A prime implicant is an implicant that cannot be made any bigger. Removing any one literal would double the rows it covers and take in a 0 of the function.
In the patch picture: a prime implicant is a safe patch that is as big as it can get. On a karnaugh map, it is a group of 1s that cannot be enlarged in any direction.
Why they matter. Every term of a minimal sop is a prime implicant. If a term were not prime, you could drop a literal and it would still cover only 1s, which is cheaper and covers at least as much.
But a minimal SOP does not always use every prime implicant. Once you have the primes, you choose the cheapest set that covers every 1 (a cover):
- A 1 covered by only one prime forces that prime into the answer. Such a prime is an essential prime implicant.
- Then add the cheapest primes that cover whatever is left.
Primes left out are redundant, often because they are the consensus of two chosen ones.
Finding all primes systematically is the first half of the quine mccluskey method.
| A\BC | 00 | 01 | 11 | 10 |
|---|---|---|---|---|
| 0 | 1m0 | 0m1 | 0m3 | 0m2 |
| 1 | 1m4 | 1m5 | 1m7 | 0m6 |