# Question about the Karnaugh maps and the number of solutions

I'm given the K-map $\sum_{(w,x,y,z)}0,2,4,5,10,12,15$ and was able to reduce it down to $\overline y\space \overline z+\overline w x \overline y +wxy+\overline x\space \overline z$

However, it says that there are two solutions. My question is what exactly does it mean to be a minimum expression? Also, what would the other "solution" be? It seems there are as many as 4+ perfectly valid expressions that would have the same number of terms. How do I know I chose the right solution?

There are a lot of expressions in which a digital logical function can be represented. But the two canonical forms of any Boolean function are:

1. Sum of minterms: This will express the function as a OR (sum) of minterms. Hence called as sum of products (SOP) form. In your case it will be:

$$F = w'x'y'z' + w'x'yz' + w'xy'z' + w'xy'z + wx'yz' + wxy'z' + wxyz$$

There will be a term corresponding to each minterm (0,2,4,5,10,12,15).

2. Product of maxterms: This will express the function as a AND (product) of maxterms. Hence called as product of sums (POS) form. In your case:

$$F = (w'+x'+y'+z)(w'+x'+y+z)(w'+x+y+z')(w'+x+y+z)(w+x'+y'+z')(w+x'+y'+z)(w+x'+y+z)(w+x+y'+z)(w+x+y+z')$$

There will be a term corresponding to each maxterm (1,3,6,7,8,9,11,13,14).

Non-canonical form: It is often the case that the canonical minterm form can be simplified to an equivalent SOP/POS form. This simplified form would still consist of a sum of product terms/product of sum terms.

And what you got here is the simplified SOP version of it. Similarly you can get the POS form of it.

Source:Wikipedia