# Laws of Boolean Algebra with Solved Examples

Boolean algebra refers to a branch of Algebra that allows the rule of numbers used in algebra to be applied to logic. Furthermore, experts also refer to it as logic algebra. Most noteworthy, Boolean algebra helps in simplifying Boolean expressions by making use of the laws of Boolean algebra.

Moreover, Boolean expressions denote combination logic circuits. Boolean algebra reduces the original expression to an expression of fewer terms. Experts make use of the laws of Boolean algebra in digital electronics. Only the binary numbers, 0 and 1, are used in Boolean algebra.

Also, there exist equations, expressions, and functions in Boolean algebra. This article deals with the laws of Boolean algebra present in the Boolean algebraic system. ## Laws of Boolean Algebra

• Associate Law
• Distributive Law
• Commutative Law
• Absorption Law

Associative laws of addition deal with OR-ing more than two variables. Furthermore, the performance of mathematical addition operation on variables will result in the returning of the same value. This takes place irrespective of the grouping of variables in shapes. Most noteworthy, Associative law using the OR operator is as follows:

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

As per the associative law of addition –

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

Associative Law of Multiplication

Associative law of multiplication revolves around AND ing more than two variables. Moreover, this law states that the performance of mathematical multiplication operation on the equation’s variables. Most noteworthy, the associative law, making use of AND operator is below:

A*(B*C) = (A*B) *C

### Distributive law

The distributive law has 2 operators: AND, OR. Most noteworthy, distributive law happens to be the most important law in Boolean algebra. There are two important statements here

Statement 1 of Distributive Law – First of all, the multiplication of two variables must take place. When the addition of the result takes place with a variable, the resulting value will be the same as the multiplication of addition with variables that are individuals. Moreover, one can write the distributive law as follows:

A + BC = (A + B) (A + C)

This is what experts refer to as the OR distributes over AND.

Statement 2 of Distributive Law – First of all, the addition of two variables takes place. Then, the multiplication of the result will take place with a variable. Most noteworthy, this would result in the same value as the addition of multiplication of the variable which comprises individual variables.

The writing of the distributive law takes place as follows

A (B+C) = (AB) + (AC)

This is what is known as the AND distributes over OR.

### Commutative Law

Commutative law states that in the Boolean equation, the interchanging of the order of operands can take place. However, its result does not change.

When using OR operator → A + B = B + A

When using AND operator → A*B = B*A

This law is significantly important in Boolean algebra.

### Absorption Law

Absorption law consists of the linking of a pair of binary operations.

1. A + AB = A
2. A (A+B) = A
3. A + ĀB = A+B
4. A.(Ā+B) = AB

3rd and 4th laws are referred to as Redundancy laws.

In Boolean algebra, the values happen to be binary values that are 0 and 1. The main aim of the laws of Boolean algebra is to simplify the logic so as to make it less complicated and easy. These laws reduce the complexities of any Boolean expression. Furthermore, Boolean algebra helps in simplifying the digital circuits. Most noteworthy, there are five major laws of Boolean algebra. These laws are- Associate Law of Addition, Associative Law of Multiplication, Distributive law, Commutative Law, and Absorption Law.

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