F(A,B,C,D) = Σm(0,2,5,6,7,8,10,12,13,14,15) — Quine–McCluskey
intro6 prime implicants, 2 essential, 4 minimal covers
Answer
B′ · D′ + C · D′ + B · D + A · D′ | B′ · D′ + C · D′ + B · D + A · B | B′ · D′ + B · D + B · C + A · D′ | B′ · D′ + B · D + B · C + A · B
4 equally minimal forms exist, all of the same cost: B′ · D′ + C · D′ + B · D + A · D′ | B′ · D′ + C · D′ + B · D + A · B | B′ · D′ + B · D + B · C + A · D′ | B′ · D′ + B · D + B · C + A · B.
Why this example is worth doing
A large ON-set whose chart is mostly resolved by essential prime implicants, which is the ordinary case and the one to see before the pathological ones. The page walks the tabulation rounds explicitly: terms grouped by the number of 1s, adjacent groups compared, every pair differing in one bit combined with a dash, anything that never combined ticked off as prime. Watching the dash positions accumulate is what makes the connection to K-map group sizes — a term with k dashes is a group of 2^k cells.
Try your own input in the Quine–McCluskey solver. Minimise past the K-map limit with the full tabular method and Petrick’s step.
How the answer is reached
Quine–McCluskey
| # | Term |
|---|---|
| 1 | B′ · D′ |
| 2 | C · D′ |
| 3 | B · D |
| 4 | B · C |
| 5 | A · D′ |
| 6 | A · B |
| # | Cover |
|---|---|
| 1 | B′ · D′ + C · D′ + B · D + A · D′ |
| 2 | B′ · D′ + C · D′ + B · D + A · B |
| 3 | B′ · D′ + B · D + B · C + A · D′ |
| 4 | B′ · D′ + B · D + B · C + A · B |
4 term(s), 8 literal(s); 2 essential prime implicant(s).
Truth table
| # | A | B | C | D | F |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 1 | 0 |
| 2 | 0 | 0 | 1 | 0 | 1 |
| 3 | 0 | 0 | 1 | 1 | 0 |
| 4 | 0 | 1 | 0 | 0 | 0 |
| 5 | 0 | 1 | 0 | 1 | 1 |
| 6 | 0 | 1 | 1 | 0 | 1 |
| 7 | 0 | 1 | 1 | 1 | 1 |
| 8 | 1 | 0 | 0 | 0 | 1 |
| 9 | 1 | 0 | 0 | 1 | 0 |
| 10 | 1 | 0 | 1 | 0 | 1 |
| 11 | 1 | 0 | 1 | 1 | 0 |
| 12 | 1 | 1 | 0 | 0 | 1 |
| 13 | 1 | 1 | 0 | 1 | 1 |
| 14 | 1 | 1 | 1 | 0 | 1 |
| 15 | 1 | 1 | 1 | 1 | 1 |