Question 1191392: Simplify the following functions using a K-map
F(X, Y,Z) = X’Y’Z’ + X’YZ + XY’Z + XYZ
Can you please explain to me the last homework question step-by-step? Thank you!
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Standard list of minterms (3 variables)
| Minterms | | X | Y | Z | Term | Designation | | 0 | 0 | 0 | X'Y'Z' | m0 | | 0 | 0 | 1 | X'Y'Z | m1 | | 0 | 1 | 0 | X'YZ' | m2 | | 0 | 1 | 1 | X'YZ | m3 | | 1 | 0 | 0 | XY'Z' | m4 | | 1 | 0 | 1 | XY'Z | m5 | | 1 | 1 | 0 | XYZ' | m6 | | 1 | 1 | 1 | XYZ | m7 |
We'll narrow our focus on these specific minterms
X'Y'Z' = m0
X'YZ = m3
XY'Z = m5
XYZ = m7
So we could say,
F(X,Y,Z) = X'Y'Z' + X'YZ + XY'Z + XYZ
F(X,Y,Z) = m0 + m3 + m5 + m7
F(X,Y,Z) = Σ(0,3,5,7)
Let's set up the K-map diagram.
First we'll need the headers.
Note: this diagram may be a bit wordy/verbose, but it might be more descriptive to see how the terms are laid out.
As figure 1 indicates, we have 0 and 1 along the left side to represent X' and X in that order.
Then along the top we have the sequence: 00, 01, 11, 10
It looks like we're counting in binary but not quite. The next term after 01 is not 10 because we want to flip 1 bit only. For more info, check out gray code sequencing
Once the headers are in place, we can locate each of the m0 through m7 terms.
You can use this informal notation to keep track
m0 = 000
m1 = 001
m2 = 010
m3 = 011
m4 = 100
m5 = 101
m6 = 110
m7 = 111
Note: the numbers in the top row (0,1,3,2) and the numbers in the bottom row (4,5,7,6) are separated by a gap of 4.
Eg: m1 and m5 are in the same column and 1+4 = 5.
Since you'll probably do a bunch of K-maps with 3 variables, it might be handy to have this somewhere on a reference page for later.
After the m0 through m7 are filled in the correct boxes, we'll then replace these m values {m0,m3,m5,m7} with 1s. The rest gets replaced with 0.
Here's one possible grouping after using these K-map grouping rules
Notice that each group has either 1 item in it or 2 (those values are powers of 2).
Furthermore, notice that we've grouped things to have the smallest number of groups while also maxing out each possible group size.
The '1' marked in red represents the term X'Y'Z' aka m0.
The '1's marked in blue represent m5+m7 = XY'Z+XYZ = XZ(Y'+Y) = XZ*1 = XZ
The '1's marked in green represent m3+m7 = X'YZ+XYZ = (X'+X)YZ = 1*YZ = YZ
Group overlapping is possible.
Therefore, we end up with X'Y'Z'+XZ+YZ which is the final answer
We could rewrite this as X'Y'Z'+Z(X+Y) if you wanted.
But I'll choose to stick to the sum of products format instead where each factor is a literal.
Verification table:
https://docs.google.com/spreadsheets/d/1woeTbKWYDyTsGeyhfUNQ4GGrDV-XjqehayOzUohHsA4/edit?usp=sharing
You don't need a google account to be able to view the spreadsheet.
A handy calculator to verify the answer:
https://www.charlie-coleman.com/experiments/kmap/
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