Simplify the expression using K-maps: F(A,B,C) = π(0,2,4,5,7).
Understand the Problem
The question is asking to simplify a boolean expression using Karnaugh maps (K-maps) for the given function defined with its product of maxterms. It requires an understanding of boolean algebra and K-map simplification techniques.
Answer
The simplified expression is \( A'B' + AC' \).
Answer for screen readers
The simplified expression is ( A'B' + AC' ).
Steps to Solve
- Identify the variables and maxterms
The function is defined as the product of maxterms: $F(A,B,C) = \pi(0, 2, 4, 5, 7)$. Identify the maxterms:
- Maxterm 0: ( A'B'C' )
- Maxterm 2: ( A'BC' )
- Maxterm 4: ( AB'C' )
- Maxterm 5: ( AB'C )
- Maxterm 7: ( ABC )
- Construct the K-map
Set up the K-map for three variables (A, B, C). The layout is typically:
| AB \ C | 00 | 01 | 11 | 10 |
|---|---|---|---|---|
| 00 | 1 | 0 | 0 | 1 |
| 01 | 1 | 1 | 0 | 1 |
| 11 | 1 | 0 | 0 | 1 |
| 10 | 0 | 0 | 0 | 0 |
- Fill in the K-map
Mark the maxterm positions with a 1 in the K-map:
- Cell 0 (00 0) = 1
- Cell 2 (00 1) = 1
- Cell 4 (10 0) = 1
- Cell 5 (10 1) = 1
- Cell 7 (11 1) = 1
- Group the ones
Look for groups of 1s to simplify:
- Group horizontal pairs and vertical pairs as required.
- The optimal grouping will lead to simplified expressions.
- Write the simplified expression
From the grouped cells, derive the simplified boolean expression:
- Grouping cells (0, 4) yields: ( A'B' + AC' )
- Grouping cells (2, 6) would not contribute further.
- The final simplified expression can be determined.
The simplified expression is ( A'B' + AC' ).
More Information
This expression represents the conditions under which the function F is true. It reduces the original expression and minimizes the number of terms, making it easier to implement in digital circuits.
Tips
- Failing to correctly identify maxterms in the K-map.
- Overlooking grouping opportunities, leading to more complex expressions.
- Incorrectly filling in values in the K-map.
