In order to deal mathematically with the digital circuits, We need to use the Boolean Algebra. It is a tool for the analysis and design of digital systems. Boolean Algebra differs in a major way from ordinary algebra because Boolean constant and variables are allowed to have only two possible values 0 or 1.

** Laws of Boolean Algebra:-** The three basic lows of Boolean Algebra are the same as in ordinary Algebra;

- Commutative Laws
- Associative Laws
- Distributive Laws

**1.Commutative Laws ** This Law state that

## A + B = B + A

**Law2: For multiplication:- ** This law state that

## A . B = B . A

**Associative laws:- **

**Law1: For Addition:-** This Law state that

## A + ( B + C ) = ( A + B ) + C

**Law2: For Multiplication:-** This law state that

## A ( B C ) = ( A B )C

**Distributive Law:-** this law state that

## A (B + C) = A.B + A.C

**Properties of Boolean Algebra:-**

**1. A + 0 = A****2. A + 1 = 1****3. A . 0 = 0****4. A . 1 = A**

**5. A + A = A****6. A + Ã = 1****7. A . A = A**

**8. A . Ã = 0**9.

**10. A + AB = A**

A(1+B) = A

A(1) = A

A = A**11. A + ÃB = A + B**

According to rule 10 put ( A+AB) in place of A

So. A+AB + ÃB = A+B

A+B ( A+Ã) = A+B

A +B(1) = A+B

A+B = A+B**12. (A+B)(A+C) = A+BC**

AA + AC + AB + BC = A+BC

A+AC+AB+BC = A+BC

A(1+C)+AB+BC = A+BC

A+AB+BC = A+BC

A (1+B) +BC = A+BC

A +BC = A+BC**Duality Theorem:- ** According to the duality Theorem a Boolean relation can be written to form another Boolean relation by

- Changing each OR sign to an AND sign.
- Changing each AND sign to an OR sign
- Complementing any 0 or 1 appearing in the expression.

For example: Distributive Law state that.

A(B+C) = AB+BC

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