Various Implicants in K-Map
Last Updated :
18 Apr, 2025
An implicant can be defined as a product/minterm term in Sum of Products (SOP) or sum/maxterm term in Product of Sums (POS) of a Boolean function. For example, consider a Boolean function, F = AB + ABC + BC. Implicants are AB, ABC, and BC.
There are various implicant in K-Map listed below :
- Prime Implicant (PI)
- Essential Prime Implicant (EPI)
- Redundant Prime Implicant (RPI)
- Selective Prime Implicant (SPI)
POS and SOP are the types of boolean expression formed according to the given K-Map. POS stands for Product of Sum created by using maxterms and SOP stands for Sum of Product created by using minterms.
Prime Implicants
A group of squares or rectangles made up of a bunch of adjacent minterms which is allowed by the definition of K-Map are called prime implicants(PI) i.e. all possible groups formed in K-Map.
Example of Prime Implicants
Here we have an example of prime implicant for better understanding given below :
Essential Prime Implicants
These are those subcubes(groups) that cover at least one minterm that can't be covered by any other prime implicant. Essential prime implicants(EPI) are those prime implicants that always appear in the final solution.
Example of Essential Prime Implicants
Redundant Prime Implicants
The prime implicants for which each of its minterm is covered by some essential prime implicant are redundant prime implicants(RPI). This prime implicant never appears in the final solution.
Example of Redundant Prime Implicants
Here, we have an example with one redundant prime implicant.
Selective Prime Implicants
The prime implicants for which are neither essential nor redundant prime implicants are called selective prime implicants(SPI). These are also known as non-essential prime implicants. They may appear in some solution or may not appear in some solution.
Example of Selective Prime Implicants
Solved Examples of Various Implicants in K-Map
Here we have examples of prime implicant for better understanding given below :
Example 1
Given F = ∑(1, 5, 6, 7, 11, 12, 13, 15), find number of implicant, PI, EPI, RPI and SPI.
Expression : BD + A'C'D + A'BC+ ACD+ABC'
No. of Prime Implicants(PI) = 5 {1,2,3,4,5}
No. of Essential Prime Implicants(EPI) = 4 {1,2,3,4}
No. of Redundant Prime Implicants(RPI) = 1 {5}
No. of Selective Prime Implicants(SPI) = 0
Example 2
Given F = ∑(0, 1, 5, 8, 12, 13), find number of implicant, PI, EPI, RPI and SPI.
Expression : A'B'C'+ C'DB + C'D'A
No. of Prime Implicants(PI) = 6 {1,2,3,4,5,6}
No. of Essential Prime Implicants(EPI) = 0
No. of Redundant Prime Implicants(RPI) = 0
No. of Selective Prime Implicants(SPI) = 6 {1,2,3,4,5,6}
Example 3
Given F = ∑(0, 1, 5, 7, 15, 14, 10), find number of implicant, PI, EPI, RPI and SPI.
No. of Prime Implicants(PI) = 6 {1,2,3,4,5,6}
No. of Essential Prime Implicants(EPI) = 2 {1,4}
No. of Redundant Prime Implicants(RPI) = 2
No. of Selective Prime Implicants(SPI) = 4 {2,3,5,6}
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Practices Problems
1. Finding Prime Implicants
- Simplify the following Boolean function using a K-Map and identify all prime implicants:
f(A, B, C, D) = ∑(0, 1, 2, 5, 8, 9, 10, 14)
2. Minimization with Don't Care Conditions
- Given the Boolean function with don't care conditions, use a K-Map to minimize it:
f(A, B, C) = ∑(0, 1, 2, 5, 7) + d(3, 6)
3. Finding Essential Prime Implicants
- For the following function, find the essential prime implicants using a K-Map:
f(A, B, C, D) = ∑(1, 3, 7, 11, 15) + d(0, 2, 5)
4. Four-Variable K-Map Minimization
- Simplify the given Boolean function using a 4-variable K-Map:
f(A, B, C, D) = ∑(1, 3, 7, 11, 15, 19, 23, 27, 31)
5. Implicants and Essential Prime Implicants
- Determine all implicants and essential prime implicants for the function:
f(A, B, C) = ∑(0, 1, 2, 3, 5, 7)
6. Three-Variable K-Map Simplification
- Use a 3-variable K-Map to simplify the Boolean function:
f(A, B, C) = ∑(1, 3, 4, 6, 7)
7. Five-Variable K-Map Minimization
- Simplify the following Boolean function using a 5-variable K-Map:
f(A, B, C, D, E) = ∑(1, 3, 7, 15, 31)
8. Minimizing Boolean Expressions with Don't Cares
- Given the function and don't care conditions, use a K-Map to minimize:
f(A, B, C, D) =∑(2, 3, 5, 7, 11, 13) + d(0, 1, 9, 15)
9. Identification of Prime and Essential Prime Implicants
- For the following Boolean function, identify all prime implicants and essential prime implicants:
f(A, B, C, D) = ∑(4, 5, 6, 7, 12, 13, 14, 15)
10. Simplification Using K-Map with Multiple Variables
- Simplify the Boolean function using a K-Map and identify any essential prime implicants:
f(A, B, C, D, E) = ∑(0, 1, 2, 5, 8, 9, 10, 14, 16, 17, 18, 21)
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