Answer:

• Boolean algebra is an important mathematical tool for designing and analysing digital systems. A basic knowledge of Boolean Algebra is indispensable to the study and analysis of logic circuits.
• As in other areas of mathematics. there are certain well developed rules and laws that must be followed in order to apply Boolean Algebra properly. Let us now study some of the important laws and rules of Boolean algebra. First we will study the laws of Boolean algebra and then the rules.

Laws of Boolean Algebra:-Generally the laws of Boolen Algebra arer classified into three forms and these are given below:-

1. Commutative laws.
2. Associative laws, and
3. Distributive laws.

All the three laws are illustrated with two or three Boolean variables but the number of Boolean variables is not limited to this.

• Cummutative laws:-There are 2 laws under this category. These are

(i) Cummutative law of addition:- This law for 2 variables is written algebraically as

• $A+B=B+A$
  • This law states that the order in which the variables are OR makes no difference. Remember the addition and the OR operation are the same when applied to logic circuits. the Cummutative law as applied to the OR gate. lt shows that it doesn't matter to which input each variable is applied.Below figure shows that :-

Fig:- Cummutative Law of addition

(ii) Cummutative law of multiplication:- This law for 2 variables is written algebraically as

• $AB=BA$
  • This law states that the order in which the variables are AND makes no difference. Below figure shows that the law applied to AND gate :-

Fig:- Cummutative Law of multiplication

Associative laws:-Like cummutative laws, there are 2 laws under this category

• Associative law of addition:- This law for 3 variables is written algebraically as
• $A+(B+C)=(A+B)+C$
  • This law states that in OR ing of more than 2 variables, the result is the same regardless of the grouping of the variables. Examples this law as applied to OR gates

Fig:- Associative Law of addition

Associative law of multiplication:- This law for 3 variables is written algebraically as

• $A(B+C)=(AB)C$
  • This law states that it makes no difference in what order the variables are grouped when AND more than 2 variables. Illustrates this law when applied to AND gates.

Fig:- Associative Law of multiplication

Distributive laws:-  This law is the same as in ordinary algebra. The distributive law for 3 variables is written as follows:-

• $A(B+C)=AB+AC$
  • This law states that ORing 2 or more variables and ANDing the result with a single variable is equivalent to ANDing the single variable with each of the two or more variables and then ORing the products. The distributive law also expresses the process of factoring in which a common variable A.Illustrates this law shown in below figure:-

Fig:- Distributive Law