K-map verification of a Boolean simplification: Consider the Boolean expression X = A*C*D + A*B*(C*D + B*C). By Boolean algebra (distributive law and idempotence), this reduces to X = A*C*(B + D). A Karnaugh-map (K-map) drawn for variables A, B, C, and D should therefore implement X = A*C*(B + D). Decide whether a K-map that shows this implementation is correct.

Introduction / Context:
Karnaugh maps (K-maps) are a visual technique used to verify and minimize Boolean expressions. When an expression is algebraically simplified, a consistent K-map grouping must reproduce the same minimized function. Here we check whether the algebraic reduction of X = A*C*D + A*B*(C*D + B*C) to X = A*C*(B + D) is reflected correctly on a K-map implementation.

Given Data / Assumptions:

• Variables: A, B, C, D with conventional binary interpretation.
• Boolean laws apply: distributive, absorption, idempotence, and commutativity.
• K-map cell adjacency uses Gray code ordering to allow grouping by powers of two.

Concept / Approach:
Start by expanding and simplifying algebraically, then map the resulting minterms on a 4-variable K-map. Consistency requires that the groups of 1-cells correspond to the product terms found in the simplified sum-of-products form.

Step-by-Step Solution:

Verification / Alternative check:
On a 4-variable K-map, mark 1s for minterms where A=1 and C=1 and (B=1 or D=1). These cells form two groups (or one combined arrangement depending on adjacency) that correspond to A*C*B and A*C*D. The K-map therefore matches the simplified expression.

Why Other Options Are Wrong:
“Incorrect” contradicts the algebraic derivation. “Correct only for POS form” is irrelevant since we are discussing SOP. “Depends on Gray code ordering” is a red herring; Gray code affects cell placement, not the underlying function.

Common Pitfalls:
Forgetting absorption (dropping ABCD when ACD is present), confusing grouping sizes, or mislabeling K-map axes leading to wrong implicants.

Final Answer:
Correct