Various Implicants in K-Map - GeeksforGeeks (2024)

Implicant is 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.

In this article, we will explore various implicants in K-Map with examples for better understanding and k-map diagram. Also, we will look into the Boolean expressions formed for each k-map.

Various Implicants in K-Map

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.

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

Here we have 2 examples of prime implicant for better understanding given below :

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 2 examples of prime implicant for better understanding given below :

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

Here we have 2 examples of prime implicant for better understanding given below :

Solved Examples of Various Implicants in K-Map

Here we have 2 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 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.

Expression : A'B'C'+ C'DB + C'D'A
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

Conclusion

In the conclusion, we have basically four types of implicants in k-map named as Prime Implicants(PI), Essential Prime Implicants(EPI), Redundant Prime Implicants(RPI) and Selective Prime Implicants(SPI). These four types provides a clear structure in the formation of groups in k-map which makes the Boolean expression formed with more clarity.

Read More:

• Introduction of K-Map (Karnaugh Map)
• 5 variable K-Map in Digital Logic – GeeksforGeeks

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)

Various Implicants in K-Map – FAQs

What is the difference between essential prime implicant and prime implicant?

An Essential Prime Implicants(EPI) is a subset of Prime Implicants(PI) where EPI contains at least one “1” or minterms that is not shared in any other PI or group made in a k-map.

What is POS and SOP used in Various Implicants in K-Map ?

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.

What is the maximum essential prime implicants?

The maximum essential prime implicants for a n-variable Boolean functionwill be 2(n – 1)