# Boolean Logic: Expression, Theorems, Operations, Examples

By Anjnee Bhatnagar|Updated : September 28th, 2022

For the binary system, Boolean logic was developed as a type of algebra. English mathematician George Boole proposed this algebra in the year 1854. The numbers 0 and 1, or True and False, are used in this application of Aristotle's propositional logic. Binary variables and logic operations are the focus of Boolean algebra.

Boolean Logic PDF

In this article, we will cover all the basics of Boolean Logic, Boolean algebra, and how Boolean expressions are used to evaluate a result into zero and one truth values.

## What is Boolean Logic?

Boolean logic is widely used in digital electronics as a form of algebra. The idea of Boolean Logic is that all the expressions or values are true or false. It mainly uses three basic Boolean operators: OR operator, AND operator, and NOT operator.

Because it works well with the binary numbering system, in which each bit can either have a value of 1 or 0, Boolean logic is particularly crucial for computer science. A different perspective would be that each bit has a value of either TRUE or FALSE.

## Boolean Expression for Logic Gates

A logical statement that can only be TRUE or FALSE is called a Boolean expression. Any form of data can be compared using Boolean expressions as long as both portions of the expression use the same fundamental data type. Data can be tested to see if it is more than, equal to, or less than other data.

A Boolean expression can consist of Boolean information or data, such as the following:

• Boolean formulas consist of Boolean variables.
• Boolean Values in the form of TRUE or False or in the binary form 1 or 0, respectively.
• Functions result in Boolean values(0 or 1) or Boolean expressions.

## Boolean Theorems

Theorems in Boolean algebra are employed to alter the form of Boolean expressions. Boolean theorems may be applied to an expression to reduce its number of minterms. Digital logic has the following Boolean algebraic theorems and laws:

• DeMorgan’s Theorem: (A.B)’ = A’ + B’
• Consensus Theorem: AB + A’C + BC= AB+ A’C
• Duality Theorem
• Redundant Theorem
• Complementary Theorem

## Boolean Logic Operations

Conjunction, disjunction, and negation are the three fundamental Boolean logic operations. The related binary operators AND, OR, and the unary operator NOT, collectively referred to as Boolean operators, are used to express these Boolean operations.

The truth table for the basic boolean algebra operations is as follows:

### OR Operation

 A B A OR B 0 0 0 0 1 1 1 0 1 1 1 1

### AND Operation

 A B A AND B 0 0 0 0 1 0 1 0 0 1 1 1

### NOT Operation

 A NOT A 0 1 1 0

## Boolean Logic Example

Find the Boolean algebra expression for the given logic system.

Solution:

The truth table for the above circuit is:

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## FAQs on Boolean Logic

• The Laws of Boolean Algebra are a set of rules or expressions that were developed to help minimize the number of logic gates required to carry out a given logic operation. These functions or theorems are also referred to as the Laws of Boolean Algebra.

• Boolean logic is a form of algebra used in digital electronics. The result of a Boolean logic results in two values or specifically two binary values that is "1" or "0" and True or False.

• The Boolean data type in computer science (often abbreviated to Bool) is meant to represent the two truth values of logic and Boolean algebra and can take one of two possible values (typically labeled true and false).

• The three main Boolean operators are: AND, OR, and NOT. These are the basis of any Boolean expression and are used to narrow or broaden an expression. Following are the functionalities of each :

• AND operator: Returns true only when all the values/ variables are true.
• OR operator: Returns true if at least one value/variable is true.
• NOT operator: Returns the complement of the Boolean value/variable.
• The majority of information databases use Boolean logic and operators (based on Boolean algebra), which enable users to find significant information by combining synonyms and variant notions. The basic Boolean connectors are AND, OR, and NOT.

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