Lecture 7: 3-Variable K-Maps

1. Structure

• A graphical tool to simplify Boolean functions.
• For 3 variables (A, B, C), the map has 8 cells ($2^3$).
• Cells are organized in Gray Code order (00, 01, 11, 10). This ensures only one bit changes between adjacent cells.

2. Grouping Rules

• Groups must be in powers of 2 (1, 2, 4, or 8 cells).
• Adjacency: Cells can group horizontally, vertically, or "wrap around" (left side with right side).
• Goal: Create the largest possible groups and use the minimum number of groups.
• Every '1' in the map must be covered by at least one group.

3. Deriving the Minimal Expression

• For each group, look at the variables on the edges.
• If a variable stays the same (always 0 or always 1) across all cells in the group, it stays in the term.
• If a variable changes (e.g., is 0 in one cell and 1 in another), it is eliminated.