Does complementing a minimized SOP expression give the minimized POS expression?

Question

I have the following K-Map:

And I was able to find the minimized SOP expression, which is:

$f_{SOP}=x_2'x_4'+x_1'x_2x_3'+x_1x_2x_3x_4$

My question is: if I want to find the minimized POS expression now, may I simply take the complement of $f_{SOP}$, or do I have to repeat the process looking at the 0s in the table now instead of the 1s?

Note: My reasoning as to why $f_{POS}=f_{SOP}'$ is mainly because of DeMorgan's Law.

Answer 1

Assuming that empty cells in the Karnaugh map stand for don't care, the minimal sum-of-products is

$$f = x_1'x_2' + x_3 + x_4$$

To get a product-of-sums, you have to look at the $0$ cells and combine them to maxterm blocks.

$$f' = x_2x_3'x_4' + x_1x_3'$$

Inverted, this results in:

$$f = (x_2'+x_3+x_4)(x_1'+x_3)$$

The example shows, that both forms do not have a direct relationship. This is especially true, if don't care cells are utilized to minimize the number of terms.