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