Implicants

Each appearance of a variable, either un-complemented or complemented, is called a literal

If a function returns the value whenever a product term is (i.e implies ), then is called an implicant of .

This function has 11 implicants; five minterms (), and six other product terms obtained by various groupings of the five minterms:

An implicant is called a prime implicant if it cannot be combined into another implicant that has fewer literals (i.e it is not encircled by a bigger group)

In the previous example:

Prime implicants are and .

Essential prime implicants are prime implicants that cover a at the output of the function that no combination of other prime implicants is able to cover.

In the previous example:

In this case, the prime implicants and are also essential prime implicants.

A collection of implicants that account for all valuations for which a given function is called a cover.

In the previous example:

Valid covers include:

Consider the following K-map

There are 5 prime implicants, which are circled. 3 of these are essential (red). These three essential prime implicants must be included in the cover (they cover all the minterms except , which we will use for).

Thus the minimum cost realization for this function is .