Properties of Boolean Algebra

Switching algebra is also known as Boolean Algebra. It is used to analyze digital gates and circuits It is logical to perform a mathematical operation on binary numbers i.e., on ‘0’ and ‘1’. Boolean Algebra contains basic operators like AND, OR, and NOT, etc. Operations are represented by ‘.’ for AND, and ‘+’ for OR. Operations can be performed on variables that are represented using capital letters eg ‘A’, ‘B’ etc. 

Properties of Switching Algebra

The Primary objective of the logic design is to solve the expression to its simplest form. This simplification process is Important to ensure that the final implementation of a logic circuit is as Simple as possible. By reducing complexity, we can increase efficiency and ease of implementation and making the overall design process more simple.

Annulment law

a variable ANDed with 0 gives 0, while a variable ORed with 1 gives 1, i.e., 

A.0 = 0 

A + 1 = 1 

Identity law

In this law variable remains unchanged it is ORed with ‘0’ or ANDed with ‘1’, i.e., 

A.1 = A 

A + 0 = A 

Idempotent law

A variable remains unchanged when it is ORed or ANDed with itself, i.e., 

A + A = A 

A.A = A 

Complement law

In this Law if a complement is added to a variable it gives one, if a variable is multiplied with its complement it results in ‘0’, i.e., 

A + A’ = 1 

A.A’ = 0 

Double Negation Law

A variable with two negations, its symbol gets cancelled out and original variable is obtained, i.e., 

((A)’)’=A 

Commutative law

A variable order does not matter in this law, i.e., 

A + B = B + A 

A.B = B.A 

Associative law

The order of operation does not matter if the priority of variables are the same like ‘*’ and ‘/’, i.e., 

A+(B+C) = (A+B)+C 

A.(B.C) = (A.B).C 

Distributive law

This law governs the opening up of brackets, i.e., 

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

(A+B)(A+C) = A + BC 

Absorption law

This law involved absorbing similar variables, i.e., 

A.(A+B) = A A + AB

A.(A+B) = A A+ A’B

A.(A+B) = A+B A(A’ + B)

A.(A+B) = AB

De Morgan law

In Demorgan law, the operation of an AND or OR logic circuit is unchanged if all inputs are inverted, the operator is changed from AND to OR, and the output is inverted, i.e., 

(A.B)’ = A’ + B’ (A+B)’ = A’.B’

Consensus theorem

AB + A’C + BC = AB + A’C

In this article, we will be going through the Properties or Laws of the Boolean algebra. So first we will start our article by defining what are the properties of Boolean Algebra, and then we will go through what are Boolean Addition and Multiplication. Then we will go through the different properties of Boolean Algebra such as Annulment, Identity law, Idempotent law, etc.