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
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 2^{n}, 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.,) 2^{3} = 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.
 ORing of the variables is represented by a plus (+) sign between them. For example ORing of A, B, C is represented as A + B + C.
 Logical ANDing 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.
Related Links 

Truth Table  Tautology 
Conjunction  Mathematical Logic 
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)
AND Law
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: \(c+\bar{BC}\)
Solution:
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
\(1+ \bar{B}\) = 1Frequently 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 nonzero 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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