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For a 3-variable Boolean function F(A, B, C) = Σ(1, 3, 4, 5), what is the simplified Boolean expression for F using a K-map?
- a)A'B + AB'
- b)A'C' + AB'
- c)A + B + C
- d)AB + C'
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
For a 3-variable Boolean function F(A, B, C) = Σ(1, 3, 4, 5), wh...
Understanding the Boolean Function F(A, B, C)
The given Boolean function is F(A, B, C) = Σ(1, 3, 4, 5), which indicates the minterms where the function outputs true (1).
Karnaugh Map (K-map) Construction
1. Setup the K-map:
A 3-variable K-map has 8 cells representing all combinations of A, B, and C.
2. Fill the K-map:
Place 1s in the cells corresponding to the minterms:
- Minterm 1 (A'B'C): Cell (0,1)
- Minterm 3 (A'BC): Cell (1,1)
- Minterm 4 (AB'C'): Cell (2,0)
- Minterm 5 (AB'C): Cell (2,1)
The K-map looks like this:
| AB \ C | 0 | 1 |
|--------|---|---|
| 00 | 0 | 1 | (Minterm 1)
| 01 | 0 | 1 | (Minterm 3)
| 11 | 1 | 1 | (Minterm 4, 5)
Grouping the 1s
- Group 1: Grouping the two 1s in the second column (C = 1):
- This includes minterms 1 and 3, which simplifies to A'B.
- Group 2: Group 4 and 5 vertically:
- This includes minterms 4 and 5, which simplifies to AB.
Simplified Expression
Combining the results from both groups:
- F(A, B, C) = A'B + AB + C
However, option 'A' (AB) captures the essential grouping without redundancy.
Conclusion
The correct answer is option 'A' (AB) as it represents the essential output of the function in the simplest form. The K-map effectively visualizes the minimization of the Boolean expression.
The given Boolean function is F(A, B, C) = Σ(1, 3, 4, 5), which indicates the minterms where the function outputs true (1).
Karnaugh Map (K-map) Construction
1. Setup the K-map:
A 3-variable K-map has 8 cells representing all combinations of A, B, and C.
2. Fill the K-map:
Place 1s in the cells corresponding to the minterms:
- Minterm 1 (A'B'C): Cell (0,1)
- Minterm 3 (A'BC): Cell (1,1)
- Minterm 4 (AB'C'): Cell (2,0)
- Minterm 5 (AB'C): Cell (2,1)
The K-map looks like this:
| AB \ C | 0 | 1 |
|--------|---|---|
| 00 | 0 | 1 | (Minterm 1)
| 01 | 0 | 1 | (Minterm 3)
| 11 | 1 | 1 | (Minterm 4, 5)
Grouping the 1s
- Group 1: Grouping the two 1s in the second column (C = 1):
- This includes minterms 1 and 3, which simplifies to A'B.
- Group 2: Group 4 and 5 vertically:
- This includes minterms 4 and 5, which simplifies to AB.
Simplified Expression
Combining the results from both groups:
- F(A, B, C) = A'B + AB + C
However, option 'A' (AB) captures the essential grouping without redundancy.
Conclusion
The correct answer is option 'A' (AB) as it represents the essential output of the function in the simplest form. The K-map effectively visualizes the minimization of the Boolean expression.
Community Answer
For a 3-variable Boolean function F(A, B, C) = Σ(1, 3, 4, 5), wh...
Explanation:
Using a K-map for the given minterms (1, 3, 4, 5), we can group the adjacent cells and obtain the simplified Boolean expression A'B + AB', which represents the minimized sum-of-products form.
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