Evaluate the claim: “Five-variable Karnaugh maps are impossible.” Consider standard K-map construction practices.
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ACorrect
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BIncorrect
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CAmbiguous as stated
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DCannot be determined without additional data
Answer
Correct Answer: Incorrect
Explanation
Introduction / Context:Karnaugh maps (K-maps) provide a visual method for simplifying Boolean expressions by grouping adjacent 1s (or 0s) according to Gray-coded adjacency. While 4-variable K-maps are commonly shown in textbooks, higher-variable maps also exist and are used in practice for instructional and limited design tasks.
Given Data / Assumptions:
- We consider the existence and usability of a 5-variable map.
- Adjacency follows Gray code ordering.
- Goal is simplification by grouping power-of-two cells.
Concept / Approach:Five-variable K-maps are typically realized as two linked 4-variable maps (or a 32-cell layout), where the fifth variable toggles between the maps. Groups can wrap across the two maps to preserve adjacency, enabling simplification just as with 3- or 4-variable cases, albeit with more care.
Step-by-Step Solution:
Represent variables as A, B, C, D in a 4x4 grid using Gray code.Introduce the fifth variable E by drawing two 4-variable maps: one for E = 0 and one for E = 1.Mark minterms in the appropriate map; consider adjacency both within each map and across the pair.Form groups of 1, 2, 4, 8, or 16 cells that respect adjacency rules, including wrap-around and inter-map adjacency.Verification / Alternative check:Numerous educational resources and CAD tools demonstrate 5-variable K-map groupings. Although 6-variable K-maps are also possible (often as four linked 4-variable maps), the method becomes unwieldy, motivating algorithmic minimization (e.g., Quine–McCluskey) for larger problems.
Why Other Options Are Wrong:
- Correct: Contradicts standard construction using two linked maps.
- Ambiguous as stated: The statement is categorical; there is no ambiguity in “impossible.”
- Cannot be determined: The existence is factual; no extra data is required.
Common Pitfalls:Assuming K-maps end at 4 variables. Usability decreases with more variables, but possibility remains. Do not confuse “impractical” with “impossible.”
Final Answer:Incorrect