Problem 39 - Discrete Mathematics with Applications

Question

Find the boolean expression represented by each Karnaugh map.

Step-by-Step Solution

Answer

To find the boolean expression represented by a Karnaugh map: 1. Identify the number of variables in the Karnaugh map. 2. Construct the Karnaugh map with appropriate grid size and label the rows/columns. 3. Fill in the Karnaugh map with the given boolean function. 4. Determine the prime implicants by finding the largest groups of 1s. 5. Determine the essential prime implicants which have at least one unique 1. 6. Pick the minimum set of prime implicants that cover all remaining 1s. 7. Write the boolean expression for the simplified function based on the prime implicants and essential prime implicants.

Step 1: Identify the number of variables in the Karnaugh map

Determine the number of variables (n) in the Karnaugh map. It can consist of any number of variables, for example, 2, 3, 4, etc.

Step 2: Construct the Karnaugh map

Construct a grid representing the Karnaugh map. For a 2-variable function, the K-map will have 2x2 cells; for a 3-variable function, it will have a 4x4 grid, and for a 4-variable function, the grid will be 4x4 or 2x8, based on the preference of representation. However, for functions with more than four variables, it can get tricky to visualize. Label the rows and columns with the boolean variables and their complements, respectively. Ensure that only one variable changes between adjacent cells, either horizontally or vertically. This adjacency is based on the Gray code.

Step 3: Fill in the Karnaugh map

Fill the Karnaugh map with the given boolean function. Place 1s in the cells corresponding to the minterms of the function, and 0s in the cells that don't represent minterms. In a Karnaugh map, there could be groups of 1s, and these groups have to be squares or rectangles containing 1, 2, 4, 8, etc., number of 1s.

Step 4: Determine the prime implicants

Find the largest groups of 1s that can be combined in the K-map. Remember that the groups must be rectangular, and the number of 1s must be a power of 2, such as 1, 2, 4, 8, etc. A 1 can be included in multiple groups if necessary. These groups are called prime implicants, and each prime implicant can be represented as a product term.

Step 5: Determine the essential prime implicants

Inspect the prime implicants found in Step 4, and find the ones that have at least one 1 that is not covered by any other prime implicant. These prime implicants are called essential prime implicants, and they will be part of the simplified boolean function.

Step 6: Pick the minimum set of prime implicants

Choose the minimum set of prime implicants that cover all the remaining 1s. Combine the essential prime implicants from Step 5 and the minimum set of prime implicants from this step to form the simplified boolean function.

Step 7: Write the boolean expression for the simplified function

Based on the groups of prime implicants and essential prime implicants, write down the simplified boolean expression. This expression will be in the Sum of Products (SOP) or Product of Sums (POS) form, depending on how the Karnaugh map was constructed and analyzed. That's it! Following these steps, you should be able to find the boolean expression represented by any given Karnaugh map.