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George Boole and Boolean Algebra

George Boole the inventor of Boolean Algebra was an Mathematician who worked in field of differential equation and algebraic logic. He was born on 2nd of November in year 1814. He was British and was born in Lincolnshire, England; he was Unitarian by religion.


Let us read What is Boolean Algebra?:

Boolean Algebra is used to analyze and simplify the digital (logic) circuits. It uses binary numbers only that is 0 and 1. Boolean Algebra was introduced by George Boole in year 1854.


Some rules of Boolean Algebra:

There are only two values that is 1 for HIGH and 0 for LOW.
If a complement have an over bar (-) then it is converted into another i.e., from HIGH (1) to LOW (0) and vice-versa.


If B(0verbar) = 1 then its value is 0

If B(0verbar) = 0 then its value is 1


Let us know more about Boolean Laws:

Boolean Laws are of six types.
1. Commutative Law
2. Associative Law
3. Distributive Law
4. AND Law
5. OR Law
6. Inversion Law


We will discuss each of them  more about them briefly.

Commutative Law:

As, we have also studied in mathematics about associative law. The same of course it is also used in Boolean Algebra.
(i) (A.B).C = A.(B.C)    (ii) (A + B) + C = A + (B + C)
This law works same as in mathematical algebra.

Truth Table of Commutative Law

Proving (A.B).C = A.(B.C)
A
B
C
A.B
(A.B).C
B.C
A.(B.C)
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
1
1
0
0
1
0
1
0
0
0
0
0
0
1
0
1
0
0
0
0
1
1
0
1
0
0
0
1
1
1
1
1
1
1

Proving (A + B) + C = A + (B + C)
A
B
C
A+B
(A + B) +C
B + C
A + (B + C)
0
0
0
0
0
0
0
0
0
1
0
1
1
1
0
1
0
1
1
1
1
0
1
1
1
1
1
1
1
0
0
1
1
0
1
1
0
1
1
1
0
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1

Distributive Law:

This law state when multiplication of (B + C) to A, then A will be distributed into B as well as C.
A.(B + C) = A.B + A.C

AND Law:

AND Law uses AND operation that is why it is called as AND Law. In AND operation the two strings are multiplied with each other. If one of them is False (0) then the result will be false too.
(i) A.0 = 0(ii) A.1 = A(iii) A.A = A(iv) A. A (Overbar) = 0

OR Law:

OR Law uses OR operation that is why it is called AND Law. In OR Law, if one strign of them is true then the resultant will be true. It uses Addition Operator (+) and work as simple addition.
(i) A + 0 = A(ii) A + 1 = 1(iii) A + A = A(iv) A. A (Overbar) = 0

Inversion Law:

Inversion law uses NOT operation. If there is double inversion (or bars) on single string then the result will be same as the string.
1 (Overbar) = 00 (Overbar) = 1A (Double Overbar) = A
Truth Table of other laws will be updated soon. Let me take breakfast made by my mother.

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