F(A,B,C,D,E) = Σm(0,1,4,5,16,17,21,25,29) — Quine–McCluskey
exam standard4 prime implicants, 3 essential
Answer
A′ · B′ · D′ + B′ · C′ · D′ + A · D′ · E
Why this example is worth doing
Five variables is where the tabular method starts to earn its place. A five-variable K-map is drawable — this site draws it — but adjacency across planes is easy to miss by eye, whereas the tabulation simply compares bit patterns and cannot overlook a pair. The page frames Quine–McCluskey as Quine's and McCluskey's answer to exactly that limitation, and notes that the algorithm's cost is exponential in the worst case, so the practical ceiling is a matter of arithmetic rather than eyesight.
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 | A′ · B′ · D′ |
| 2 | B′ · C′ · D′ |
| 3 | B′ · D′ · E |
| 4 | A · D′ · E |
| # | Cover |
|---|---|
| 1 | A′ · B′ · D′ + B′ · C′ · D′ + A · D′ · E |
3 term(s), 9 literal(s); 3 essential prime implicant(s).
Truth table
| # | A | B | C | D | E | F |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 1 | 1 |
| 2 | 0 | 0 | 0 | 1 | 0 | 0 |
| 3 | 0 | 0 | 0 | 1 | 1 | 0 |
| 4 | 0 | 0 | 1 | 0 | 0 | 1 |
| 5 | 0 | 0 | 1 | 0 | 1 | 1 |
| 6 | 0 | 0 | 1 | 1 | 0 | 0 |
| 7 | 0 | 0 | 1 | 1 | 1 | 0 |
| 8 | 0 | 1 | 0 | 0 | 0 | 0 |
| 9 | 0 | 1 | 0 | 0 | 1 | 0 |
| 10 | 0 | 1 | 0 | 1 | 0 | 0 |
| 11 | 0 | 1 | 0 | 1 | 1 | 0 |
| 12 | 0 | 1 | 1 | 0 | 0 | 0 |
| 13 | 0 | 1 | 1 | 0 | 1 | 0 |
| 14 | 0 | 1 | 1 | 1 | 0 | 0 |
| 15 | 0 | 1 | 1 | 1 | 1 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 | 1 |
| 17 | 1 | 0 | 0 | 0 | 1 | 1 |
| 18 | 1 | 0 | 0 | 1 | 0 | 0 |
| 19 | 1 | 0 | 0 | 1 | 1 | 0 |
| 20 | 1 | 0 | 1 | 0 | 0 | 0 |
| 21 | 1 | 0 | 1 | 0 | 1 | 1 |
| 22 | 1 | 0 | 1 | 1 | 0 | 0 |
| 23 | 1 | 0 | 1 | 1 | 1 | 0 |
| 24 | 1 | 1 | 0 | 0 | 0 | 0 |
| 25 | 1 | 1 | 0 | 0 | 1 | 1 |
| 26 | 1 | 1 | 0 | 1 | 0 | 0 |
| 27 | 1 | 1 | 0 | 1 | 1 | 0 |
| 28 | 1 | 1 | 1 | 0 | 0 | 0 |
| 29 | 1 | 1 | 1 | 0 | 1 | 1 |
| 30 | 1 | 1 | 1 | 1 | 0 | 0 |
| 31 | 1 | 1 | 1 | 1 | 1 | 0 |