Understanding K-Map Symbols - Get Help Now!

[k31453]

Hi, below image. My teacher use symbol in K-MAP. But not sure how it uses.

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 ?

 

[collinsmark]

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!