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?


1 Answer 1


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.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.