Implicants are groups of adjacent 1s (for SOP) or 0s (for POS) in a Karnaugh Map (K-Map) that help simplify Boolean expressions. Different types of implicants determine which groups must be selected to obtain the minimum Boolean expression.
- Prime Implicant (PI)
- Essential Prime Implicant (EPI)
- Redundant Prime Implicant (RPI)
- Selective Prime Implicant (SPI)
Prime Implicants
A Prime Implicant (PI) is the largest possible group of adjacent cells that cannot be combined into a larger group. It represents one possible simplified term of the Boolean expression.
Example: Here we have an example of prime implicant for better understanding given below:
Essential Prime Implicants
An Essential Prime Implicant (EPI) is a prime implicant that covers at least one minterm not covered by any other prime implicant. It must be included in the final simplified expression.
Example:
Redundant Prime Implicants
A Redundant Prime Implicant (RPI) is a prime implicant whose minterms are already covered by essential prime implicants. It is not required in the final expression.
Example: Here, we have an example with one redundant prime implicant.
Selective Prime Implicants
A Selective Prime Implicant (SPI) is a non-essential prime implicant that may be selected to cover the remaining uncovered minterms after choosing all essential prime implicants.
Example
Solved Examples
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'
- Prime Implicants (PI): 5 → {1, 2, 3, 4, 5}
- Essential Prime Implicants (EPI): 4 → {1, 2, 3, 4}
- Redundant Prime Implicants (RPI): 1 → {5}
- 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'D'B + C'D'A
- Prime Implicants (PI): 6 → {1, 2, 3, 4, 5, 6}
- Essential Prime Implicants (EPI): 0
- Redundant Prime Implicants (RPI): 0
- 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 {3,5}
- No. of Redundant Prime Implicants (RPI) = 0
- No. of Selective Prime Implicants (SPI) = 4 {1,2,4,6}
Read More:
Practice Problems
- Find all Prime Implicants for the given K-Map.
- Identify the Essential Prime Implicants from the given groups.
- Determine the Redundant and Selective Prime Implicants.
- Write the minimum Boolean expression using the selected implicants.
- Verify whether a given implicant is Prime, Essential, Redundant, or Selective.