Product of Sum(POS) or Sum of Product(SOP) after Karnaugh map simplification

Question

I am just curious on how do I determine whether a simplified boolean expression is in a SOP form or POS form. for example this question: Question

the answer to this expression is : NOT B.D/ ⌝B.D and this is in SOP form Can anyone explain why?

Answer 1

I think this should be a 'philosophical' argument. ⌝B.D is the special case where the number of elements to be summed up becomes one.

You can think of ⌝B.D = ⌝B.D + ⌝B.B + ⌝D.D + 0.(anything) which makes it an SOP.

Answer 2

Terminology:

First the theory, you can further study on Wikipedia (DNF, CNF):

• Sum of products = DNF (Disjunctive normal form) = disjunction (+) of conjunctions (·) ~ "the disjunction is not inside any bracket, but only as the root operator(s)".

• Product of sums = CNF (Conjunctive normal form) = conjuction of disjunctions ~ "the conjunction is not inside a bracket, but only as the root operator(s)".

• Full/Complete CNF/DNF = the terms (products/sums) contains all given variables in either direct or negated form; the terms are then maxterms/minterms.

Finding the right circles:

You can see, that the four circles in the Karnaugh map correspond to the four products in the original function in the same order (top to bottom, left to right).

Given function as a SOP:

The function is now in a form of sum of products, because you can literally see, that there are four products.

It is also in the form of sum of maxterms, because the four parts contain all variables in their direct or negated form.

f(a,b,c,d) = ¬a·¬b·¬c·d + ¬a·¬b·c·d + a·¬b·c·d + a·¬b·¬c·d

For example the first term: ¬a·¬b·¬c·d ~ if the variables a, b and c are logical zeros and only d is true, then the function's output is logical 1.

Minimized function as a SOP:

You can see, that the maxterms can be grouped and that creates the minimal sum of product: f(a,b,c,d) = ¬b·d, because all cells, where b is a logical 0 and d is a logical 1 are included.

The minimized function is indeed a SOP/DNF, because it certainly does contain only one product — the ¬b·d — and there is no + (disjunction) operator inside that product.

Minimized function as a POS:

The surprise might come when you realize, that circling and writing the function as a product of sums yields the same minimal form: f(a,b,c,d) = (¬b)·(d), because there are exactly two terms: ¬b (orange circle) and d (red circle).

Both are sums with only one operand. Because of that the minimized function is a product of sum.

Conclusion:

The minimized function f(a,b,c,d) = ¬b·d is both a SOP and a POS. You can check the correct solution by using wolframalpha.com.