Minterm is the product of N distinct literals where each literal occurs exactly once. The output of the minterm functions is 1. Maxterm is the sum of N distinct literals where each literals occurs exactly once. The output of the maxterm functions is 0. The conversion from minterm to maxterm involves changing the representation of the function from a Sum of Products (SOP) to a Product of Sums (POS).
In this article, we will cover prerequisites like minterm, maxterm, minterm designation, maxterm designation, conversion from Cardinal form to Minterm Expression, and conversion from Cardinal form to Maxterm Expression with a detailed explanation of conversion from minterm expression to maxterm expression with solved examples and FAQs.
What is Minterm?
Minterm is also a fundamental part of Boolean algebra. Minterm is the product of various 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:
A’B + AC + BC
Read more, Minterm
What is Maxterm?
Maxterm is also 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’)
Read more, Maxterm
Minterm Designation
This is another way to represent a minterm. A binary pattern of min term is formed by using 1 and 0 corresponding to the variable and complement of variable respectively. The decimal equivalent of binary pattern used as the subscript of letter m, is known as Minterm designation.
Steps to obtain Minterm designation:
- Find binary pattern by writing 1 for the variable and 0 for the complement of the variable.
- Find decimal equivalent of the binary pattern.
- Use decimal number as a subscript of the letter ‘m’.
Example:
To find Minterm Designation of (ABC’)
Binary pattern of minterm (ABC’) : 110
Decimal equivalent of binary pattern 110 : 6
Hence, minterm designation of (ABC’) is: m6
Find Minterm Designation of A’B’C
Binary pattern of min term A’B’C : 001
Decimal equivalent of binary pattern 001 : 1
Hence, minterm designation of A’B’C is: m1
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
Conversion from Cardinal Form to Minterm Expression
A minterm expression can be obtained from a given cardinal form of boolean function by using the following steps:
- Convert the minterm designation into binary pattern having number of bits same as the number of variables used in the function.
- Write the product of variable or complement of variable corresponding to the bits 1 or 0 of the bit pattern respectively.
- Add the product terms (minterms) to get min term expression.
Example:
Find the maxterm expression for the boolean function: F(A,B,C) = ∑(5,1,3,6)
Binary of 5 is 101: AB’C
Binary of 1 is 001: A’B’C
Binary of 3 is 011: A’BC
Binary of 6 is 110: ABC’
Hence, minterm expression is: (AB’C)+(A’B’C)+(A’BC)+(ABC’)
Find the maxterm expression for the boolean function: F(A,B,C) = ∑(3,7)
Binary of 3 is 011: A’BC
Binary of 7 is 111: ABC
Hence, minterm expression is: (A’BC)+(ABC)
Conversion from Cardinal Form to Maxterm 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) = Π(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) = Π(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’)
Conversion from Minterm Expression to Maxterm Expression
- Find the minterm designations: For each min term, find its designation.
- Identify the missing minterms: A boolean function with n variables will have 2n possible minterms. Identify the minterms that are not present in the given expression. These will be the maxterms in the maxterm expression.
- Find the maxterm designations: For each missing minterm, find its maxterm designation. The maxterm designation is the complement of the minterm designation.
- Write the maxterms from their designations: Convert each maxterm designation into a term by replacing 0 with the uncomplemented variable and 1 with the complemented variable.
- Form the max termexpression: Finally, form the maxterm expression by taking the product of all the maxterms.
Solved Examples:
1. Convert the Minterm Expression to Maxterm Expression f(x, y, z) = x’yz + xy’z + x’y’z’
Minterm designation of x’yz = 011 = 3 = m3
Minterm designation of xy’z = 101 = 5 = m5
Minterm designation of x’y’z’ = 000 = 0 = m0
The function mentioned above has 3 variables. There can be 23 = 8 possible terms, out of which only three terms are used to form the expression. It means, the maxterm expression should have remaining 5 terms. Which are M1 , M2 , M4 , M6 , M7
Maxterm for M1 = 001 = (x + y + z’)
Maxterm for M2 = 010 = (x + y’ + z)
Maxterm for M4 = 100 = (x’ + y + z)
Maxterm for M6 = 110 = (x’ + y’ + z)
Maxterm for M7 = 111 = (x’ + y’ + z’)
The boolean function using maxterms will be: f(x, y, z) = (x + y + z’)(x + y’ + z)(x’ + y + z)(x’ + y’ + z)(x’ + y’ + z’)
2. Convert the Minterm Expression to Maxterm Expression f(x, y, z) = xyz’ + xy’z’ + x’yz’ + xyz
Minterm designation of xyz’ = 110 = 6 = m6
Minterm designation of xy’z’ = 100 = 4 = m4
Minterm designation of x’yz’ = 010 = 2 = m2
Minterm designation of xyz = 111 = 7 = m7
The function has 3 variables. There can be 23 = 8 possible terms, out of which only four terms are used to form the expression. It means, the maxterm expression should have the remaining 4 terms, which are M0, M1, M3, M5.
Maxterm for M0 = 000 = (x + y + z)
Maxterm for M1 = 001 = (x + y + z’)
Maxterm for M3 = 011 = (x + y’ + z’)
Maxterm for M5 = 101 = (x’ + y + z’)
The boolean function using maxterms will be: f(x, y, z) = (x + y + z)(x + y + z’)(x + y’ + z’)(x’ + y + z’)
3. Convert the Minterm Expression to Maxterm Expression f(x, y) = xy + xy’
Minterm designation of xy = 11 = 3 = m3
Minterm designation of xy’ = 10 = 2 = m2
The function has 2 variables. There can be 22 = 4 possible terms, out of which only two terms are used to form the expression. It means, the maxterm expression should have the remaining 2 terms, which are M0, M1.
Maxterm for M0 = 00 = (x + y)
Maxterm for M1 = 01 = (x + y’)
The boolean function using maxterms will be: f(x, y) = (x + y)(x + y’)
Conversion From Minterm Expression to Maxterm Expression – FAQs
Why are minterms used for?
Minterms is used for canonical representation of Boolean functions.
How to represent minterms in K-map?
The minterms in K-map are represented by m.
Why are maxterms used for?
Maxterms is used for canonical representation of Boolean functions.
How we represent maxterms in K-maps?
Maxterms are represented by ‘M’ in K-maps. The output result of maxterm function is 0.
What is the difference between Minterm and 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.
Refer this article for a detailed answer : minterm vs maxterm
Write the two standard forms to represent Boolean expression used in K-Map.
The two standard forms to represent Boolean expression used in K-Map are SOP and POS.