# SOP and POS Digital Logic Designing with solved examples

Table of Contents

**Description:**

**Sop and Pos digital Logic designing-** In this tutorial you will learn about the SOP “Sum of Product” and POS “Product of Sum” terms in detail. We will discuss each one in detail and we will also solve some examples. The **SOP** (Sum of Product) and **POS** (Product of Sum) are the methods for deducing a particular logic function. In other words, these are the ways to represent the deduced reduced logic function. We can use the deduced logic function in designing a logic circuit.

If you want to learn about the Logic Gates in digital Electronics then I highly recommend read my previous article on Logic Gates in Digital Electronics Complete Guide.

Without any further delay let’s get started!!!

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**Logic Design:**

The logic gates which are combined for specific Boolean function is called logic design. So, Logic Design is the basic organization of the circuitry of a digital computer. All digital computers are based on a two-valued logic system 1/0, ON/OFF, YES/NO. Computers perform calculations using components called logic gates. Which are made up of integrated circuits that receive an input signal, process it, and change it into an output signal. The components of the gates pass or block a clock pulse as it travels through them, and the output bit of the gates control other gates or output the result. There are three basic kinds of logic gates, called “and”, “or”, and “not”. By connecting logic gates together, a device can be constructed that can perform basic arithmetic functions.

As I have already explained all the logic gates in detail in my previous article, so in this article we will only talk about the **SOP and POS** in digital logic designing.

**SOP and POS:**

The terms “product” and “sum” have been borrowed from mathematics to describe AND and OR logic operations. Any logic system can be represented in one of these two logic ways. As it will be explained in this section, the two forms are equivalent ways of expressing a logic system; however, some logic systems lend themselves to one rather than the other.

A product term is defined as an AND relationship between any number of variables, and a sum term is defined as an OR relationship between any number of logic variables. Any logic system can be represented in two logically equivalent ways: as the OR’ing of AND’ed terms, known as the Sum of Products (SOP) form; or as the AND’ing of OR’ed terms, known as the Product of Sums (POS) form. The two forms are interchangeable, and one form can be transformed to the other following a few basic rules. As an example, consider the XOR relationship YSOP:

YSOP=A⋅B+A⋅B

This SOP relationship can be expressed in POS form as:

YPOS=(A+B)⋅(A+B)

In this example, the POS and SOP forms are equally simple, but this is not always the case. For circuits with more than two inputs, it may turn out that one form is simpler that the other. If a circuit is to be constructed, it makes sense to evaluate both forms so that the simplest one can be constructed.

**Sum of Product (SOP) or (AND-OR implementation):**

In SOP several product terms are logically added, AND-OR gates are used.

**Example: **

Implement the Function given below using the SOP.

F = ABC + DEF

**Sol:**

**Example:**

Implement the function given below using SOP.

F = AB + CD + EF + GH

**Sol:**

Isn’t it simple? how easily we can construct SOP circuits from given logic equations. Let’s dig deeper and this time we will construct SOP from a Truth Table.

**Construct SOP from a Truth Table:**

There are certain rules that you need to follow while constructing a logic circuit from any truth table. These rules for the SOP circuits are given below:

- A circuit for a truth table with N input columns can use AND gates with N inputs, and each row in the truth table with a ‘1’ in the output column requires one N-input AND gate.
- Inputs to the AND gate are inverted if the input shows a ‘0’ on the row, and not inverted if the input shows a ‘1’ on the row.
- All AND terms are connected to an M-input OR gate, where M is the number of ‘1’ output rows.
- The output of the OR gate is the function output.

SOP implementation from a truth table using AND-OR Gates combination.

### Product of Sums (POS)

**Construct POS from a Truth Table:**

Unlike the SOP the POS also has certain rules which are given below.

- A circuit for a truth table with N input columns can use OR gates with N inputs, and each row in the truth table with a ‘0’ in the output column requires one N-input OR gate.
- Inputs to the OR gate are inverted if the input shows a ‘1’ on the row, and not inverted if the input shows a ‘0’ on the row.
- All OR terms are connected to an M-input AND gate, where M is the number of ‘1’ output rows.
- The output of the AND gate is the function output.

POS implementation from a truth table given above using OR-AND Gates combination.

**Boolean Variable:**

Such type of a variable which have one value among two possible values (0,1).

**Package:**

Package is a combination of similar gates.

**Minterms:**

A binary variable may be either in its normal form (x) or in it’s complement form x’. Consider three binary variables x, y, and z. The possible combinations will be eight (8).

2^{n }= 2^{3 }= 8

Where n = number of variables.

Combining all combinations with AND gate in such a way that we take each variable being primed if the corresponding bit is zero and unprimed if the corresponding bit is 1.

Each of these 8 AND terms represents one of the distinct area in the Venn diagram and is called a Minterm or a standard product. Minterms are represented by small letter (m).

**Maxterm:**

Consider three binary variables x, y, and z. We will have eight possible combinations.

2^{n }= 2^{3 = }8

Where n is the number of variables.

Combining all combinations with OR gate in such a way that, we take each variable being primed if the corresponding bit is 1 and unprimed if the corresponding bit is zero (0). Each of these 8 OR terms represents one of the distinct area in the Venn diagram and is called a Maxterm or a standard sums. Note that each Maxterm is the compliment of its corresponding Minterm and Vice Versa. Minterms and Maxterms for three variables shown in the table.

**Canonical form:**

Expressing a Boolean function in SOP or POS is called Canonical form.

**Standard form:**

Simplified form of the Boolean function for example

F = x + y

As we know x + yz = (x+y) (x+z)

F = (x + ) (x + y)

As x + = 1

So,

F = x + y

This is the standard form, because further it cannot be simplified. So it’s the standard form.

**Use of Minterms:**

For Minterms consider 1’s in the truth table given above

∑ Symbol is only taken in Minterms and it also called. Now for the Maxterms we will consider 0’s from the table.

For Maxterms consider 0’s in the truth table given above.

Maxterm is also called POS (Product of sum).

**Example:**

Express the Boolean function

in a product of Maxterm form.

**Sol:**

In the given function F, z is missing in the first term and y is missing in the 2^{nd} term. So,

From the truth table below.

For Maxterms we consider 0’s in the table. We also know that each Maxterm is the complement of the corresponding Minterms.

**Conversion between Canonical forms:**

We know that each Maxterm is the compliment of its corresponding Minterm. The conversion between the canonical forms can be best explained with the help of the following example.

Let’s say for example if we are told to convert

F(X,Y,Z) = ∑(1,4,5,6,7) into Maxterm.

This conversion can be easily done with the help of a truth table I already explained.

From the equation F(X,Y,Z) = ∑(1,4,5,6,7) we know that we have three variables A, B, and C. So we will have 8 possible combinations which you can see in the truth table given above, as 2^{n}= 2^{3} = 8.

As you know for the Minterms we select 1’s in the truth table while for the Maxterms we select 0’s in the truth table. So the truth table will become.

As per the function we marked 1’s in the truth table. As you know each Maxterm is the compliment of its corresponding Minterm and you also know for Maxterms we select 0’s in the truth table which are represented by the tick signs. So the function will become.

Excellent