Prime implicant: meaning and examples

A prime implicant is an implicant that cannot be made any larger: removing any literal from it would make it cover a row where the function is false. On a Karnaugh map, it is a group of 1-cells that no bigger legal group contains.

Prime implicants are the candidates from which every shortest sum of products is assembled, because any implicant that is not prime can be replaced by a larger one that covers more rows with fewer literals. The Quine–McCluskey method finds them by merging minterms that differ in one bit until nothing merges further; a Karnaugh map finds them by drawing the largest rectangles. The engine's map reports how many there are and derives a cover from them, checked exactly on every row.

The artefact on this page is Σm(A, B, C; 1, 3, 5, 7), true on the four odd-numbered rows, which are exactly the rows where C is true. The map finds a single group of four cells, 1, 3, 5 and 7, and derives the one-literal expression C; the prime implicant count is 1. That term cannot be enlarged, since the only larger group would be the whole map, and it cannot be shrunk without losing cells, which makes it prime. The equivalence of C with the original holds on every assignment.

Prime implicants are the vocabulary of logic minimisation courses and of synthesis tools, where the question “which prime implicants are essential?” decides the cover. In testing they describe the weakest sufficient conditions of a rule. The term is confused with implicant (any group, not necessarily largest) and with essential prime implicant (a prime implicant that some true row needs, because no other prime implicant covers that row).

Example

f := Σm(A, B, C; 1, 3, 5, 7) · output

Exact checks

Always true (tautology)? No Counterexample: A = F, B = F, C = F · Exhaustively checked and solver cross-checked · 1 rows

Always false (contradiction)? No Counterexample: A = F, B = F, C = T · Exhaustively checked and solver cross-checked · 2 rows

Can be true? Yes Example assignment: A = F, B = F, C = T · Exhaustively checked and solver cross-checked · 2 rows

Truth table (8 rows)

Prime implicant: meaning and examples
#	A	B	C	f
0	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	TRUE	TRUE
2	FALSE	TRUE	FALSE	FALSE
3	FALSE	TRUE	TRUE	TRUE
4	TRUE	FALSE	FALSE	FALSE
5	TRUE	FALSE	TRUE	TRUE
6	TRUE	TRUE	FALSE	FALSE
7	TRUE	TRUE	TRUE	TRUE

Plain-language reading

Beginnerf is true exactly when the truth-table row of (A, B, C) is one of the rows 1, 3, 5, and 7.
In every other case, f is false.
About the operators:
- Σm lists the truth-table rows (minterms) where the formula is true. Rows are numbered from 0 with the first listed input as the most significant bit.

Intermediatef is true exactly when the truth-table row of (A, B, C) is one of the rows 1, 3, 5, and 7.
In every other case, f is false.

Advancedf is true exactly when the truth-table row of (A, B, C) is one of the rows 1, 3, 5, and 7 [canonical sum of minterms (DNF)].
In every other case, f is false.
Formal notation: Σm(A, B, C; 1, 3, 5, 7)

Karnaugh map

A \ BC

0m0

1m1

1m3

0m2

0m4

1m5

1m7

0m6

cells m1, m3, m5, m7C

C · Derived C. Exactly equivalent on every assignment.

The common slip

Assuming a prime implicant must appear in the final expression. Prime means it cannot be enlarged, not that it is needed; the implicant page shows a function with 6 prime implicants whose cover uses only 3. Only essential prime implicants are guaranteed a place.

Questions

How does the engine decide that a group is prime?

It enumerates the legal rectangles of 1-cells (with any don't-care cells counted as 1) and keeps those not contained in a larger one. The count it reports is the number of such groups, and the derived cover is built from them and checked on every row.

Practice

Glossary

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.