Boolean Algebra

Boolean algebra is the category of algebra in which the variable’s values are the truth values, true and false, ordinarily denoted 1 and 0 respectively. It is used to analyze and simplify digital circuits. It is also called as Binary Algebra or logical Algebra. It has been fundamental in the development of digital electronics and is provided for in all modern programming languages. It is also used in set theory and statistics.

The important operations performed in boolean algebra are – conjunction (∧), disjunction (∨) and negation (¬). Hence, this algebra is far way different from elementary algebra where the values of variables are numerical and arithmetic operations like addition, subtraction is been performed on them.

Table of contents:

Boolean Algebra Operations

The basic operations of Boolean algebra are as follows:

  • Conjunction or AND operation
  • Disjunction or OR operation
  • Negation or Not operation

Boolean Algebra


Below is the table defining the symbols for all three basic operations.

Operator Symbol Precedence
NOT ‘ (or) ¬ Highest
AND . (or) ∧ Middle
OR + (or) ∨ Lowest

Suppose A and B are two boolean variables, then we can define the three operations as;

  • A conjunction B or A AND B, satisfies A ∧ B = True, if A = B = True or else A ∧ B = False.
  • A disjunction B or A OR B, satisfies A ∨ B = False, if A = B = False, else A ∨ B = True.
  • Negation A or ¬A satisfies ¬A = False, if A = True and ¬A = True if A = False

Boolean Algebra Terminologies

Now, let us discuss the important terminologies covered in Boolean algebra.

Boolean Algebra: Boolean algebra is the branch of algebra that deals with logical operations and binary variables.

Boolean Variables: A boolean variable is defined as a variable or a symbol defined as a variable or a symbol, generally an alphabet that represents the logical quantities such as 0 or 1.

Boolean Function: A boolean function consists of binary variables, logical operators, constants such as 0 and 1, equal to the operator, and the parenthesis symbols.

Literal: A literal may be a variable or a complement of a variable.

Complement: The complement is defined as the inverse of a variable, which is represented by a bar over the variable.

Truth Table: The truth table is a table that gives all the possible values of logical variables and the combination of the variables. It is possible to convert the boolean equation into a truth table. The number of rows in the truth table should be equal to 2n, where “n” is the number of variables in the equation. For example, if a boolean equation consists of 3 variables, then the number of rows in the truth table is 8. (i.e.,) 23 = 8.

Boolean Algebra Truth Table

Now, if we express the above operations in a truth table, we get;

A B A ∧ B A ∨ B
True True True True
True False False True
False True False True
False False False False
A ¬A
True False
False True

Boolean Algebra Rules

Following are the important rules used in Boolean algebra.

  • Variable used can have only two values. Binary 1 for HIGH and Binary 0 for LOW.
  • The complement of a variable is represented by an overbar. Thus, complement of variable B is represented as \(\bar{B}\). Thus if B = 0 then \(\bar{B}\)=1 and B = 1 then \(\bar{B}\) = 0.
  • OR-ing of the variables is represented by a plus (+) sign between them. For example OR-ing of A, B, C is represented as A + B + C.
  • Logical AND-ing of the two or more variable is represented by writing a dot between them such as A.B.C. Sometimes the dot may be omitted like ABC.

Laws of Boolean Algebra

There are six types of Boolean algebra laws. They are:

  • Commutative law
  • Associative law
  • Distributive law
  • AND law
  • OR law
  • Inversion law

Those six laws are explained in detail here.

Commutative Law

Any binary operation which satisfies the following expression is referred to as a commutative operation. Commutative law states that changing the sequence of the variables does not have any effect on the output of a logic circuit.

  • A. B = B. A
  • A + B = B + A

Associative Law

It states that the order in which the logic operations are performed is irrelevant as their effect is the same.

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

Distributive Law

Distributive law states the following conditions:

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


These laws use the AND operation. Therefore they are called AND laws.

  • A .0 = 0
  • A . 1 = A
  • A. A = A
  • \(A. \bar{A}= 0\)

OR Law

These laws use the OR operation. Therefore they are called OR laws.

  • A  + 0 = A
  • A + 1 = 1
  • A + A = A
  • \(A + \bar{A}= 1\)

Inversion Law

This law uses the NOT operation. The inversion law states that double inversion of variable results in the original variable itself.

  • \(A+\bar{\bar{A}}=1\)

Example of Boolean Algebra Simplication

Question: Simplify the following expression: \(c+\bar{BC}\)


Given: \(C+\bar{BC}\)

According to Demorgan’s law, we can write the above expressions as

\(C+(\bar{B}+ \bar{C})\)

From Commutative law:

\((C+\bar{C})+ \bar{B}\)

From Complement law

\(1+ \bar{B}\) = 1

Therefore, \(C+\bar{BC} = 1\)

Frequently Asked Questions on Boolean Algebra

What is meant by Boolean algebra?

In Mathematics, boolean algebra is called logical algebra consisting of binary variables that hold the values 0 or 1, and logical operations.

Why do we use Boolean algebra?

In electrical and electronic circuits, boolean algebra is used to simplify and analyze the logical or digital circuits.

What are the three main Boolean operators?

The three important boolean operators are:
AND (Conjunction)
OR (Disjunction)
NOT (Negation)

Is the value 0 represents true or false?

In boolean logic, zero (0) represents false and one (1) represents true. In many applications, zero is interpreted as false and a non-zero value is interpreted as true.

Mention the six important laws of boolean algebra.

The six important laws of boolean algebra are:
Commutative law
Associative law
Distributive law
Inversion law
AND law
OR law

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  1. Understood

  2. Shedrack Barnabas kashindi

    Mathematics is simple if you simplify it.

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