Introduction / Context:
Karnaugh maps (K-maps) visualize adjacency in Boolean spaces, enabling grouping of 1-cells to simplify expressions. The map wraps around at the edges so that cells differing in only one variable are considered neighbors even across boundaries.
Given Data / Assumptions:
- K-map uses Gray-coded ordering along rows and columns.
- Edge adjacency follows toroidal wraparound (left-right and top-bottom).
- Adjacency implies one-bit Hamming distance between cells.
Concept / Approach:
Because the variable ordering is Gray coded, the first and last rows (and columns) differ by only one variable for corresponding columns (or rows). Therefore, the top-row cell and the bottom-row cell directly beneath/above it are adjacent, allowing valid grouping across the border.
Step-by-Step Solution:
Consider a 4-variable K-map with rows AB and columns CD in Gray code.Row AB = 00 (top) and AB = 01, 11, 10 (middle), and AB = 00 wraps to AB = 10 (bottom) with one-bit difference.Thus, top/bottom cells align as neighbors for grouping 2, 4, 8, etc. cells.This enables larger implicants and greater simplification.
Verification / Alternative check:
Draw a 3- or 4-variable K-map and confirm that opposite edges correspond to single-bit changes; edges are conceptually “touching.”
Why Other Options Are Wrong:
Incorrect: Ignores the Gray code and toroidal adjacency.Only true for 2-variable / don’t-care cells: Adjacency rules are general; don’t-cares can be used, but they are not required for edge adjacency.
Common Pitfalls:
Treating the K-map as a flat grid without wraparound, which prevents optimal grouping.Misaligning Gray code order, breaking single-bit adjacency.
Final Answer:
Correct
Discussion & Comments