# Mastering Boolean Algebra: Definitions, Operations, Laws, and Example

When using algebra One area of algebra that deals with the logical operation of variables is called Boolean algebra. A variable in Boolean algebra has only two possible values: true or false. In other words, only possible value of variables i.e. either 0 or 1. Conjunction, disjunction, and negation are three main logical operations.

Boolean algebra, which is often used in computer science, digital electronics, and numerous areas for decision-making and logical analysis, offers a framework for describing and manipulating binary data and logical connections using operations like AND, OR, and NOT.

The definition, various operations, rules, and terminology of Boolean algebra will all be covered in this article. Additionally, we'll use thorough examples to help the reader understand the subject as we explain it.

## Definition

Boolean Algebra is a mathematical structure that represents logical operations and relationships using binary variables and logical operators. The variables in Boolean Algebra can only take on two values: 0 (false) or 1 (true).

While mathematical expressions in Boolean algebra represent truth values, they are primarily used to indicate numbers in elementary algebra. Binary variables or bits "1" and "0" are used by the truth values to indicate the input and output statuses. The three fundamental Boolean operators are the logical operators AND, OR, and NOT.

## Operations of Boolean algebra

In this type of algebra, three main types are involved.

1. AND operation or Conjunction (^)

2. OR operation or Disjunction (v)

3. Not operation or Negation (-)

When we want to find the sum, difference, and transpose of two variables these Boolean operators are frequently used.

Now we discuss these operations one by one.

### AND operation or Conjunction (^)

AND operation or conjunction operation is used to multiply the algebra. The symbol of conjunction operation is the (^) symbol. In conjunction, the operation gives a value true if both values are true otherwise all cases give false value. We can write two variables in this form A.B or A^B.

 P Q P^Q 0 0 0 0 1 0 1 0 0

### OR operation or Disjunction (v)

OR operation or disjunction operation is used to add variables. (v) is symbol of OR operation. In solution, if both values are false OR operation gives the result false, and in all other cases it gives the true value in the result.

 P Q P v Q 0 0 0 0 1 1 1 0 1

### Not operation or Negation (-)

Not operation or Negation gives the negation value of truth value. If a given value is false it transposes or the negation value is true. The symbol (-) is the symbol of the negation operation.

 P -P P 0 1 0 1 0 1

## Laws of Boolean Algebra:

A mathematical structure and set of operations used in logic design and computer science is called boo Boolean algebra. George Boole introduced Boolean algebra in the middle of the 1800s. Binary logic is the foundation of Boolean algebra; a variable can only contain the values 1 (true) and 0 (false). We now go over the significance of the laws and rules governing Boolean algebra.

 Name of Laws Law Identity Laws A + 0 = A A. 1 = A Domination Laws A + 1 = 1 A. 0 = 0 Idempotent Laws A + A = A A. A = A Complement Laws A + A =1 A .A =0 Distributive Laws (A. (B + C) = (A.B) + (A.C)) (A + (B. C) = (A + B) c. (A + C)

## Example section of Boolean algebra

### Example number 1

Let’s suppose – (P × (-P + Q)) +Q. Examine the Boolean algebra with the help of Boolean algebra.

Solution

Given data

– (P × (-P + Q)) +Q

By using the Boolean theorem, we solve the given expression

Step 1:

By using De Morgen’s Law

(AB) = (A) + (B)

Q + (Q)P +P

Step 2:

Again using De Morgen law

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

Q + (Q)P +P

Table of Given expression Boolean algebra

 P Q -P -P+Q P(-P+Q) -(P(-P+Q)) -(P(-P+Q))+Q 0 0 1 1 0 1 1 0 1 1 1 0 1 1 1 0 0 0 0 1 1

To find the Boolean expression with laws and truth tables, you can also use a Boolean calculator.

### Example number 2

Let suppose (A*-B) *C+(-A*B*C). With the help of law determine the expression of Boolean algebra.

Solution

Given data

(A*-B) *C+(-A*B*C)

With the help of a Boolean expression table, we easily solve the given expression.

 A B C -B (A*-B) ((A*-B)*C) -A (-A*B) ((-A*B)*C) (((A*-B)*C)+((-A*B)*C)) 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 1 0 0 1 1 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0

## Boolean algebra: Terminology

• ### Complement

The complement of a Boolean variable (A) is denoted as (A) or (A') and represents the opposite or negation of (A).

• ### Canonical Form

An expression in which the logical operations are arranged in a specific standard order, such as the Sum-of-Products (SOP) or Product-of-Sums (POS) form.

• ### Digital Circuit

A circuit designed using Boolean algebra principles to perform logical operations, commonly found in computers and electronic devices.

• ### Boolean Expression

A combination of Boolean variables and operators (such as AND, OR, NOT) that represents a logical relationship.

• ### Truth Value

The result of evaluating a Boolean expression, which can be either True (1) or False (0).

• ### Boolean Function

A logical operation or combination of operations that takes one or more Boolean inputs and produces a Boolean output.

• ### Simplification

The process of reducing a Boolean expression to its simplest form by applying Boolean algebra laws and rules.

## Warp Up

The definition, methods, guidelines, and jargon of Boolean algebra have all been clarified in this article. We go over a thorough example of Boolean algebra in detail. With the aid of our thorough example and their solution, we hope you will be able to defend this topic anywhere.

## FAQs

### Q 1.

What are Boolean variables?

Boolean variables are symbolic representations (often denoted as A, B, C, etc.) that can take on binary values, typically True (1) or False (0).

### Q 2.

What does the NOT operator do?

The NOT operator negates or reverses the truth value of its operand. If the operand is True, NOT returns False, and vice versa.

### Q 3.

How is Boolean algebra used in digital circuit design?