Prime Implicants and Explicit Implicants

Implicants play a crucial role in Boolean logic, as they form the building blocks for both SOP and POS expressions. An implicant can be thought of as a product term in SOP or a sum term in POS representing a Boolean function. Essentially, implicants encapsulate the various input combinations (minterm or maxterm) for which the Boolean function evaluates to true.

Sum of Products

In digital electronics, SOP is an acronym for Sum of Products. SOP is a logical expression that represents a Boolean function in terms of the logical OR operation (sum) and the logical AND operation (products).

Example:

For example, consider a Boolean function F(A, B, C, D) defined as:

F(A, B, C, D) = (A AND B) OR (A AND C) OR (B AND D)

In SOP form, this function can be represented as:

F(A, B, C, D) = AB + AC + BD

In the above function A,B,C,D are literals or the inputs of the combinational circuit. Each product term can be one or more than one literals ANDed together.

Product of Sums

In digital electronics, POS is acronym for Product of Sum. It is a method of representing and simplifying Boolean expressions. The POS form is the complement of the SOP (Sum of Products) form.

Example:

For example, consider a Boolean function F(A, B, C, D) defined as:

F(A, B, C, D) = (A OR B OR C’) AND (A OR B’ OR C) AND (A OR C OR D’)

In POS form, this function can be represented as:

POS(A, B, C, D) = (A + B + C’) * (A + B’ + C)*(A + C + D’)

Implicants

Implicant is a product/minterm term in Sum of Products (SOP) or sum/maxterm term in Product of Sums (POS) of a Boolean function.

Example:

For example., consider a boolean function F(A, B, C) defined as:

F = AB + ABC + BC.

its implicants are AB, ABC, and BC. 

Now let’s understand prime implicants and explicit implicants.

Prime implicants

A prime implicant is an implicant that cannot be further reduced or combined with other implicants while still covering all the minterms (input combinations) for which the Boolean function is true. Prime implicants are important in the process of logical minimization.

Example:

Lets take an example of prime Implicants:

prime implicants

Explicit implicants

Explicit implicants represent the minimal set of terms needed to cover all the output 1s in the truth table. They are used in logic optimization techniques such as Karnaugh maps and Quine-McCluskey method to simplify the logic function.

Example:

A	B	C	X
0	0	0	1
0	0	1	0
0	1	0	0
0	1	1	1
1	0	0	0
1	0	1	1
1	1	0	0
1	1	1	1

minterms = (0,3,5,7)

Lets make group of this minterms , there are several possible groupings, lets make two groups (0,3) and (5,7)

Therefore, (A’ B’ C’ + A’ B C) and (A B’ C + A B C) are explicit implicants.

Conclusion

In conclusion, understanding prime implicants, explicit implicants, and the application of SOP and POS expressions in Boolean logic is fundamental for digital circuit design and optimization. These concepts helps to create efficient digital systems.

Frequently Asked Questions

1. When should I use SOP (Sum of Products) and when should I use POS (Product of Sums) expressions in Boolean logic design?

SOP expressions are used when we are minimizing expressions for functions with mainly 1s in the truth table, while POS expressions are used when we are optimizing functions with mainly 0s

2. How are explicit implicants different from prime implicants?

Explicit implicants basically represents minimal set of terms which are required to cover all the output 1s in the truth table while prime implicants are implicants that cannot be further simplified and used as intermediate step in minimization.