Interpreting K-map entries — meaning of a '1' in a cell. In a Karnaugh map used for SOP minimization, each '1' placed in a square represents what condition?

Introduction / Context:
Karnaugh maps are populated with 1s, 0s, and sometimes Xs (don’t care) to represent the output of a Boolean function for every input combination. Understanding exactly what a '1' means is essential for forming correct groups when obtaining a minimized sum-of-products expression.

Given Data / Assumptions:

• K-map arranged with Gray-coded rows/columns.
• We are targeting an SOP (sum-of-products) simplification.
• Entries may be 1 (true), 0 (false), or X (don’t care).

Concept / Approach:
For SOP, we group adjacent 1s (including wrap-around) in sizes of 2^k to form product terms that evaluate to 1 for those input combinations. Each 1 corresponds to a minterm in the truth table where the function’s output is HIGH.

Step-by-Step Solution:

Verification / Alternative check:
Reconstruct the truth table from the minimized SOP and verify that it yields 1 for exactly the minterms marked by 1s in the K-map and 0 elsewhere (except Xs which are don’t care conditions).

Why Other Options Are Wrong:

• All LOW input combinations: not specific; only certain input combinations produce 1s.
• LOW for all HIGH inputs: contradicts SOP mapping.
• DON’T CARE for all inputs: Xs are don’t care; 1s are definite HIGHs.
• Hazard-free region: hazards are timing-related; a 1 cell does not guarantee hazard-free behavior.

Common Pitfalls:

• Treating Xs as mandatory 1s; Xs are optional and used for larger groupings.
• Forgetting wrap-around adjacency on map edges.

Final Answer:
a HIGH for each input truth table condition that produces a HIGH output.