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# Various Implicants in K-Map

Implicant refers to any product/minterm term in the SOP (Sum of Products) or sum/maxterm term in the POS (Product of Sums) of a Boolean function. For example, if we consider a boolean function, F = MN + MNO + NO, the implicants here would be MN, MNO and NO.

In this article, we will take a look at the Various Implicants in K-Map according to the GATE Syllabus for CSE (Computer Science Engineering). Read ahead to learn more.

## Various Implicants in K-Map

An implicant refers to the product/minterm term in the SOP (Sum of Products) or the sum/maxterm term in the POS (Product of Sums) of a Boolean function. For example, let us consider any boolean function, F = MN + MNO + NO, then implicants are MN, MNO and NO. Before reading ahead, learn more about the basics of K-Map and Boolean Algebra here.

## Prime Implicants

A group of squares or rectangles made up of a bunch of adjacent minterms that is allowed by definition of a Karnaugh Map are known as prime implicants or PI, i.e. all the possible groups that are formed in K-Map.

## Redundant Prime Implicants

The redundant prime implicants or RPI refer to the prime implicants for which every one of its minterms gets covered by some important prime implicants. This type of prime implicant never happens to appear in the final solution.

## Essential Prime Implicants

These refer to those subcubes or groups that cover at least one of the minterms that canâ€™t get covered by another prime implicant. The EPI or essential prime implicants refer to the prime implicants that always appear in the final solution.

## Selective Prime Implicants

The SPI or selective prime implicants refer to those prime implicants for which neither the redundant nor essential prime implicants are there. They are also called non-essential prime implicants. These may appear in certain types of solutions or may not even appear in some solutions at all.

### Example

#### Example-1

Find the number of implicants, EPI, PI, RPI and SPI if F = âˆ‘(1, 5, 6, 7, 11, 12, 13, 15)

Expression : QS + P’R’S + P’QR+ PRS + PQR’

Total number of Implicants = 8

Total number of PI or Prime Implicants = 5

Total number of EPI or Essential Prime Implicants = 4

Total number of RPI or Redundant Prime Implicants = 1

Total number of SPI or Selective Prime Implicants = 0

#### Example-2

Find the number of implicants, EPI, PI, RPI and SPI if F = âˆ‘(0, 1, 5, 8, 12, 13)

Expression : P’Q’R’ + R’SQ + R’D’P

Total number of Implicants = 6

Total number of PI or Prime Implicants = 6

Total number of EPI or Essential Prime Implicants = 0

Total number of RPI or Redundant Prime Implicants = 0

Total number of SPI or Selective Prime Implicants = 6

#### Example-3

Find the number of implicants, EPI, PI, RPI and SPI if F = âˆ‘(0, 1, 5, 7, 15, 14, 10)

Total number of Implicants = 7

Total number of PI or Prime Implicants = 6

Total number of EPI or Essential Prime Implicants = 2

Total number of RPI or Redundant Prime Implicants = 2

Total number of SPI or Selective Prime Implicants = 4

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