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.Â 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 (âˆ§), disjunction (âˆ¨) and negation (Â¬). Hence, this algebra is far way different from elementary algebra where the values of variables are numerical and arithmetic operations like addition, subtraction are been performed on them.
Basic operations of Boolean Algebra
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  â€˜ (or)Â Â¬  Highest 
AND  . (or)Â âˆ§  Middle 
OR  + (or)Â âˆ¨  Lowest 
Suppose A and B are two boolean variables, then we can define the three operations as;
 A conjunction B or A ANDÂ B, satisfies AÂ âˆ§ B = True, if A = B = True or else A âˆ§ B = False.
 A disjunction B or A OR B, satisfies AÂ âˆ¨ B = False, if A = B = False, else AÂ âˆ¨ B = True.
 Negation A orÂ Â¬A satisfiesÂ Â¬A = False, if A = True andÂ Â¬A = True if A = False
Now, if we express the above operations in a truth table, we get;
A  B  A âˆ§ B  A âˆ¨ B 
True  True  True  True 
True  False  False  True 
False  True  False  True 
False  False  False  False 
A  Â¬A 
True  False 
False  True 
Also, read:
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Truth Table  Tautology 
Conjunction  Mathematical Logic 
Rules in Boolean Algebra
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.
Laws of Boolean Algebra
There are six types of Boolean laws. Th0se 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.
 AÂ + 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.
 \(A+\bar{\bar{A}}=1\)
Boolean Algebra Example
Question:
Simplify the following expression:Â \(c+\bar{BC}\)
Solution:
Given:Â \(C+\bar{BC}\)
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
\(1+ \bar{B}\) = 1Therefore,Â \(C+\bar{BC} = 1\)
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