Minterm & Maxterm | Digital Circuits - Electronics and Communication Engineering (ECE) PDF Download

Introduction

There are two ways in which we can put the Boolean function. These ways are the minterm canonical form and maxterm canonical form.

Literal

A Literal signifies the Boolean variables including their complements. Such as B is a boolean variable and its complements are ~B or B', which are the literals.

Minterm

The product of all literals, either with complement or without complement, is known as minterm.

Example:
The minterm for the Boolean variables A and B is:

A.B  
A.~B  
~A.B  

The complement variables ~A and ~B can also be written as A' and B' respectively. Thus, we can write the minterm as:

A.B'  
A'.B  

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Minterm from values

Using variable values, we can write the minterms as:

• If the variable value is 1, we will take the variable without its complement.
• If the variable value is 0, take its complement.

Example: 
Let's assume that we have three Boolean variables A, B, and C having values

A = 1
B = 0
C = 0

Now, we will take the complement of the variables B and C because these values are 0 and will take A without complement. So, the minterm will be:
Minterm=A.B'C'
Let's take another example in which we have two variables B and C having the value

B = 0
C = 1
Minterm = B'C

Shorthand Notation for Minterm

We know that, when Boolean variables are in the form of minterm, the variables will appear in the product. There are the following steps for getting the shorthand notation for minterm.

• In the first step, we will write the term consisting of all the variables
• Next, we will write 0 in place of all the complement variables such as ~A or A'.
• We will write 1 in place of all the non-complement variables such as A or b.
• Now, we will find the decimal number of the binary formed from the above steps.
• In the end, we will write the decimal number as a subscript of letter m(minterm). Let's take some example to understand the theory of shorthand notation

Example 1: Minterm = AB'

• First, we will write the minterm:
  Minterm = AB'
• Now, we will write 0 in place of complement variable B'.
  Minterm = A0
• We will write 1 in place of non-complement variable A.
  Minterm = 10
• The binary number of the minterm AB' is 10. The decimal point number of (10)₂ is 2. So, the shorthand notation of AB' is
  Minterm = m₂

Example 2: Minterm = AB'C'

• First, we will write the minterm:
  Minterm = AB'C'
• Now, we will write 0 in place of complement variables B' and C'.
  Minterm = A00
• We will write 1 in place of non-complement variable A.
  Minterm = 100
• The binary number of the minterm AB'C' is 100. The decimal point number of (100)₂ is 4. So, the shorthand notation of AB'C' is
  Minterm = m₄

Maxterm

The sum of all literals, either with complement or without complement, is known as maxterm.

Example:
The maxterm for the Boolean variables A and B will be:

A+B  
A+~B  
~A+B  

We know that the complement variables ~A and ~B can be written as A' and B' respectively. So, the above maxterm can be written as

A+B'  
A'+B  

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Maxterm from values

Using the given variable values, we can write the maxterm as:

• If the variable value is 1, then we will take the variable without a complement.
• If the variable value is 0, take the complement of the variable.

Example:
Let's assume that we have three Boolean variables A, B., and C having values

A = 1
B = 0
C = 0

Now, we will take the complement of the variables B and C because these values are 0 and will take A without complement. So, the maxterm will be:

Maxterm = A + B' + C'

Let's take another example in which we have two variables B and C having the value

B = 0
C = 1

Maxterm = B' + C

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Shorthand Notation for Maxterm

We know that, when Boolean variables are in the form of maxterm, the variables will appear in sum. The steps for the maxterm are same as minterm:

• In the first step, we will write the term consisting of all the variables
• Next, we will write 0 in place of all the complement variables such as ~A or A'.
• We will write 1 in place of all the non-complement variables such as A or b.
• Now, we will find the decimal number of the binary formed from the above steps.
• In the end, we will write the decimal number as a subscript of letter Here, M denotes maxterm.

Let's take some example to understand the theory of shorthand notation

Example 1: Maxterm = A+B'

• First, we will write the minterm:
  Maxterm = A+B'
• Now, we will write 0 in place of complement variable B'.
• We will write 1 in place of non-complement variable A.
• The binary number of the maxterm A+B' is 10. The decimal point number of (10)₂ is 2. So, the shorthand notation of A+B' is
  Maxterm = M₂

Example 2: Maxterm = A+B'+C'

• First, we will write the maxterm:
• Maxterm = A+B'+C'
• Now, we will write 0 in place of complement variables B' and C'.
• We will write 1 in place of non-complement variable A.
• The binary number of the maxterm A+B'+C' is 100. The decimal point number of (100)₂ is 4. So, the maxterm of A+B'+C' is m₄.