Boolean Algebra: Theorems and Postulates

Boolean theorems are the fundamental tools of Boolean Algebra. Boolean theorems are either used to transform the expression or can be used to simplify the terms of the expression. In this article, we will discuss the important boolean postulates and theorems and Laws of Boolean Algebra in detail. 

Boolean algebra is based on logical reasoning and hence removes the uncertainty of answers as being based on objectivity rather than subjectivity. Boolean algebra forms the basis of computer programming and binary systems.

Table of Contents

• Boolean Laws and Theorems: An Introduction

• History of Augustus De Morgan

• Basic Theorems of Boolean Algebra

• Laws of Boolean Algebra

• Limitations of Boolean Theorem

• Applications of Boolean Theorem

History of Augustus De Morgan

Augustus De Morgan

Image credit: Wikimedia

Name: Augustus De Morgan

Born:  27 June 1806

Died: 18 March 1871

Field: Mathematics

Nationality: British

Basic Theorems of Boolean Algebra

There are two most important Boolean Theorems also known as De Morgan’s Theorem or Laws.

• De Morgan’s First Theorem: According to De Morgan’s First Theorem, the complement of the product of the variables is equal to the sum of the complements of variables separately.

Mathematically:

$(A . B)^{\prime}=A^{\prime}+B^{\prime}$

Truth table for First Theorem for verification of above expression:

Truth Table for First Theorem

From the last two columns, we have: $(A . B)^{\prime}=A^{\prime}+B^{\prime}$.

Hence, De Morgan’s First Theorem is proved.

• De Morgan’s Second Theorem: According to De Morgan’s Second Theorem, the complement of the sum of the variables is equal to the product of the complements of the variables separately.

Expressing it mathematically:

$(A+B)^{\prime}=A^{\prime} . B^{\prime}$

Truth table for Second Theorem for verification of above expression:

Truth Table for Second Theorem

From the last two columns, we have:  $(A+B)^{\prime}=A^{\prime} . B^{\prime}$

Hence, De Morgan’s second Theorem is proved.

Laws of Boolean Algebra

Commutative Law
Any binary operations satisfying the following expression are commutative operations. Commutative law does not have any effect on the output of a logic circuit.

• $A.B=B.A$ 

• $A+B=B+A$

Commutative law

Associative Law

According to this law, 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
The operations satisfying the following conditions are said to satisfy this law:

• A.(B+C)=(A.B)+(A.C)

AND Law
The laws using the AND operation are called AND laws.

AND Operator

• $A .0=0$

• $A \cdot 1=A$

• $A . A=A$

• $A . \bar{A}=0$
  
  

OR Law

The laws using the OR operation are called OR laws.

OR Operator

• $A+0=A$

• $A+1=1$

• $A+A=A$

• $A+\bar{A}=1$

Inversion Law
The inversion law states that double inversion of a variable will give the original variable.

Limitations of Boolean Theorem

Boolean theorems are not applicable in case of ternary coding which uses base 3 instead of 2 as in binary coding.

Applications of Boolean Theorem

• Boolean Theorem is a fundamental tool of Boolean Algebra.

• Digital world is completely based on Boolean algebra. Everything works on binary coding.

• Electronic systems and electronic circuits wholly work in principle of Boolean Algebra.

Solved Examples

1. Find the complement of $\bar{A} B+C \bar{D}$
Ans: $\overline{\bar{A} B+C \bar{D}}=\overline{(\bar{A} B}) \cdot(\overline{C \bar{D}})$

$\Rightarrow (A+\bar{B}) \cdot(\bar{C}+D)$ 

2. Find the complement of $A B+C D=0$

Ans: $A B+C D=0$

Taking complement on both sides.

$\Rightarrow \overline{A B+C D}=\bar{O} $

$\Rightarrow \overline{A B} \cdot \overline{C D}=1 $

$\Rightarrow (\bar{A}+\bar{B}) \cdot(\bar{C}+\bar{D})=1$

3. Simplify the Boolean expressions $(X+Y)(X+\bar{Y})(\bar{X}+Z)$

Ans: $(X+Y)(X+\bar{Y})=X X+X \bar{Y}+Y X+Y \bar{Y} $

$\Rightarrow X+X \bar{Y}+Y X+O, \text { as } X X=X \text { and } Y \overline{Y}= 0$

$\Rightarrow X+X(\bar{Y}+Y), \text { as } \bar{Y}+Y=1$

$\Rightarrow X+X \cdot 1, \text { as } X \cdot 1=X $

$\Rightarrow X+X $

$\Rightarrow X$

Now
$(X+Y)(X+\bar{Y})(\bar{X}+Z) $

$\Rightarrow X(\bar{X}+Z) $

$\Rightarrow X \bar{X}+X Z, \text { by distributive law }$

$\Rightarrow 0+X Z $

$\Rightarrow X Z$

Conclusion

In the article, we have discussed the detailed proof of Boolean Theorem and Boolean Laws. In the world of digitisation, everything works in the binary system and hence in Boolean algebra. So, for the growth of human society Boolean Algebra is a great tool and eases our day to day life.

Important Formulas to Remember

•  $(A . B)^{\prime}=A^{\prime}+B^{\prime}$

•  $(A + B)^{\prime}=A^{\prime}.B^{\prime}$

Important Points to Remember

• Boolean Algebra works on logical reasoning.

• Boolean Algebra has two trues either TRUE or FALSE i.e., $0,1$.

List of Related Links

• Boolean Algebra

• Introduction to Truth table