- Boolean algebra which deals with variables and binary numbers.
- This algebra is named after mathematician George Boole who developed this in 1854.
- The variables considered for simplification are called Boolean Variables.
- The value of voltage when HIGH is considered 1 and when LOW is considered as 0.

## Basic Laws of Boolean Algebra

Here, the Boolean postulates and basic laws that are used are given underneath.

**Boolean Postulates**

- Considering the binary numbers 0 and 1, Boolean variable (x) and its complement (x’).
- They are known as
**literal**. - The possible
**logical OR**operations are:

x + 0 = x

Also, x + 1 = 1

x + x = x

And x + x’ = 1

- Similarly, the possible
**logical AND**operations are:

x.1 = x

x.0 = 0

x.x = x

x.x’ = 0

We can verify these postulates by simply replacing the Boolean variables with 0 or 1.

## Basic Laws

The basic laws of Boolean Algebra are:

- Commutative law
- Associative law
- Distributive law

**Commutative Law**

According to this law changing the sequence of the variables does not affect the output.

The logical OR & logical AND operations between x & y are shown below

x + y = y + x

x.y = y.x

The OR operation is represented by ‘+’ and AND operation by ‘.’

This law exists for logical OR and AND operation.

**Associative Law**

This law states that the order of operation does not matter as the result remains unaffected. The equations below make it more clear.

x + (y + z) = (x + y) + z

x.(y.z) = (x.y).z

This law exists for logical OR and AND operation.

**Distributive Law**

Basically this law helps to remove the brackets and solve the logical expression. The brackets can have either AND or OR logic. The distribution of logical OR & logical AND operations between variables x, y & z are :

x.(y + z) = x.y + x.z

x + (y.z) = (x + y).(x + z)

It works good for logical OR and logical AND operations.

The above laws are basic laws and can be verified by substituting the Boolean variable with ‘0’ or ‘1’.

## Theorems of Boolean Algebra

Basic theorems of Boolean Algebra are

- Duality theorem
- De Morgan’s theorem

**Duality Theorem**

As the name says we find the dual of the given Boolean expression. We replace AND operation with OR and vice versa. We replace all zeros by 1. There is always a dual function for every Boolean function.

.

As seen from the above table there are two Boolean equations and are dual to one other.

With the help of the duality theorem we can easily verify the Boolean equations of each group.

**De Morgan’s Theorem**

- De Morgan’s Theorem helps us to find the complement of Boolean functions.
- According to this law the complement of the sum of variable is equal to product of complement of each variable.
- We can understand this law using two variables x and y as

(x + y)’ = x’.y’

- The second way or the dual of the above law with variables x and y is

(x.y)’ = x’ + y’

- The complement of the product of two variables x and y is equal to the sum of complement of each variable.
- This theorem is applicable for more than two variables as well.