Full adder, carry-out K-map — adders
corethe majority function, three terms
Answer
B · C + A · C + A · B
Why this example is worth doing
Mapping the carry-out column gives three overlapping pairs and a symmetric three-term answer — the majority function, true when at least two of the three inputs are. The page uses it as the site's showcase K-map because the groups overlap in a way that is obvious on the map and invisible in the algebra, and because the result is a named function a student will meet again in voting circuits and in carry-lookahead.
Try your own input in the Half adder & full adder. Truth tables, K-maps and circuits for both adders, and the ripple-carry chain.
How the answer is reached
Step by step
A · B + A · C + B · C is already a minimum sum-of-products form (3 terms, 6 literals), proved by Quine–McCluskey — no law can shorten it. It is the only minimum-cost form.
A · B + A · C + B · C is already a minimum sum-of-products form (3 terms, 6 literals), proved by Quine–McCluskey — no law can shorten it. It is the only minimum-cost form.
Quine–McCluskey
| # | Term |
|---|---|
| 1 | B · C |
| 2 | A · C |
| 3 | A · B |
| # | Cover |
|---|---|
| 1 | B · C + A · C + A · B |
3 term(s), 6 literal(s); 3 essential prime implicant(s).
Truth table
| # | A | B | C | F |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 |
| 2 | 0 | 1 | 0 | 0 |
| 3 | 0 | 1 | 1 | 1 |
| 4 | 1 | 0 | 0 | 0 |
| 5 | 1 | 0 | 1 | 1 |
| 6 | 1 | 1 | 0 | 1 |
| 7 | 1 | 1 | 1 | 1 |