Various Implicants in K-Map

**Prerequisite –** K – Map (Karnaugh Map)

Implicant is a product/minterm term in Sum of Products (SOP) or sum/maxterm term in Product of Sums (POS) of a Boolean function. E.g., consider a boolean function, F = AB + ABC + BC. Implicants are AB, ABC and BC.

**Prime Implicants –**

A group of square or rectangle made up of bunch of adjacent minterms which is allowed by definition of K-Map are called**prime implicants(PI)**i.e. all possible groups formed in K-Map.**Example:****Essential Prime Implicants –**

These are those subcubes(groups) which cover atleast one minterm that can’t be covered by any other prime implicant.**Essential prime implicants(EPI)**are those prime implicants which always appear in final solution.**Example:****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 final solution.**Example:****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:**

**Example-1:** Given F = ∑(1, 5, 6, 7, 11, 12, 13, 15), find number of implicant, PI, EPI, RPI and SPI.

No. of Implicants = 8 No. of Prime Implicants(PI) = 5 No. of Essential Prime Implicants(EPI) = 4 No. of Redundant Prime Implicants(RPI) = 1 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.

No. of Implicants = 6 No. of Prime Implicants(PI) = 6 No. of Essential Prime Implicants(EPI) = 0 No. of Redundant Prime Implicants(RPI) = 0 No. of Selective Prime Implicants(SPI) = 6

**Example-3:** Given F = ∑(0, 1, 5, 7, 15, 14, 10), find number of implicant, PI, EPI, RPI and SPI.

No. of Implicants = 7 No. of Prime Implicants(PI) = 6 No. of Essential Prime Implicants(EPI) = 2 No. of Redundant Prime Implicants(RPI) = 2 No. of Selective Prime Implicants(SPI) = 4

**Example-4:** GATE IT 2006 | Question 35