Karnaugh Map from Boolean Expression

[nabelekt]

Hi,

I'm trying to figure out a few questions on a practice exam that I'm working on for my Intro to Logic Systems class and could use some help.

One of the questions (and the others are similar) says:

Determine the minimized realization in the sum-of-produicts form using literals of the function:
f(A,B,C) = $\Sigma$m(4,6) + $\Sigma$d(2,3,7)

The given answer is f(A,B,C) = AC'.

I know that $\Sigma$m(4,6) can be represented by AB'C' + ABC' and that $\Sigma$d(2,3,7) can be represented by A'BC' + A'BC + ABC, but beyond that I am not sure what to do. I think that I need to construct a Karnaugh map from the expression, but am not sure how to do it.

Any help is greatly appreciated. Thanks!

 

[NascentOxygen]

f(A,B,C) = $\Sigma$m(4,6) + $\Sigma$d(2,3,7)

Before embarking on a K-map, I suggest that you construct a truth table for your Boolean expression for f(A,B,C), and compare it with AC', to make sure you have that right.

 

[nabelekt]

Thanks for the reply. I'm pretty sure that I know how to construct a Karnaugh map given a truth-table. But I do not know how to construct a truth-table given an expression like that. Can you help me with that?

 

[NascentOxygen]

Don't you just OR those two expressions:
AB'C' + ABC' with A'BC' + A'BC + ABC?

I assume that is the way to go.

 

[rezeew]

Hi there,

I can understand your confusion with this question. Karnaugh maps are a useful tool for simplifying Boolean expressions, as they help visualize the logical relationships between variables. In this case, the given expression can be represented by the following Karnaugh map:

| | 00 | 01 | 11 | 10 |
|---|----|----|----|----|
| 0 | 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 | 1 |

To construct this map, we first need to identify the variables in the expression. In this case, A, B, and C are the variables. Next, we need to determine the minterms (represented by 1s) and don't care terms (represented by Xs) in the expression. In your case, m(4,6) and d(2,3,7) are the minterms, while m(0,1,2,5) are the don't care terms.

Next, we fill in the Karnaugh map by placing 1s in the corresponding squares for the minterms and don't care terms. The remaining squares can be filled with 0s. Then, we look for groups of adjacent 1s in the map. These groups represent possible simplified terms in the expression. In this case, we can see that the group of 1s in the top left corner represents the term AB'C'. Similarly, the group of 1s in the bottom right corner represents the term ABC.

Using the simplified terms from the Karnaugh map, we can then write the expression as f(A,B,C) = AB'C' + ABC. However, we can further simplify this expression by using the Boolean algebra rule A(A+B) = A. This means that we can combine the terms AB'C' and ABC to get f(A,B,C) = AC'.

I hope this explanation helps you understand how to construct a Karnaugh map from a Boolean expression and how to use it to simplify the expression. Let me know if you have any further questions. Good luck on your exam!

Best,