# Karnaugh Maps

#### Prototape

Joined Feb 12, 2011
10
I'm having a hard time understanding a few concepts with Karnaugh Maps. I know how to fill out a truth table with the SOP and POS expressions, then fill out a Karnaugh Map based on said expressions. But from there I kind of stumble.

I've drawn up something to show exactly what I have right now:

I'm not too clear on 2 major parts:
1. Taking those groups and turning them into a boolean expression.
2. Simplifying the boolean expression (not using boolean algebra, I was told that was the point of the Karnaugh map is to skip that step essentially).

Help would be greatly appreciated. I don't expect people to be giving me answers to my homework, I just need help understanding this.

#### Georacer

Joined Nov 25, 2009
5,181
First of all, take a look in our AAC e-book, in the chapter that refers to Karnaugh maps:

The general idea is that, after you fill the map, you try to include all the 1s, if you are going for an SOP (or the 0s, for an POS), with as few sets as you can. In a 4-by-4 map, like your example, the sets can include either 1 ace, 2 adjacent 1s, 4 adjacent ones forming a square, 8 1s forming an 2-by-4 rectangle or all 16 squares.
Remember that the map is toroidal, and its edges circle around from left to right and top to bottom. But it seems you understand that.

That said, you only need to draw the set that include the positions (starting from 0): {1,3}, {8,10} and {9,13}. These sets cover all the aces of the map.

Furthermore, they are translated in 3 products, that contain the variables that are ALWAYS true in each one of them. Respectively, these products will be A'B'D, AB'D' and AC'D.

#### t_n_k

Joined Mar 6, 2009
5,447
I think in your posted example you have some redundancy in your selected groupings of terms.

Are you simply highlighting a range of possibilities of which you are uncertain how to express as a product term?

Also one term doesn't appear to be "grouped" - i.e. the single isolated '1' at A'BCD'

The resulting SOP probably needs 4 terms maximum.

#### Prototape

Joined Feb 12, 2011
10
First of all, take a look in our AAC e-book, in the chapter that refers to Karnaugh maps:

The general idea is that, after you fill the map, you try to include all the 1s, if you are going for an SOP (or the 0s, for an POS), with as few sets as you can. In a 4-by-4 map, like your example, the sets can include either 1 ace, 2 adjacent 1s, 4 adjacent ones forming a square, 8 1s forming an 2-by-4 rectangle or all 16 squares.
Remember that the map is toroidal, and its edges circle around from left to right and top to bottom. But it seems you understand that.

That said, you only need to draw the set that include the positions (starting from 0): {1,3}, {8,10} and {9,13}. These sets cover all the aces of the map.

Furthermore, they are translated in 3 products, that contain the variables that are ALWAYS true in each one of them. Respectively, these products will be A'B'D, AB'D' and AC'D.

Thanks for the link, I'll read what I can. Any material helps, I've been having a hard time finding good resources for this sort of information.

I think in your posted example you have some redundancy in your selected groupings of terms.

Are you simply highlighting a range of possibilities of which you are uncertain how to express as a product term?

Also one term doesn't appear to be "grouped" - i.e. the single isolated '1' at A'BCD'

The resulting SOP probably needs 4 terms maximum.
I'm just highlighting all possible groups. I know it's best to group in the largest groups possible, but I'm pretty new to this idea so I'm just trying to be thorough so I don't miss any possible expressions, as I don't really have an 'eye' for it yet.

I thought each group had to be at least a group of 2, and that you can only group things that are next to each other? So I'm not exactly sure how to group an isolated digit, I just figured you left them alone. It wasn't really covered in my lecture.

Edit: I understand it now. Thank you both for the help!

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#### Georacer

Joined Nov 25, 2009
5,181
I think in your posted example you have some redundancy in your selected groupings of terms.

Are you simply highlighting a range of possibilities of which you are uncertain how to express as a product term?

Also one term doesn't appear to be "grouped" - i.e. the single isolated '1' at A'BCD'

The resulting SOP probably needs 4 terms maximum.
Oops! It seems I have missed this one. Thanks for the heads up!