Difference between SOP and POS in Digital Logic

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’. 

Representation of Boolean expression can be primarily done in two ways. They are as follows: 

1. Sum of Products (SOP) form 
2. 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

 
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 = [Tex]\Sigma  [/Tex]_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: 

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) = [Tex]\Sigma  [/Tex]_m(1, 3, 6) 
= A’.B’.C + A’.B.C + A.B.C’ 

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 = [Tex]\Pi  [/Tex]_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: 

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) = [Tex]\Pi[/Tex]_M(0, 2, 4, 5, 7) = (A+B+C).(A+B'+C).(A'+B+C).(A'+B+C').(A'+B'+C')

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.

What is Minterm and Maxterm in SOP and POS?

SOP uses minterms to create a Boolean expression which is product of Boolean variables whereas POS uses maxterms to create a Boolean expression which is sum of Boolean variables.

What is the symbol for SOP and POS?

SOP and POS are two forms of Boolean expression where SOP is denoted with the sign summation ∑ and POS is denoted by pi notation Π.

What is the example of SOP and POS expressions?

Example of SOP form = AB + BC + CA

Example of POS form = (A + B)(B + C)(C + A)