Boolean Algebra Laws and Theorem

By Aina Parasher|Updated : May 4th, 2022

Boolean Algebra Laws: Before discussing various Boolean algebra laws, we will discuss a brief history, importance, and its evaluation as the most significant tool in modern-day digital electronics. In his book, “The Mathematical Analysis of Logic (1847)” George Boole laid the foundation for the study of Boolean Algebra as a branch of mathematics. In modern-day electronic automation, efficient representation of Boolean functions plays a vital role in logic synthesis and formal verification.  

There are certain axioms and laws of Boolean algebra that can help in simplifying the complex Boolean expressions to reduce the number of logic gates required for the implementation of a particular logic design. In this article, our focus is to discuss the Boolean algebra laws along with some axioms, and theorems that can be helpful in reduction.  

 

Table of Content

 Axioms of Boolean Algebra

There is a set of axioms or postulates in Boolean algebra that we accept without any proof. We can build a bunch of useful theorems using these axioms. In the following table, I have listed some useful axioms that can be helpful in our further requirements. 

S.No. 

Axiom 

1 

0.0=0

2 

0.1=0

3 

1.0=0

4 

1.1=1

5 

0+0=0

6 

0+1=1

7 

1+0=1

8 

1+1=1

9 

x̄=0 (if x=1)

10 

x̄=1 (if x=0)

What are the Boolean Algebra Laws?

The Boolean Algebra laws are a series of rules or expressions that have been developed to assist minimize the number of logic gates required to complete a given logic operation, resulting in a list of functions or theorems that are usually referred to as the Laws of Boolean Algebra. Further, let us discuss these in detail in the upcoming sections.

Law of Complementation

The term complement means ‘to invert’, every variable attains its inverse form after the implementation of the complement operation. The following statements can be made from the law of complementation. 

Statement 1: Whenever we apply the AND operation on a Boolean variable and its complement, it results in logic 0. 

Statement 2: Whenever we apply the OR operation on a Boolean variable and its complement, it results in logic 1. 

S.No. 

Law 

1 

X.X̅=0

2 

X+X̅=1

Law of Annulment

This Boolean algebra law states that whenever we apply the AND operation on a Boolean variable and 0, it will result in logic 0. After the AND operation with 0, irrespective of the value of the Boolean variable it’s got annulled and results in logic 0. 

Similarly, whenever we apply the OR operation on a Boolean variable and 1, it will result in logic 1. After the OR operation with 1, irrespective of the value of the Boolean variable it’s got annulled and results in logic 1. 

S.No. 

Law 

1 

X.0=0

2 

X+1=1

Law of Identity

The application of OR operation on a Boolean variable and 0 results in the same Boolean variable. Similarly, the application AND operation on a Boolean variable and 1 result in the Boolean variable. In both cases, the result is identical to the Boolean variable. 

S.No.

Law 

1

X+0=X

2

X.1=X

Idempotent Law

Idempotent denotes an element of a set that is unchanged in value when multiplied or otherwise operated on by itself. This Boolean algebra law states that for any Boolean variable if we apply AND operation with itself results in the same variable. Similarly, if we apply OR operation on a Boolean variable with itself results in the same variable.  

S.No. 

Law 

1 

X+X=X

2 

X.X=X

Commutative Law

Commutative means the condition that a group of quantities connected by operators gives the same result whatever the order of the quantities involved. If we apply AND operation on two Boolean variables, then there is no significance for the order of application. Similarly in the OR operation of Boolean variables, the order of application is insignificant. 

S.No.

Law 

1 

X+Y=Y+X

2 

X.Y=Y.X

Distributive Law

This Boolean algebra law allows the multiplication or factorization outside the Boolean expression. This law is applicable for AND as well as OR operators in Boolean algebra. 

S.No. 

Law 

1 

X(Y+Z)=X.Y+Y.Z  (OR distributive law)

2 

X+(Y.Z)=(X+Y).(Y+Z) (AND distributive law)

Associative Law

This Boolean algebra law allows the grouping of variables. This law is defined for AND as well as OR operators in Boolean algebra. 

S.No. 

Law 

1 

X+(Y+Z)= (X+Y)+Z= X+Y+Z  (OR associative law)

2 

X(Y.Z)= (X.Y)Z= XYZ  (AND associative law) 

Double Negation Law

This law of Boolean algebra states the double application of inversion on any Boolean variable results in the same variable. It is denoted by the following formula.

X̿ = X

Law of Absorption

This Boolean algebra law helps in the elimination of similar variables; hence it enables the reduction of complex terms into simplified expressions. This law is applicable for both AND as well as OR operators. 

S.No 

Law 

1 

X+(X.Y)= (X.1)+(X.Y)= X(1+Y)=X (OR absorption law)

2 

X(X+Y)= (X+0).(X+Y)= X+(0.Y)= X (AND absorption law)

De Morgan’s Theorem

De Morgan’s theorem is a powerful tool of reduction in Boolean algebra. There are two separate statements for theorem. 

Statement 1: The complement of the sum of the variables is equal to the product of their individual compliments. 

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We can prove the above statement using a two-variable truth table as shown below. 

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Statement 2: 

The complement of the product of the variables is equal to the sum of their individual complements. 

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The above statement can be proved by using the two-variable truth table as shown below.

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FAQs

  • In Boolean algebra there are various laws to help in the reduction of complex Boolean expressions, they are namely commutative law, distributive law, associative law, the law of complementation, law of annulment, Idempotent law, Double negation law, etc. Along with this DE Morgan’s theorem is also helpful in the reduction of complex Boolean expressions.

  • In modern-day electronic automation, efficient representation of Boolean functions plays a vital role in logic synthesis and formal verification. There are certain axioms and Boolean algebra laws that can help in simplifying the complex Boolean expressions to reduce the number of logic gates required for the implementation of a particular logic design.

  • If X and Y are any two Boolean variables then 


    X+Y=Y+X

    X+Y=Y+X

    And  

    X.Y=Y.X

    X.Y=Y.X

     

    This is also known as the law of commutation in Boolean algebra. 

  • If X is a Boolean variable, then, the idempotent law of Boolean algebra states that, 


    X+X=X

    X+X=X

    And  

    X.X=X

    X.X=X

  • If X and Y are two Boolean variables then according to De Morgan’s theorem, the complement of the sum of the variables is equal to the product of their individual compliments. 

    And the complement of the product of the variables is equal to the sum of their individual complements. 


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