Karnaugh map (K-map) grouping rules — which is invalid? In K-map simplification, which of the following cell combinations cannot be grouped as valid adjacencies to form implicants?
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ACorners in the same row
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BCorners in the same column
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CDiagonal corners
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DOverlapping combinations
Answer
Correct Answer: Diagonal corners
Explanation
Introduction / Context:Karnaugh maps exploit adjacency of minterms or maxterms to eliminate variables and simplify Boolean expressions. Understanding valid adjacencies (including wrap-around at edges) prevents mistakes that lead to incorrect simplifications or missed opportunities for larger groups.
Given Data / Assumptions:
- Standard Gray-code ordering is used on K-map rows/columns.
- Adjacency exists between cells that differ in exactly one variable.
- Edge wrap-around connects left-right edges and top-bottom edges.
Concept / Approach:Valid K-map groups are rectangles with areas that are powers of two (1, 2, 4, 8, ...). Cells on opposite edges are adjacent due to wrap-around. However, diagonal cells differ in two variables and are not adjacent, so they cannot be grouped directly without intermediate cells.
Step-by-Step Solution:
1) Same-row corners: adjacent via horizontal wrap-around → valid.2) Same-column corners: adjacent via vertical wrap-around → valid.3) Overlapping groups: allowed if each group is a valid power-of-two rectangle that helps cover all 1s redundantly to increase simplification.4) Diagonal corners: differ in two bits → not adjacent → invalid grouping.Verification / Alternative check:Write Gray-coded labels and compute Hamming distance between diagonal cells; distance = 2 → no adjacency.
Why Other Options Are Wrong:
- Row/column corners: legal due to wrap-around.
- Overlapping: legal and often useful for creating larger implicants.
Common Pitfalls:Forgetting wrap-around adjacency or forcing diagonal groupings that skip required intermediate cells, leading to incorrect minimized expressions.
Final Answer:Diagonal corners