Use K-map to create the Boolean function (based on 3 variables X,Y,Z) in POS; Draw logical diagram using logical gates. YZ 00 01 11 10 1
Use K-map to create the Boolean function (based on 3 variables X,Y,Z) in POS; Draw logical diagram using logical gates. YZ 00 01 11 10 1
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
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Transcribed Image Text:### Boolean Function Creation Using a K-map
**Instruction:**
Use a Karnaugh map (K-map) to create the Boolean function (based on 3 variables X, Y, Z) in Product of Sums (POS) form; Draw a logical diagram using logical gates.
**K-map Explanation:**
The given K-map is a 3-variable map with variables X, Y, and Z. The cell arrangement for YZ across the top is in Gray code order: 00, 01, 11, 10.
**K-map Details:**
| YZ | 00 | 01 | 11 | 10 |
|:----:|:--:|:--:|:--:|:--:|
| **X**=0 | 0 | 0 | 0 | 0 |
| **X**=1 | 0 | | | |
- Each cell represents a minterm. The value inside the cell shows whether the minterm is included (1) or not included (0) in the function.
- The top row corresponds to X=0, and the bottom row corresponds to X=1.
- Since all cells containing zeros are present, the POS expression will be derived by identifying common variables across zeros.
**POS Expression Derivation:**
1. **From X=0 row (covering all columns):**
- (X' + Y + Z)
- (X' + Y' + Z)
- (X' + Y' + Z')
- (X' + Y + Z')
Since all outputs are zero for X=0, all combinations of Y and Z with X=0 need to be considered.
2. **From X=1 row (only one zero is marked, at 00 position):**
- (X + Y + Z)
**Logical Diagram:**
To draw the logical diagram using gates:
- Each POS term represents an OR gate.
- All terms are combined using an AND gate.
Final POS Expression:
- (X' + Y + Z)(X' + Y' + Z)(X' + Y' + Z')(X' + Y + Z')(X + Y + Z)
Create OR gates for each term and combine their outputs using an AND gate to form the logical diagram.
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