# Boolean Algebra

Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. Boolean Algebra is used to analyze and simplify the digital circuits.

Boolean algebra is also called as Binary Algebra or logical Algebra. It has been fundamental in the development of digital electronics, and is provided for in all modern programming languages. It is also used in set theory and statistics.

## Basic operations of Boolean Algebra

The basic operations of Boolean calculus are as follows:

 Operator Symbol Precedence NOT ‘ Highest AND . Middle OR + Lowest

## Rule in Boolean Algebra

Following are the important rules used in Boolean algebra.

• Variable used can have only two values. Binary 1 for HIGH and Binary 0 for LOW.

• Complement of a variable is represented by an overbar. Thus, complement of variable B is represented as B. Thus if B = 0 then B=1 and B = 1 then B= 0.

• OR-ing of the variables is represented by a plus (+) sign between them. For example OR-ing of A, B, C is represented as A + B + C.

• Logical AND-ing of the two or more variable is represented by writing a dot between them such as A.B.C. Sometimes the dot may be omitted like ABC.

## Boolean Laws

There are six types of Boolean Laws.

Commutative law: Any binary operation which satisfies the following expression is referred to as commutative operation. Commutative law states that changing the sequence of the variables does not have any effect on the output of a logic circuit.

$\large (i)\;A.B=B.A\;\;\;\;(ii)\;A+B=B+A$

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

$\large (i)\;\left(A.B\right).C=A.\left(B.C\right)\;\;\;\;\;\;(ii)\; \left(A+B\right)+C=A+\left(B+C\right)$

Distributive law: States the following condition.

$\large A.\left(B+C\right)=A.B+A.C$

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

$\large (i)A.0=0$

$\large (ii)A.1=A$

$\large (iii)A.A=A$

$\large (iv)A.\bar{A}= 0$

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

$\large (i)A+0=A$

$\large (ii)A+1=1$

$\large (iii)A+A=A$

$\large (iv)A+\bar{A}=1$

INVERSION law:

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

$\large (iv)A+\bar{\bar{A}}=1$<

#### Practise This Question

If A is a 3×3 non-singular matrix such that AA=AA and B=A1A then BB' equals