13. 2- and 3-Variable K-Map Minimization
Let's apply the grouping technique to small K-Maps.
2-Variable K-Map Grouping
We group powers of 2 (1, 2, 4) adjacent '1's. A group of 2 eliminates 1 variable, a group of 4 eliminates 2 variables.
Example: Simplify $F(A, B) = \Sigma(0, 2, 3)$
| $\overline{B}$ | B | |
|---|---|---|
| $\overline{A}$ | 1 ($m_0$) | 0 ($m_1$) |
| A | 1 ($m_2$) | 1 ($m_3$) |
- Group 1: $m_2$ and $m_3$. Variable A is fixed (1), B changes. Term: A.
- Group 2: $m_0$ and $m_2$. Variable B is fixed (0), A changes. Term: $\overline{B}$.
- Result: $F = A + \overline{B}$.
3-Variable K-Map Structure (8 cells)
We typically arrange the variables as $A$ (rows) and $BC$ (columns, using Gray code: 00, 01, 11, 10).
| A\BC | 00 | 01 | 11 | 10 |
|---|---|---|---|---|
| 0 | $m_0$ | $m_1$ | $m_3$ | $m_2$ |
| 1 | $m_4$ | $m_5$ | $m_7$ | $m_6$ |
Remember wrap-around adjacency! $m_0$ is adjacent to $m_2$, and $m_4$ is adjacent to $m_6$ (on the left/right boundaries).