F(A,B,C,D) = Σm(1,3,7,11,15) + d(0,2,5) — Karnaugh map
coredon't-cares used to enlarge groups; 2 minimal covers
Answer
A′ · B′ + C · D | A′ · D + C · D
2 equally minimal forms exist, all of the same cost: A′ · B′ + C · D | A′ · D + C · D.
Why this example is worth doing
The standard textbook don't-care problem, and the one that teaches what a don't-care actually is. The three X cells are not extra 1s you must cover and not 0s you must avoid; they are cells you may claim when claiming them makes a group bigger, and ignore otherwise. Here the tool takes 0 and 2 to build A̅B̄ and leaves 5 unused, and there is a second equally minimal cover using A̅D instead. Solving it without don't-cares gives three terms rather than two.
Try your own input in the Karnaugh map solver. Group a 2- to 6-variable map yourself and have every group marked right or wrong.
How the answer is reached
Quine–McCluskey
| # | Term |
|---|---|
| 1 | A′ · B′ |
| 2 | A′ · D |
| 3 | C · D |
| # | Cover |
|---|---|
| 1 | A′ · B′ + C · D |
| 2 | A′ · D + C · D |
2 term(s), 4 literal(s); 1 essential prime implicant(s).
Truth table
| # | A | B | C | D | F |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | X |
| 1 | 0 | 0 | 0 | 1 | 1 |
| 2 | 0 | 0 | 1 | 0 | X |
| 3 | 0 | 0 | 1 | 1 | 1 |
| 4 | 0 | 1 | 0 | 0 | 0 |
| 5 | 0 | 1 | 0 | 1 | X |
| 6 | 0 | 1 | 1 | 0 | 0 |
| 7 | 0 | 1 | 1 | 1 | 1 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 9 | 1 | 0 | 0 | 1 | 0 |
| 10 | 1 | 0 | 1 | 0 | 0 |
| 11 | 1 | 0 | 1 | 1 | 1 |
| 12 | 1 | 1 | 0 | 0 | 0 |
| 13 | 1 | 1 | 0 | 1 | 0 |
| 14 | 1 | 1 | 1 | 0 | 0 |
| 15 | 1 | 1 | 1 | 1 | 1 |