Boolean Algebra

Table of contents

  1. Introduction
  2. Rule in Boolean Algebra
  3. Boolean Laws
    1. AND law
    2. OR law
    3. INVERSION law
    4. Commutative law
    5. Associative law
    6. Distributive law
  4. DeMorgan’s Theorem

Introduction

Boolean Algebra is used to analyze and simplify the digital (logic) circuits. It uses only the binary numbers i.e. 0 and 1. It is also called as Binary Algebra or logical Algebra. Boolean algebra was invented by George Boole in 1854.

Rule in Boolean Algebra

Following are the important rules used in Boolean algebra.

  1. Variable used can have only two values. Binary 1 for HIGH and Binary 0 for LOW.
  2. Complement of a variable is represented by an overbar (-) or (!). Thus, complement of variable B is represented as B Bar. Thus if B = 0 then !B = 1 and B = 1 then !B = 0.
  3. ORing of the variables is represented by a plus (+) sign between them. For example ORing of A, B, C is represented as A + B + C.
  4. Logical ANDing of the two or more variable is represented by writing a dot between them such as A.B.C. Sometime the dot may be omitted like ABC.

Boolean Laws

There are six types of Boolean Laws.

AND law

These laws use the AND operation. Therefore they are called as AND laws.

Example:  
1. A.0 = 0
1. A.1 = A
1. A.A = A
1. A.!A = 0

OR law

These laws use the OR operation. Therefore they are called as OR laws.

Example:  
1. A+0 = A
2. A+1 = 1
3. A+A = A
4. A+!A = 1

INVERSION law

This law uses the NOT operation. The inversion law states that double inversion of a variable results in the original variable itself.

Example:  !!A = A

Commutative law

Any binary operation which satisfies the following expression is referred to as commutative operation.

Example:  A.B = B.A                         A+B = B+A

Associative law

This law states that the order in which the logic operations are performed is irrelevant as their effect is the same.

Example: (A.B).C = A.(B.C)                (A+B)+C = A+(B+C)

Distributive law

Distributive law states the following condition.

Example:  A.(B+C) = A.B + A.C

DeMorgan’s Theorem

This theorem is useful in finding the complement of Boolean function. It states that the complement of logical OR of at least two Boolean variables is equal to the logical AND of each complemented variable.

DeMorgan’s theorem with 2 Boolean variables x and y can be represented as

  (x + y)’ = x’.y’

The dual of the above Boolean function is

  (x.y)’ = x’ + y’

Therefore, the complement of logical AND of two Boolean variables is equal to the logical OR of each complemented variable. Similarly, we can apply DeMorgan’s theorem for more than 2 Boolean variables also.

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