Karnaugh Map Examples
Master Karnaugh maps with step-by-step solved examples for 2, 3, and 4 variables. Learn grouping techniques and Boolean simplification.
2-Variable K-Map Example
Problem: F(A,B) = Σm(1,3)
Step 1: Truth Table
| A | B | F |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Step 2: K-Map
| B=0 | B=1 | |
|---|---|---|
| A=0 | 0 | 1 |
| A=1 | 0 | 1 |
Step 3: Grouping
Group the two adjacent 1s vertically (column B=1).
Result: F = B
3-Variable K-Map Example
Problem: F(A,B,C) = Σm(1,2,3,5,7)
Step 1: Truth Table
| A | B | C | F |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 |
Step 2: K-Map Layout
| BC=00 | BC=01 | BC=11 | BC=10 | |
|---|---|---|---|---|
| A=0 | 0 | 1 | 1 | 1 |
| A=1 | 0 | 1 | 1 | 0 |
Step 3: Grouping Strategy
- Group 1: Vertical pair (BC=01 column) → A'C + AC = C
- Group 2: Horizontal pair (A=0 row, BC=11,10) → A'B
- Individual: Single cell (A=1, BC=11) → ABC
Result: F = C + A'B + ABC
4-Variable K-Map Example
Problem: F(A,B,C,D) = Σm(0,1,2,5,8,9,10)
Step 1: 4-Variable K-Map
| CD=00 | CD=01 | CD=11 | CD=10 | |
|---|---|---|---|---|
| AB=00 | 1 | 1 | 0 | 1 |
| AB=01 | 0 | 1 | 0 | 0 |
| AB=11 | 0 | 0 | 0 | 0 |
| AB=10 | 1 | 1 | 0 | 1 |
Step 2: Optimal Grouping
- Group 1: Top row (AB=00) → A'B'
- Group 2: Bottom row (AB=10) → AB'
- Group 3: Single cell → A'BC'D
Result: F = A'B' + AB' + A'BC'D = B'(A' + A) + A'BC'D = B' + A'BC'D
Advanced K-Map Techniques
Don't Care Conditions
Don't care conditions (X or d) can be treated as either 0 or 1 to achieve maximum simplification.
Example: F(A,B,C) = Σm(1,3,7) + Σd(0,2,5)
Use don't cares strategically to create larger groups and achieve better simplification.
Grouping Rules
- Group sizes: Must be powers of 2 (1, 2, 4, 8, 16)
- Shape: Groups must be rectangular
- Adjacency: Include wrap-around edges
- Overlap: Groups can overlap
- Coverage: All 1s must be covered
- Minimality: Use fewest, largest groups
Common Mistakes to Avoid
❌ Wrong
- • Diagonal grouping
- • Non-rectangular groups
- • Forgetting wrap-around
- • Missing larger groups
- • Including 0s in groups
✅ Correct
- • Horizontal/vertical grouping
- • Rectangular shapes only
- • Consider edge wrap-around
- • Maximize group sizes
- • Group only 1s and don't cares
Practice Problems
Try These Examples:
- Problem 1: F(A,B,C) = Σm(0,2,4,6) - Hint: Look for a pattern
- Problem 2: F(A,B,C,D) = Σm(0,1,4,5,10,11,14,15) - Hint: Four groups of 2
- Problem 3: F(A,B,C) = Σm(1,2,3,5,6,7) + Σd(0,4) - Hint: Use don't cares
Summary
Karnaugh maps provide a visual method for Boolean function simplification. The key is to identify the largest possible groups of adjacent 1s, following the grouping rules. With practice, K-maps become an intuitive tool for logic optimization and circuit design.