Minterms and Maxterms are important parts of Boolean algebra. Minterm is the product of N distinct literals where each literal occurs exactly once. The output of the minterm functions is 1. Minterm is represented by m. Maxterm is the sum of N distinct literals where each literals occurs exactly once. The output of the maxterm functions is 0. Maxterm is represented by M. In this article we will learn about minterms and maxterms, their difference, why we use minterms and maxterms along with the solved examples.
What is Minterm?
Minterm is the product of various different literals in which each literal occurs exactly once. The output result of the minterm functions is 1. It is represented by m. To represent a function, we perform a sum of minterms also called the Sum Of Products (SOP).
Example of SOP:
A’B + AC + BC
Table for Two-Variable Minterm
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Variable
|
Minterm
|
|
A
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B
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Term
|
Representation
|
|
0
|
0
|
A’B’
|
m0
|
|
0
|
1
|
A’B
|
m1
|
|
1
|
0
|
AB’
|
m2
|
|
1
|
1
|
AB
|
m3
|
Minterms for Values
Minterms for values is the minterms obtained by the values of the Boolean variable.
Steps for Obtaining Minterms from Values
1. If the value of the Boolean variable is 1 then we will take the variable without complementing it.
2. If the value of the Boolean variable is 0 then we will take the variable by complementing it.
Minterm Example:
If there are four Boolean variables A, B, C, D with the values A = 1, B = 0, C = 0 and D = 1. Find the minterms for values given.
Solution:
Given the values of the Boolean variables:
A = 1, B = 0, C = 0 and D = 1
The required minterm is given by = AB’C’D
We complemented B and C as its value is 0.
What is Maxterm?
Maxterm is the sum of various different literals in which each literal occurs exactly once. The output result of the maxterm functions is 0. It is represented by M. To represent a function, we perform product of maxterms also called Product of Sum (POS).
Example of POS:
(A+B). (C+D)
Table for Two Variable Maxterm
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Variable
|
Maxterm
|
|
A
|
B
|
Term
|
Representation
|
|
0
|
0
|
A+B
|
M0
|
|
0
|
1
|
A+B’
|
M1
|
|
1
|
0
|
A’+B
|
M2
|
|
1
|
1
|
A’+B’
|
M3
|
Maxterms for Values
Maxterms for values is the maxterms obtained by the values of the Boolean variable.
Steps for obtaining maxterms from values:
1. If the value of the Boolean variable is 0 then we will take the variable without complementing it.
2. If the value of the Boolean variable is 1 then we will take the variable by complementing it.
Maxterm Example:
If there are four Boolean variables A, B, C, D with the values A = 1, B = 0, C = 0 and D = 1. Find the maxterm for values given.
Solution:
Given the values of the Boolean variables:
A = 1, B = 0, C = 0 and D = 1
The required maxterm is given by = A’+ B + C + D’
We complemented A and D as its value is 1 and B and C are 0 therefore there are no change.
Minterm vs Maxterm
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Minterm
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Maxterm
|
|
Minterm is the term with the product of N literals occurring exactly once.
|
Maxterm is the term with the sum of N literals occurring exactly once.
|
|
It is represented by m.
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It is represented by M.
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| It is logical AND of distinct literals. |
It is logical OR of distinct literals. |
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The sum of minterms forms SOP (Sum of Product) functions.
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The product of maxterms forms POS (Product of Sum) functions.
|
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The output result of minterm function is 1.
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The output result of maxterm function is 0.
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| It works on active high. |
It works on active low. |
| Example: AB + A’B’ |
Example: (A+B). (A’+B’) |
Below mentioned is the image of Four Variable K-Map.
Why we use Minterms and Maxterms ?
Minterms and Maxterms are used to express the Boolean functions in its canonical forms. The minterms and maxterms are the basic building blocks for representing the Boolean functions in the canonical forms. The minterms are the terms which are used when we require active high output. The maxterms are the terms which are used when we require active low output.
Solved Examples on Minterms and Maxterms
Example 1: For the Boolean variables A = 0, B = 1 and C = 0 obtain the minterm and maxterm for the variables.
Solution:
Given the values of Boolean variables as:
A = 0, B = 1 and C = 0
The required minterm for above values = A’BC’
The required maxterm for above values = A + B’ + C
Logic gate diagram for above minterm and maxterm
Logic Gate diagram for Minterm and Maxterm
Example 2: Simplify the POS form and obtain the result in POS form only.
F(A, B, C, D) = (A + B + C + D)(A + B’ + C + D’)(A + B’ + C’ + D)(A + B’ + C’ + D’)(A’ + B + C + D)(A’ + B + C’ + D’)(A’ + B’ + C’ + D’)
Solution:
We draw K-Map to simplify the given POS
The simplified POS form F(A, B, C, D) = (B + C + D)(A + B’ + D’ )(A + B’ + C’)(A’ + C’ + D’)
Logic gate diagram for above maxterm:
Logic gate diagram for maxterm
Example 3: Simplify the SOP form and obtain the result in SOP only.
F(A, B, C, D) = A’BC’D’ + A’BC’D + A’BCD + AB’C’D’ + AB’C’D + ABC’D’ + ABCD’
Solution:
We draw K-Map to simplify the given SOP
The Simplified SOP form is F(A, B, C, D) = ABD’ + AB’C’ + A’BC’ + A’BD
Logic gate diagram for above minterm:
Logic gate diagram for minterm
FAQs on Minterm Vs Maxterm
Q.1: Why are minterms and maxterms used for?
Answer:
Minterms and maxterms are used for canonical representation of Boolean functions.
Q.2: How we represent minterms in K-maps?
Answer:
Minterms are represented by m in K-maps. The output result of minterm function is 1.
Q.3: Write the two standard forms to represent the boolean expression used in K-map?
Answer:
The sum of minterms in a function is called SOP (Sum of Product) and the product of maxterms in a function is called POS (Product of Sum).