CHAPTER 4 The Karnaugh Map The Karnaugh Map
![CHAPTER 4 The Karnaugh Map CHAPTER 4 The Karnaugh Map](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-1.jpg)
![The Karnaugh Map • Provides a systematic method for simplifying Boolean expressions • Produces The Karnaugh Map • Provides a systematic method for simplifying Boolean expressions • Produces](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-2.jpg)
![The 3 -Variable K-Map The 3 -Variable K-Map](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-3.jpg)
![The 4 -Variable K-Map The 4 -Variable K-Map](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-4.jpg)
![K-Map SOP Minimization • A 1 is placed on the KMap for each product K-Map SOP Minimization • A 1 is placed on the KMap for each product](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-5.jpg)
![Mapping Standard SOP Expression Example: Map the following standard SOP expression on a K-Map: Mapping Standard SOP Expression Example: Map the following standard SOP expression on a K-Map:](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-6.jpg)
![Map the following standard SOP expression on a K-Map: Example: Solution: Map the following standard SOP expression on a K-Map: Example: Solution:](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-7.jpg)
![Exercise: Map the following standard SOP expression on a K-Map: Exercise: Map the following standard SOP expression on a K-Map:](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-8.jpg)
![K-Map Simplification of SOP Expressions A group must contain either 1, 2, 4, 8 K-Map Simplification of SOP Expressions A group must contain either 1, 2, 4, 8](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-9.jpg)
![Example: Group the 1 s in each K-Maps Example: Group the 1 s in each K-Maps](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-10.jpg)
![Determining the minimum SOP Expression from the Map • Groups the cells that have Determining the minimum SOP Expression from the Map • Groups the cells that have](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-11.jpg)
![Determining the minimum SOP Expression from the Map • For 3 -variable map: 1) Determining the minimum SOP Expression from the Map • For 3 -variable map: 1)](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-12.jpg)
![Determining the minimum SOP Expression from the Map • For 4 -variable map: 1) Determining the minimum SOP Expression from the Map • For 4 -variable map: 1)](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-13.jpg)
![Example: Determine the product term for the K-Map below and write the resulting minimum Example: Determine the product term for the K-Map below and write the resulting minimum](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-14.jpg)
![](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-15.jpg)
![Example: Use a K-Map to minimize the following standard SOP expression Example: Use a K-Map to minimize the following standard SOP expression](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-16.jpg)
![Example: Use a K-Map to minimize the following standard SOP expression Example: Use a K-Map to minimize the following standard SOP expression](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-17.jpg)
![Mapping Directly from a Truth Table Mapping Directly from a Truth Table](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-18.jpg)
![Don’t Care (X) Conditions • A situation arises in which input variable combinations are Don’t Care (X) Conditions • A situation arises in which input variable combinations are](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-19.jpg)
![Don’t Care (X) Conditions Example of the use of “don’t care” conditions to simplify Don’t Care (X) Conditions Example of the use of “don’t care” conditions to simplify](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-20.jpg)
![Exercise: Use K-Map to find the minimum SOP from 1 2 Exercise: Use K-Map to find the minimum SOP from 1 2](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-21.jpg)
![K-Map Simplification of POS Boolean Expression • A 0 is placed on the K-map K-Map Simplification of POS Boolean Expression • A 0 is placed on the K-map](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-22.jpg)
![Example: F=(A+B+C)(A+B+C)(A+B+C) The combination of binary values: (0 + 0)(0 + 1)(0 + 1 Example: F=(A+B+C)(A+B+C)(A+B+C) The combination of binary values: (0 + 0)(0 + 1)(0 + 1](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-23.jpg)
![Converting Between POS and SOP Using the K-Map • For a POS expression, all Converting Between POS and SOP Using the K-Map • For a POS expression, all](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-24.jpg)
![Example: • Using a K-map, convert the following standard POS expression into a minimum Example: • Using a K-map, convert the following standard POS expression into a minimum](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-25.jpg)
![](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-26.jpg)
- Slides: 26
![CHAPTER 4 The Karnaugh Map CHAPTER 4 The Karnaugh Map](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-1.jpg)
CHAPTER 4 The Karnaugh Map
![The Karnaugh Map Provides a systematic method for simplifying Boolean expressions Produces The Karnaugh Map • Provides a systematic method for simplifying Boolean expressions • Produces](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-2.jpg)
The Karnaugh Map • Provides a systematic method for simplifying Boolean expressions • Produces the simplest SOP or POS expression • Similar to a truth table because it presents all of the possible values of input variables
![The 3 Variable KMap The 3 -Variable K-Map](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-3.jpg)
The 3 -Variable K-Map
![The 4 Variable KMap The 4 -Variable K-Map](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-4.jpg)
The 4 -Variable K-Map
![KMap SOP Minimization A 1 is placed on the KMap for each product K-Map SOP Minimization • A 1 is placed on the KMap for each product](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-5.jpg)
K-Map SOP Minimization • A 1 is placed on the KMap for each product term in the expression. • Each 1 is placed in a cell corresponding to the value of a product term
![Mapping Standard SOP Expression Example Map the following standard SOP expression on a KMap Mapping Standard SOP Expression Example: Map the following standard SOP expression on a K-Map:](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-6.jpg)
Mapping Standard SOP Expression Example: Map the following standard SOP expression on a K-Map: Solution:
![Map the following standard SOP expression on a KMap Example Solution Map the following standard SOP expression on a K-Map: Example: Solution:](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-7.jpg)
Map the following standard SOP expression on a K-Map: Example: Solution:
![Exercise Map the following standard SOP expression on a KMap Exercise: Map the following standard SOP expression on a K-Map:](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-8.jpg)
Exercise: Map the following standard SOP expression on a K-Map:
![KMap Simplification of SOP Expressions A group must contain either 1 2 4 8 K-Map Simplification of SOP Expressions A group must contain either 1, 2, 4, 8](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-9.jpg)
K-Map Simplification of SOP Expressions A group must contain either 1, 2, 4, 8 or 16 cells. Each cell in group must be adjacent to one or more cells in that same group but all cells in the group do not have to be adjacent to each other Always include the largest possible number 1 s in a group in accordance with rule 1 Each 1 on the map must be included in at least one group. The 1 s already in a group can be included in another group as long as the overlapping groups include noncommon 1 s To maximize the size of the groups and to minimize the number of groups
![Example Group the 1 s in each KMaps Example: Group the 1 s in each K-Maps](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-10.jpg)
Example: Group the 1 s in each K-Maps
![Determining the minimum SOP Expression from the Map Groups the cells that have Determining the minimum SOP Expression from the Map • Groups the cells that have](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-11.jpg)
Determining the minimum SOP Expression from the Map • Groups the cells that have 1 s. Each group of cells containing 1 s create one product term composed of all variables that occur in only one form (either uncomplemented or complemented) within the group. Variable that occurs both uncomplemented and complemented within the group are eliminated. These are called contradictory variables.
![Determining the minimum SOP Expression from the Map For 3 variable map 1 Determining the minimum SOP Expression from the Map • For 3 -variable map: 1)](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-12.jpg)
Determining the minimum SOP Expression from the Map • For 3 -variable map: 1) A 1 cell group yields a 3 variable product term 2) A 2 -cell group yields a 2 -variable product term 3) A 4 cell group yields a 1 variable term 4) An 8 cell group yields a value of 1 for the expression
![Determining the minimum SOP Expression from the Map For 4 variable map 1 Determining the minimum SOP Expression from the Map • For 4 -variable map: 1)](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-13.jpg)
Determining the minimum SOP Expression from the Map • For 4 -variable map: 1) A 1 cell group yields a 4 variable product term 2) A 2 -cell group yields a 3 -variable product term 3) A 4 cell group yields a 2 variable term 4) An 8 cell group yields a 1 variable term 5) A 16 cell group yields a value of 1 for the expression
![Example Determine the product term for the KMap below and write the resulting minimum Example: Determine the product term for the K-Map below and write the resulting minimum](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-14.jpg)
Example: Determine the product term for the K-Map below and write the resulting minimum SOP expression 1
![](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-15.jpg)
![Example Use a KMap to minimize the following standard SOP expression Example: Use a K-Map to minimize the following standard SOP expression](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-16.jpg)
Example: Use a K-Map to minimize the following standard SOP expression
![Example Use a KMap to minimize the following standard SOP expression Example: Use a K-Map to minimize the following standard SOP expression](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-17.jpg)
Example: Use a K-Map to minimize the following standard SOP expression
![Mapping Directly from a Truth Table Mapping Directly from a Truth Table](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-18.jpg)
Mapping Directly from a Truth Table
![Dont Care X Conditions A situation arises in which input variable combinations are Don’t Care (X) Conditions • A situation arises in which input variable combinations are](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-19.jpg)
Don’t Care (X) Conditions • A situation arises in which input variable combinations are not allowed • Don’t care terms either a 1 or a 0 may be assigned to the output
![Dont Care X Conditions Example of the use of dont care conditions to simplify Don’t Care (X) Conditions Example of the use of “don’t care” conditions to simplify](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-20.jpg)
Don’t Care (X) Conditions Example of the use of “don’t care” conditions to simplify an expression
![Exercise Use KMap to find the minimum SOP from 1 2 Exercise: Use K-Map to find the minimum SOP from 1 2](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-21.jpg)
Exercise: Use K-Map to find the minimum SOP from 1 2
![KMap Simplification of POS Boolean Expression A 0 is placed on the Kmap K-Map Simplification of POS Boolean Expression • A 0 is placed on the K-map](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-22.jpg)
K-Map Simplification of POS Boolean Expression • A 0 is placed on the K-map for each sum term in the POS expression. • Mapping a Standard POS Expression – Step 1: Determine the binary value of each sum term in the standard POS expression. – Place a 0 on the K-map in the corresponding cell.
![Example FABCABCABC The combination of binary values 0 00 10 1 Example: F=(A+B+C)(A+B+C)(A+B+C) The combination of binary values: (0 + 0)(0 + 1)(0 + 1](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-23.jpg)
Example: F=(A+B+C)(A+B+C)(A+B+C) The combination of binary values: (0 + 0)(0 + 1)(0 + 1 +0)(0 + 1)(1 + 0) The resulting minimum POS is: A ( B + C)
![Converting Between POS and SOP Using the KMap For a POS expression all Converting Between POS and SOP Using the K-Map • For a POS expression, all](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-24.jpg)
Converting Between POS and SOP Using the K-Map • For a POS expression, all the cells that do not contain 0 s contain 1 s, from which the SOP expression is derived. • Likewise, for an SOP expression, all the cells that do not contain 1 s contain 0 s, from which the POS expression is derived.
![Example Using a Kmap convert the following standard POS expression into a minimum Example: • Using a K-map, convert the following standard POS expression into a minimum](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-25.jpg)
Example: • Using a K-map, convert the following standard POS expression into a minimum POS expression, a standard SOP expression, and a minimum SOP expression. • ( A + B + C + D)(A + B + C+ D)(A+ B + C + D)(A + B + C + D)
![](https://slidetodoc.com/presentation_image/4ac89ae355129171dc99328c879f6d00/image-26.jpg)
Karnaugh map is made up of
Penyederhanaan k-map 4 variabel
Penyederhanaan dengan karnaugh map f = ab' + a'b + ab
Karnaugh map probability
Pos k map
Don't care condition in k map
Dr karnaugh
Karnaugh map method
Five variable k map
Karnaugh map
Boole
Maurice karnaugh
Contoh soal peta karnaugh
Maurice karnaugh
Sistemas digitales
Circuitos
Biểu đồ karnaugh 4 biến
Wyświetlacz 7-segmentowy tablica karnaugh
Karnaugh haritası soruları
Karnaugh diagramm 5 variablen
Peta karnaugh 3 peubah
Mappe di karnaugh esempi
Diagrama karnaugh 4 variabile
Karnaugh kmap xor patterns
Mapa de karnaugh maxtermos
5 változós karnaugh tábla
Karnaugh diyagramı