Difference between SOP and POS in Digital Logic
Last Updated : 28 Dec, 2024
In digital logic, the inputs and output of a function are in the form of binary numbers (Boolean values) i.e., the values are either zero (0) or one (1). Therefore, digital logic is also known as 'Boolean logic'. These inputs and output can be termed as 'Boolean Variables'. The output Boolean variable of a digital signal can be expressed in terms of input Boolean variables which forms the 'Boolean Expression'.
SOP Vs POS in Digital Logic
Representation of Boolean expression can be primarily done in two ways. They are as follows:
- Sum of Products (SOP) form
- Product of Sums (POS) form
NOTE
If the number of input variables are n, then the total number of combinations in Boolean algebra is 2n.
If the input variable (let A) value is :
- Zero (0) - a is LOW -It should be represented as A' (Complement of A)
- One (1) - a is HIGH -It should be represented as A
In Boolean logic,
AND is represented as '.'
A AND B is written as 'A.B'
OR is represented as '+'
A OR B is written as 'A+B'
For example, Considering number of input variables =3, Say A, B and C.
Total number of combinations are: 23=8.
Truth Table Representation
| A | B | C |
|---|
| 0 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
| 1 | 1 | 1 |
What is SOP?
It is one of the ways of writing a Boolean expression. As the name suggests, it is formed by adding (OR operation) the product terms. These product terms are also called as 'min-terms'. Min-terms are represented with 'm', they are the product(AND operation) of Boolean variables either in normal form or complemented form.
Therefore, SOP is sum of minterms and is represented as:
F in SOP = \Sigma m(0, 3)
Here, F is sum of minterm0 and minterm3.
Example of SOP
For Example:
A=0, B=0, C=0 Minterm is A'.B'.C'
A=1, B=0, C=1 Minterm is A.B'.C
SOP Truth Table
Consider a function X, whose truth table is as follows:
| A | B | C | X |
|---|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 0 |
The function X can be written in SOP form by adding all the min-terms when X is HIGH(1).
While writing SOP, the following convention is to be followed:
If variable A is Low(0) - A'
A is High(1) - A
X (SOP) = \Sigma m(1, 3, 6)
= A'.B'.C + A'.B.C + A.B.C'
What is POS?
As the name suggests, it is formed by multiplying(AND operation) the sum terms. These sum terms are also called as 'max-terms'. Maxterms are represented with 'M', they are the sum (OR operation) of Boolean variables either in normal form or complemented form.
Therefore, POS is product of maxterms and is represented as:
F in POS = \Pi M(1, 2) Here, F is product of maxterm1 and maxterm2.
Example of POS
For Example:
A=0, B=1, C=0 Maxterm is A+B'+C
A=1, B=1, C=1 Maxterm is A'+B'+C'
POS Truth Table
Consider a function X, whose truth table is as follows:
| A | B | C | X |
|---|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 0 |
The function X can be written in POS form by multiplying all the max-terms when X is LOW(0).
While writing POS, the following convention is to be followed:
If variable A is Low(0) - A
A is High(1) - A'
X (POS) = \PiM(0, 2, 4, 5, 7)
= (A+B+C).(A+B'+C).(A'+B+C).(A'+B+C').(A'+B'+C')
Difference between SOP and POS in Digital Logic
| S.No. | SOP | POS |
|---|
| 1. | A way of representing Boolean expressions as sum of product terms. | A way of representing Boolean expressions as product of sum terms. |
| 2. | SOP uses minterms. Minterm is product of Boolean variables either in normal form or complemented form. | POS uses maxterms. Maxterm is sum of Boolean variables either in normal form or complemented form. |
| 3. | It is sum of minterms. Minterms are represented as 'm' | It is product of maxterms. Maxterms are represented as 'M' |
| 4. | SOP is formed by considering all the minterms, whose output is HIGH(1) | POS is formed by considering all the maxterms, whose output is LOW(0) |
| 5. | While writing minterms for SOP, input with value 1 is considered as the variable itself and input with value 0 is considered as complement of the input. | While writing maxterms for POS, input with value 1 is considered as the complement and input with value 0 is considered as the variable itself. |
Conclusion
In the conclusion, SOP and POS are two forms of Boolean expression by using 2 or more variables. SOP stands for Sum of Product which is sum of minterms or number of 1s whereas POS stands for Product of Sum which is product of maxterms or number of 0s.