The Sum of Product (SOP) and Product of Sum (POS) forms are two of the most commonly used Boolean algebra expressions in digital circuit design. SOP and POS forms are used to simplify complex Boolean expressions and reduce the number of gates needed in a circuit. In this article, we will explore the different types of SOP and POS forms, including the canonical and non-canonical forms, as well as the minimal forms. We will also discuss the schematic design of SOP and POS and the conversion between different forms.

## Sum of Products | Sum of Products Calculator | Sum of Products Boolean Algebra Examples | Product to Sum Formulas

• SoP is a form of Boolean algebra expression that represents logical functions.
• It consists of a sum of multiple products, where each product is made up of variables and their complements.
• SoP is also known as disjunctive normal form (DNF).
• SoP is the complement of the product of sums (PoS) expression.
• SoP expressions can be derived from truth tables, Karnaugh maps, or Boolean algebraic manipulations.
• SoP expressions can be used to simplify complex logical functions and reduce the number of gates needed to implement them.
• SoP expressions are used extensively in digital circuit design, computer architecture, and other fields that require logical reasoning.
• SoP expressions can be evaluated using basic logical operators such as AND, OR, and NOT.
• SoP expressions can be converted into other forms, such as sum of maxterms (SoMT) or product of maxterms (PoMT).
• SoP expressions can be extended to multiple dimensions, where each dimension represents a different input variable.

## Types of Sum of Product (SOP) Forms:

There are three types of Sum of Product (SOP) forms:

### Canonical SOP Form: The canonical SOP form is the standard form of representing a Boolean algebraic expression in SOP. In this form, the variables in each term of the SOP expression are combined using AND operation, and the terms are combined using OR operation.

For example, the canonical SOP form of the Boolean expression A.B + C.D.E + F.G is: (A' + B).(C' + D' + E).(F + G')

#### Non-Canonical SOP Form: Non-canonical SOP forms are SOP expressions that are not in the standard canonical form. These forms are usually more complex and difficult to work with than canonical SOP forms.

For example, the non-canonical SOP form of the Boolean expression A.B.C + D.E + F.G.H is:

(A.B + C).(D + E).(F.G + H)

### Minimal SOP Form: The minimal SOP form is the simplest form of representing a Boolean algebraic expression in SOP. In this form, there is only one term for each possible combination of input variables that results in a logical 1 output.

For example, the minimal SOP form of the Boolean expression A.B + A.C + B.C is:

A.B + A.C + B.C

## Schematic Design of Sum of Product (SOP):

The schematic design of Sum of Product (SOP) is a graphical representation of a Boolean algebraic expression using logic gates. The schematic design shows the logic gates used to implement the Boolean expression in a circuit.

## Conversion Techniques for Sum of Product (SOP):

Conversion from Minimal SOP to Canonical SOP Form: The conversion from minimal SOP to canonical SOP form involves the following steps:

• Determine the minterms for which the output of the Boolean expression is 1.
• Write the minterms in canonical SOP form by combining the input variables using AND operation.
• Combine the minterms using OR operation to obtain the canonical SOP form.
• Conversion from Canonical SOP to Canonical POS: The conversion from canonical SOP to canonical POS involves the following steps:
• Obtain the complement of the Boolean expression in canonical SOP form.
• Write the complement in canonical POS form.
• Obtain the complement of the resulting canonical POS form.

#### Conversion from Canonical SOP to Minimal SOP: The conversion from canonical SOP to minimal SOP involves the following steps:

• Expand the canonical SOP form using Boolean algebraic identities.
• Identify the essential prime implicants.
• Write the minimal SOP form using the essential prime implicants.

## Types of Product Of Sum Forms

The Product Of Sum (POS) form is another type of Boolean expression that can be used to represent a logic function. It is the complement of the Sum Of Product (SOP) form. The POS expression is formed by taking the product of the complements of the inputs and then summing those terms. There are different types of POS forms which are as follows:

#### Canonical POS FormNon-Canonical FormMinimal POS FormCanonical POS Form

The canonical POS form is the most widely used form of a POS expression. A POS expression is said to be in canonical form if it is expressed as the sum of products of literals, where each term includes every input variable or its complement. In this form, each term is called a "maxterm", which is the complement of a minterm in the SOP expression.

A truth table is a table that shows the output of a Boolean function for all possible combinations of its inputs. The output of a Boolean function can be either 0 or 1, depending on the input values.

The Canonical POS form is a form of representing a Boolean function using product terms. The truth table for a Boolean function in Canonical POS form can be constructed by following these steps:

Write down all the minterms for the function.

Write a 1 in the output column for each minterm, and a 0 for all other input combinations.

Write the minterm next to each 1 in the output column.

For example, let's consider the Boolean function f(A,B,C) = (A+B)(B+C). The minterms for this function are A'B', AB', B'C', and BC'. The truth table for this function in Canonical POS form would look like:

In this truth table, we can see that the minterms A'B', AB', B'C', and BC' correspond to the rows where the output is 1. These minterms can be combined using OR operations to obtain the Canonical POS form of the function, which is (A'B') + (AB') + (B'C') + (BC').

### Non-Canonical Form:

A POS expression that is not in the canonical form is called a non-canonical form. Non-canonical forms are sometimes used in circuit design, but they are not as useful as canonical forms. The non-canonical form of a POS expression can be obtained by taking the product of the non-minterms, i.e., the terms that do not contain all input variables.

### Minimal POS Form:

A POS expression is said to be in minimal form if it contains the minimum number of terms required to represent the function. A minimal POS expression can be obtained by using the Quine-McCluskey algorithm, just like the minimal SOP expression. In the minimal POS form, each term is a product of literals, just like the minimal SOP form.

## Schematic Design of Product of Sum (POS)

The schematic design of a POS expression is similar to that of an SOP expression. The only difference is that the AND gates and OR gates are reversed. In a POS expression, the inputs are first complemented, and then they are ANDed together to form a product term. These product terms are then ORed together to form the final output.

## Conversion from Minimal POS to Canonical form POS:

To convert a minimal POS expression to canonical form, we follow the same procedure as we did for the minimal SOP expression. We first write the minimal POS expression in a truth table and then use the Quine-McCluskey algorithm to obtain the canonical form.

### Conversion From Canonical POS to SOP:

To convert a canonical POS expression to SOP form, we first complement each input variable and then apply DeMorgan's theorem to obtain the SOP expression. For example, suppose we have the following POS expression:

F(A,B,C) = (A+B)(A+C)(B+C)

We first complement each input variable:

F(A,B,C) = (A'+B')(A'+C')(B'+C')

Then we apply DeMorgan's theorem to obtain the SOP expression:

F(A,B,C) = (A'+B')(A'+C')(B'+C')

= (A+B+C)(A+B'+C)(A'+B+C)(A'+B'+C)

### Canonical to Minimal POS:

To convert a canonical POS expression to minimal form, we use the Quine-McCluskey algorithm, just like we did for the minimal SOP expression. We first write the canonical POS expression in a truth table and then use the Quine-McCluskey algorithm to obtain the minimal form.

## Conclusion

In conclusion, Sum of Product (SOP) and Product of Sum (POS) are two important Boolean expressions used in digital electronics. They are used to represent logical functions in a simple and concise way. SOP expressions consist of multiple product terms that are combined using the logical OR operation, while POS expressions consist of multiple sum terms that are combined using the logical AND operation. The canonical form of both expressions is the most useful as it is unique and can be easily converted to other forms. The conversion process involves grouping the minterms or maxterms based on the number of literals and using them to write the canonical, minimal or non-canonical form of the expression.

## FAQs

### Q. What is a Sum of Product (SOP) expression?

A. A Sum of Product (SOP) expression is an expression in which multiple product terms are combined using a logical OR operation.

#### Q. What is a Product of Sum (POS) expression?

A. A Product of Sum (POS) expression is an expression in which multiple sum terms are combined using a logical AND operation.

#### Q. What is the canonical form of an expression?

A. The canonical form of an expression is the most useful form as it is unique and can be easily converted to other forms.

#### Q. What is the conversion process for converting an expression from the minimal form to the canonical form?

A. The conversion process involves grouping the minterms or maxterms based on the number of literals and using them to write the canonical form of the expression.

#### Q. Can a canonical SOP expression be converted to a canonical POS expression?

A. Yes, a canonical SOP expression can be converted to an equivalent canonical POS expression using De Morgan's theorem.

### Q. What is a minterm?

A. A minterm is a product term in which all the input variables appear once in either true or complemented form.

#### Q. What is a maxterm?

A. In Boolean algebra, a maxterm is an expression formed by taking the complement of each variable in a minterm and then ORing the resulting terms. It is the complement of the minterm and represents the maximum set of input conditions that result in an output of 0 for a given function.

A maxterm is also known as a sum of literals, because it is a sum of variables, where each variable can be either negated or non-negated. The sum of literals is called a maxterm because it represents the maximum number of input conditions that can result in an output of 0.

For example, the maxterm for the Boolean function f(A,B,C) = (A+B)(B+C) would be (A'+B')+(B'+C'). This is because the minterms for the function are A'B', AB', B'C', and BC', and the maxterm is formed by taking the complement of each of these terms and ORing them together.