# Simplifying a (Sum of Products) Don't care Function with a (Product of Sums) Function using (K-Map)

The functions:

f(u,v,w,z)=π(0,4,8,12,9)
d(u,v,w,z)=E(1,5,3)

π represents the product of sums(pos) a.k.a the product of maxterms
E represents the sum of products(sop) a.k.a the sum of minterms
f represents the regular function
d represents the don't care function

The K-Map:

uv\wx
|00  |01 |11 |10
00 |0   |d  |1  |1
01 |0   |d  |1  |1
11 |0   |d  |1  |1
10 |0   |0  |1  |1

So I know for pos you're supposed to group the 0's in the k-map. I know in the don't care function, the designer picks if they want to make it a 1 or a 0, my prof told the class if you can group a d in the k-map then do it. I know for sop you're supposed to group the 1's in the k-map.

Since the second function is a don't care function does the E(sum of products) become irrelevant in solving this problem? So I would try to group the 0's in the k-map together since the first function is a π(product of sums function)? I've never come across an example like this and couldn't find any similar examples online. Any help would be greatly appreciated.

• I've looked at your functions and your table and I do not understand your notation. Any way I look at it, the two statements at the top do not match the table you've produced. For example, the "don't care's" are 1, 3, and 5 (I assume 1, 3, 5 is the same to you as 1, 5, 3 would be); or 0001, 0011, and 0101 (as I read it.) But I cannot see how 0011 shows up as a 'd' in your table. – jonk Oct 12 '18 at 17:41
• Yes, 1,3,5 is the same to me as 1,5,3 for the "don't care's", this is the way my prof wrote the problem. I did incorrectly label my K-Map, it should be like this. Also, is there any other notation you would like me to clarify? – F. Ryan Oct 12 '18 at 19:01
• The don't care function should be: d(u,v,w,z)=E(1,5,13). I incorrectly, wrote my don't care function wrong in my original question. Sorry, for the confusion. My K-Map is correct when the don't care function is d(u,v,w,z)=E(1,5,13), as it should be. – F. Ryan Oct 12 '18 at 19:04
• I do now follow your don't care values. I do NOT now follow the rest. I'd normally get: $$\begin{array}{rl}\begin{smallmatrix}\begin{array}{r|cccc}&\overline{w}\:\overline{z}&\overline{w}\: z&w \:z&w \:\overline{z}\\\hline\overline{u}\:\overline{v}&1&d&0&0\\\overline{u}\:v&1&d&0&0\\u\: v&1&d&0&0\\u\:\overline{v}&1&1&0&0\end{array}\end{smallmatrix}\end{array}$$ But I think you actually mean that the $\pi$ function says where the 0's are at, not the 1's?? Is that it? – jonk Oct 13 '18 at 23:45