolic Logic was invented, it was used much later when in the 20th century. Claude Shannon discovered the similarity of structure between it and telephone switching circuits. His paper “A Symbolic Analysis of Relay and Switching Circuits” made an important contribution to the use of Boolean algebra towards the designing of modern Business Machine based on Binary Number.

There are three basic operations in Boolean algebra AND, OR and NOT. These are symbolized by ∩, U and {} respectively in case of the theory of sets. In this chapter the more common symbolic plus+, dot ‘.’ and prime ( ′ ) would be used for the three operations respectively. The similarity would become obvious in the way the present chapter would synthesize and generalize what we have studied earlier and apply it to the end purpose of the designing of the electric circuits. Given below is a table showing the operations of symbolic logic to the three more or less corresponding system.

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