k31453 said:
Hi, below image. My teacher use symbol in K-MAP. But not sure how it uses.
View attachment 64044
Myattempt : i got
0 0 0 0
0 0 1 1
0 0 1 1
0 1 1 0 --> Using his technique.
Can somebody help me out ?
I'm not sure how you are organizing your variables. But in the figure that you attached, WY represent the columns, and XZ represent the rows.
For example, in the attachment, the first column (the leftmost column) is
W'Y'. The second column is
W'Y. The third column is
WY. And the rightmost column is
WY'. Notice they are organized in Gray code.
The unsimplified expression is given to you in the form of product of sums (POS). This is not the more familiar sum of products (SOP) that you want. That's why the variables get used.
There are three terms multiplied together in the original expression,
f= (X + Y + Z')(W + Z')(W + X' + Y + Z),
or putting it another way,
f = \alpha \beta \gamma
where we assign
\alpha \equiv X + Y + Z'
\beta \equiv W + Z'
\gamma \equiv W + X' + Y + Z.
Now here is what we do. Let's start with the equation \alpha \equiv X + Y + Z'. That means \alpha is a 1 whenever
X is a 1 OR whenever
Y is a 1 OR whenever
Z' is a 1 (same as saying
Z is 0).
So let's put that into your k-map. We know that \alpha is a 1 whenever
X is a 1.
X being 1 corresponds to the bottom two rows in the k-map (using the k-map in the attachment). So put an \alpha into each of these eight boxes. Just write the symbol in the corner. We know that \alpha is 1 in each of these boxes because
X is 1.
Now do the same thing for
Y (the middle two columns). We know that \alpha is 1 in each of these boxes because
Y is 1. Write the symbol in the corner of each box if there isn't an \alpha there already.
Repeat for
Z' (the top row and the bottom row). We know that \alpha is 1 in each of these boxes because
Z' is 1 (same thing as saying
Z is 0).
Now move on to the next equation, \beta \equiv W + Z'. Write a \beta symbol in each box corresponding to
W, and again for each box corresponding to
Z'
Do you get the idea? Finally finish up by doing the same sort of thing with \gamma \equiv W + X' + Y + Z.
So now the table is filled in with a bunch of symbols. Recall,
f = \alpha \beta \gamma.
In other words,
f is 1 if \alpha is 1 AND if \beta is 1 AND if \gamma is 1.
So from the k-map,
f is 1 in any given box only if that box contains
all three symbols, \alpha, \beta and \gamma. So that's where you label each box 1 or 0. If a box has all three symbols in it, label it a 1. Otherwise it's a 0.
Now make your circles in the standard way. Good luck!
