What is Product of Sum (POS) Form?

The POS (Product Of Sum) form is a procedure by which we can simplify any Boolean Expression. From the name itself, we can understand that the POS form is the product of all the sums. Here, the sum does not mean traditional addition, the sum here refers to the 'OR' operation, and the product here refers to the 'AND' operation. Thus, in POS form, we perform the OR of multiple variables and then perform the AND operation between them. This form is exactly opposite to the SOP (Sum Of Product) form. POS form is also referred to as conjunctive normal form.

Maxterm

In maxterm, each variable whose value is assigned to 1 is represented in the complemented form. The variable whose value is assigned to 0 is represented in the un-complimented form. We can represent maxterm with 'M'.

We have two types of POS form:

• Canonical POS form
• Minimal POS form.

In canonical POS form, we have all the variables present in a complimented or un-complimented form in each maxterm. In contrast, in minimal POS form, we do not have all the variables in the complimented or uncomplimented form in each maxterm.

Let us take some examples to find and understand the POS form.

Example 1:

For the given truth table, find and then minimize the POS form.

In POS form , we consider those outputs which are low or 0. In the above table , we can see that there are three cases when the output is 0 (output is represented in the column F).Thus, we consider only those occurrences.

We follow the convention of maxterms in POS form i.e., 0 represents un-complimented form and 1 represents complimented form.

F = (X + Y + Z) . (X + Y + Z') . (X + Y' + Z') This is the canonical POS form.

Now , applying several laws of Boolean Algebra we will minimize the POS form.

F = (X + Y + Z) . (X + Y + Z') . (X + Y' + Z')

F = ((X + Y) + (Z.Z')) . (X + Y' + Z') [ Distributive law -> (X + Y) . (X + Z) = X + YZ ]

F = (X + Y) . (X + Y' + Z') [ Z.Z' = 0 ]

F = X + Y.(Y' + Z') [ Distributive law -> (X + Y) . (X + Z) = X + YZ ]

F = X + Y.Y' +Y.Z'

F = X + Y.Z' [ Y.Y' = 0 ]

F = (X + Y) . (X + Z') [ Distributive law -> (X + Y) . (X + Z) = X + YZ ]

F = (X + Y) . (X + Z') This is the minimal POS form.

Now, we will see the logic diagram for the minimal POS form.

Example 2 :

Suppose we have a Boolean function named F which is defined by two variables A and B. We have represented the maxterms for the function F as follows -

F = \Pi (0 , 1 , 3)

Find the POS form and minimize it.

Therefore , we can write F as -

F = \Pi ( M₀ . M₁ . M₃ )

Thus, we can construct the truth table from the above represented form -

Thus, the canonical POS form from the above truth table is -

F = (A + B) . (A + B') . (A' + B')

F = (A + B.B') . (A' + B') [ Distributive law -> (A + B) . (A + B') = A + BB' ]

F = A . (A' + B') [ B.B' = 0 ]

F = A.A' + A.B' [ A.A' = 0 ]

F = A.B'

This is the minimized POS form.

Now, we will see the logic diagram for the minimal POS form.

Conclusion

Thus, from the above examples and explanations, we can easily conclude that the Product Of Sum (POS) is an alternative way of expressing Boolean functions where we perform the product of all sums. Here, product refers to AND operation and sum refers to OR operation.