# Is Karnaugh Map possible for Maxterms?

I read about Minterms i.e. sums of products, simplification using Karnaugh Graph. Can this graph be used for Maxterms, i.e. products of sums, as well? If yes, then how?

If not, then is there some other similar way to use, for simplification of Maxterms? I know, one can always convert Maxterms to Minterms and then use K-Map. I mam looking for some direct way.

• Please only use quote blocks for quotes and $...$ for mathematics. If you need italics, use *...* but italicizing almost every noun in your post makes it less readable, not more. – David Richerby Jan 3 '15 at 11:37

Yes you can do it directly on the K-map. In order to get a minimal product of maxterms (aka product-of-sum [POS], aka conjunctive-normal-form [CNF]), you simply circle/unify the zeros that appear in the K-map instead of the ones. (It doesn't matter how you got the K-map, it can be from any representaiton of the function, it's just a table of the function's output after all). You can also build the K-map from a (not necessarily minimal) POS by adding a zero the K-map for every maxterm (with the remaining squares being ones.) This should be said to varying extents in most textbooks, e.g.

• Digital Fundamentals by Thomas L. Floyd, 9th edition, pp. 221-224 has a very detailed example. (I'm guessing the latest, 11th edition has it too, but I haven't checked.)
• The Electronics Handbook, Second Edition edited by Jerry C. Whitake, pp. 55-57 has an example too, but this isn't as detailed.
• Introduction to Digital Electronics by Reid and Dueck, p. 114 also has an example.
• Digital Logic Techniques, 3rd Edition by John Stonham, p. 43.

That this approach is correct comes from De Morgan's laws and double negation. It's the same as if you flipped all zeros and ones in the K-map and SOP-minimized the negated function and then finally applied De Morgan to the result double-negating the function. This is explained on an example in

• Principles of Modern Digital Design by P.K. Lala, p. 70.

and probably other textbooks as well.

minterms & maxterms are complements. see eg Minterms, Maxterms, and K-Maps which explains the interrelationship in more detail. see also DeMorgans law used for complements.

• I'm guessing nobody upvoted this because your 1st link is now a dead link (at least from outside uiuc). – SX welcomes ageist gossip Mar 5 '15 at 19:40

Yes, a Karnaugh Map for maxterms is possible. I have a easy and quick method to show.

Summary: Take its complement, and you'll get immediately deduce the minterm expression.

Golden Rule: We know that the maxterms are the opposite for minterms. To draw the a maxterm expression on the Karnaugh map, all you have to do is simply deduce its minterm and draw on the Karnaugh map.

Example: Draw a Karnaugh Map for this maxterm expression:

As you see, this equals to: f4(x,y,z) = M (0,5)

Now, you take the opposite (or complement), which is the minterm: f4(x,y,z) = Sigmam(1,2,3,4,6,7).

Now it is easy, you're working on a minterm. The Karnaugh map for the function would be: