Essential prime implicant: meaning and examples

An essential prime implicant is a prime implicant that covers at least one true row no other prime implicant covers. Every cover of the function must include it; the non-essential ones are a matter of choice.

After the prime implicants of a function are known, the cover is built in two steps: take every essential prime implicant, because some minterm has no other home, then choose among the remaining prime implicants until every true row is covered. A function can have no essential prime implicants at all, which happens when every true row lies in two or more prime implicants; the choice is then entirely free, and several equally short covers exist. Such functions are called cyclic.

The artefact on this page is exactly that case. Σm(A, B, C; 0, 1, 2, 5, 6, 7) has 6 prime implicants, each a pair of adjacent cells, and each of its six true cells belongs to two of them, so none is essential. The engine's map derives (NOT A AND NOT B) OR (A AND C) OR (B AND NOT C) from the groups 0 and 1, 5 and 7, and 2 and 6, checked equivalent on every assignment; the three prime implicants it did not choose would cover the function equally well.

Essential prime implicants are the first thing a designer marks on a map and the first step of the Quine–McCluskey cover table. In reviewing a rule they identify the clauses that cannot be dropped without changing behaviour. The term is confused with prime implicant (largest, not necessarily needed) and with a term that merely appears in the chosen expression, which may be one of several interchangeable options, as on this page.

Example

f := Σm(A, B, C; 0, 1, 2, 5, 6, 7) · output

Exact checks

Always true (tautology)? No Counterexample: A = F, B = T, C = T · Exhaustively checked and solver cross-checked · 4 rows

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

Can be true? Yes Example assignment: A = F, B = F, C = F · Exhaustively checked and solver cross-checked · 1 rows

Truth table (8 rows)

Essential prime implicant: meaning and examples
#	A	B	C	f
0	FALSE	FALSE	FALSE	TRUE
1	FALSE	FALSE	TRUE	TRUE
2	FALSE	TRUE	FALSE	TRUE
3	FALSE	TRUE	TRUE	FALSE
4	TRUE	FALSE	FALSE	FALSE
5	TRUE	FALSE	TRUE	TRUE
6	TRUE	TRUE	FALSE	TRUE
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 0, 1, 2, 5, 6, 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 0, 1, 2, 5, 6, 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 0, 1, 2, 5, 6, and 7 [canonical sum of minterms (DNF)].
In every other case, f is false.
Formal notation: Σm(A, B, C; 0, 1, 2, 5, 6, 7)

Karnaugh map

A \ BC

1m0

1m1

0m3

1m2

0m4

1m5

1m7

1m6

cells m0, m1NOT A AND NOT B
cells m5, m7A AND C
cells m2, m6B AND NOT C

(NOT A AND NOT B) OR (A AND C) OR (B AND NOT C) · Derived (NOT A AND NOT B) OR (A AND C) OR (B AND NOT C). Exactly equivalent on every assignment.

The common slip

Assuming the expression on the page is the only correct one. When no prime implicant is essential, as here, a different choice of three groups gives a different expression with the same term count, and both are exact. The engine labels its cover irreducible, not unique.

Questions

How do I spot an essential prime implicant on a map?

Find a 1-cell that lies in only one of the largest rectangles. That rectangle is essential. If every 1-cell lies in two or more, as on this page, there are none, and the cover is chosen among equally short alternatives.

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.