Sum Of Product (SOP)
Sum of Product is the abbreviated form of SOP. Sum of product form is a form of expression in Boolean algebra in which different product terms of inputs are being summed together. This product is not arithmetical multiply but it is Boolean logical AND and the Sum is Boolean logical OR.
To understand better about SOP, we need to know about min term.
Min Term
Minterm means the term that is true for a minimum number of combination of inputs. That is true for only one combination of inputs.
Since AND gate also gives True only when all of its inputs are true so we can say min terms are AND of input combinations like in the table given below.
Types of Sum Of Product (SOP) Forms
There are few different forms of Sum of Product.
- Canonical SOP Form
- Non-Canonical SOP Form
- Minimal SOP Form
Canonical SOP Form
This is the standard form of Sum of Product. It is formed by O Ring the minterms of the function for which the output is true. This is also known as Sum of Min terms or Canonical disjunctive normal form (CDNF). It is just a fancy name. “canonical” means “standardized” and “disjunctive” means “Logical OR union”.
Canonical SOP expression is represented by summation sign ∑ and minterms in the braces for which the output is true. For example, a functions truth table is given below.
For this function the canonical SOP expression is F = ∑( m1, m2, m3, m5 )
Which means that the function is true for the min-terms {1, 2, 3, 5}.
By expanding the summation we get, F = m1 + m2 + m3 + m5
Now putting min-terms in the expression F = A̅B̅C + A̅BC̅ + A̅BC + AB̅C
Canonical form contains all inputs either complemented or non-complemented in its product terms.
Product of Sum
Product of Sum abbreviated for POS.
The product of Sum form is a form in which products of different sum terms of inputs are taken. These are not arithmetic product and sum but they are logical Boolean AND and OR respectively.
To better understand about Product of Sum, we need to know about Max term.
Max Term
Maxterm means the term or expression that is true for a maximum number of input combinations or that is false for only one combination of inputs.
Since OR gate also gives false for only one input combination. So Maxterm is OR of either complemented or non-complemented inputs.
Max terms for 3 input variables are given below.
3 inputs have 8 different combinations so it will have 8 maxterms. Maxterms are denoted by capital M and decimal combination number In the subscript as shown in the table given above.
Types of Product Of Sum Forms
There are different types of Product of Sum forms.
- Canonical POS Form
- Non – Canonical Form
- Minimal POS Form
Canonical POS Form
It is also known as Product of Max term or Canonical conjunctive normal form (CCNF). Canonical means standard and conjunctive means intersection.
In this form, Maxterms are AND together for which output is false.
Canonical POS expression is represented by ∏ and Maxterms for which output is false in brackets as shown in the example given below.
F = ∏ (M0, M4, M6, M7)
Expanding the product
F = M0.M4.M6.M7
Putting Max terms
F = (A+B+C)(A̅+B+C)(A̅+B̅+C)(A̅+B̅+C̅)
