Maxterms are defined as the sum of distinct literals, and they are used to represent Boolean functions that equal 0. In this article we will know What is Maxterm, how to find Maxterm designations and their cardinal form. This article covers two-variable, three-variable, four-variable maxterm with their K-Map, conversion from cardinal form to Maxterm expressions with examples and also the advantages and disadvantages of using Maxterms.
What is Maxterm?
Maxterm is a fundamental part of Boolean algebra. Maxterms are the sum of various distinct literals in which each literal occurs exactly once (in either its complement or un-complement form). The output result of the maxterm function is 0. It is represented by M. To represent a function, we perform the product of maxterms which is called the Product of Sum (POS) or maxterm expression.
Example:
(A+B).(A'+B).(A'+B')
The expression above, contains each term with the combination of both the variables used in the expression. Hence, it is a canonical Product of Sum (POS) expression of maxterm expression.
Maxterm Designation
This is another way to represent a maxterm. A binary pattern of maxterm is formed by using 0 corresponding to the variable and 1 corresponding to the complement of variable. The decimal equivalent of binary pattern used as subscript with letter M, is known as Maxterm designation.
Steps to obtain Maxterm designation:
- Find binary pattern by writing 0 for the variable and 1 for the complement of the variable.
- Find decimal equivalent of the binary pattern.
- Use decimal number as a subscript of the letter M.
Example:
Find Maxterm Designation of (A' + B + C)
Binary pattern of maxterm (A' + B + C): 100
Decimal Equivalent of binary pattern 100: 4
Hence, maxterm designation of (A' + B + C) is: M4
Find Maxterm Designation of (A + B + C)
Binary pattern of maxterm (A + B + C): 000
Decimal Equivalent of binary: 0
Hence, maxterm designation of (A + B + C) is: M0
Two-Variable Maxterm
Maxterms for two variables are called two-variable maxterm.
Two variable Maxterm Table:
|
|
0
| 0
| A+B
| M0
|
0
| 1
| A+B'
| M1
|
1
| 0
| A'+B
| M2
|
1
| 1
| A'+B'
| M3
|
Two variable Maxterm K-Map:
Two Variable K-Map
Three Variable Maxterm
Maxterms for three variables are called three variable maxterm.
Three variable Maxterm Table:
|
|
0
| 0
| 0
| A+B+C
| M0
|
0
| 0
| 1
| A+B+C'
| M1
|
0
| 1
| 0
| A+B'+C
| M2
|
0
| 1
| 1
| A+B'+C'
| M3
|
1
| 0
| 0
| A'+B+C
| M4
|
1
| 0
| 1
| A'+B+C'
| M5
|
1
| 1
| 0
| A'+B'+C
| M6
|
1
| 1
| 1
| A'+B'+C'
| M7
|
Three variable Maxterm K-Map:
Three variable Maxterm K-Map
Four Variable Maxterm
Maxterms for four variables are called four variable maxterm.
Four variable Maxterm Table:
|
|
0
| 0
| 0
| 0
| A+B+C+D
| M0
|
0
| 0
| 0
| 1
| A+B+C+D'
| M1
|
0
| 0
| 1
| 0
| A+B+C'+D
| M2
|
0
| 0
| 1
| 1
| A+B+C'+D'
| M3
|
0
| 1
| 0
| 0
| A+B'+C+D
| M4
|
0
| 1
| 0
| 1
| A+B'+C+D'
| M5
|
0
| 1
| 1
| 0
| A+B'+C'+D
| M6
|
0
| 1
| 1
| 1
| A+B'+C'+D'
| M7
|
1
| 0
| 0
| 0
| A'+B+C+D
| M8
|
1
| 0
| 0
| 1
| A'+B+C+D'
| M9
|
1
| 0
| 1
| 0
| A'+B+C'+D
| M10
|
1
| 0
| 1
| 1
| A'+B+C'+D'
| M11
|
1
| 1
| 0
| 0
| A'+B'+C+D
| M12
|
1
| 1
| 0
| 1
| A'+B'+C+D'
| M13
|
1
| 1
| 1
| 0
| A'+B'+C'+D
| M14
|
1
| 1
| 1
| 1
| A'+B'+C'+D'
| M15
|
Four variable Maxterm K-Map:
Four variable Maxterm K-Map
Cardinal Form of Maxterm Expression
A boolean function which is defined by using maxterm designations is known as cardinal form of maxterm expression.
Syntax:
f(variables) = Î (max term designations)
Example:
Let an expression be: (A+B'+C)(A+B'+C')(A'+B'+C)
Binary pattern of maxterm (A+B'+C) : 010 : 2
Binary pattern of maxterm (A+B'+C') : 011 : 3
Binary pattern of maxterm (A'+B'+C) : 110 : 6
Equivalent maxterm designation is: M2 . M3 . M6
Its cardinal form will be: f(A, B, C)= Î (2, 3, 6)
(where, Î denotes product)
Conversion from Cardinal Form to Max Term Expression
A maxterm expression can be obtained from a given boolean function in cardinal form by using the following steps:
- Convert maxterm designation into binary form keeping the total number of bits same as the number of variables used in the function.
- For 0 take the variable and for 1 take the complement of the variable.
- Add the variable and the complement for every bit to obtain the maxterm
- Multiply the maxterms to get maxterm expression.
Example:
Find the maxterm expression for the boolean function: F(A, B, C) = Î (7, 3)
Binary of 7 is 111: (A' + B' + C')
Binary of 3 is 011: (A + B' + C')
Hence, Max Term expression: (A' + B' + C')(A + B' + C')
Find the maxterm expression for the boolean function: F(A, B, C) = Î (0, 3, 5)
Binary of 0 is 000: (A + B + C)
Binary of 3 is 011: (A + B' + C')
Binary of 5 is 101: (A' + B + C')
Hence, Max Term expression: (A + B + C)(A + B' + C')(A' + B + C')
Advantages:
- Maxterms can be used to represent all possible Boolean functions.
- They can be used to simplify Boolean expressions.
Disadvantages:
- Maxterms can lead to more complex expressions than minterms.
- They can be more difficult to understand and analyze than minterms.
Q.1: Why are maxterms used for?
Answer:
Maxterms is used for canonical representation of Boolean functions.
Q.2: What is the difference between Minterm and Maxterm?
Answer:
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.
Refer this article for a detailed answer : minterm vs maxterm
Q.3: How we represent maxterms in K-maps?
Answer:
Maxterms are represented by 'M' in K-maps. The output result of maxterm function is 0.
Q.4: What is Product of Sum (POS) Form?
Answer:
It is way of representing boolean expressions as product of maxterms.
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