Implicant: meaning and examples

An implicant of a function is a product term that implies it: wherever the term is true, the function is true. On a Karnaugh map, an implicant is any legal group of cells that all contain 1.

A term t is an implicant of f when t -> f is a tautology, which is the same as saying that every row covered by t is a true row of f. Every minterm of a true row is an implicant, and so is every larger term obtained by merging adjacent minterms. A sum of products for f is simply a set of implicants whose rows together cover every true row, and the art of simplification is choosing few, large ones. An implicant that cannot be enlarged is a prime implicant.

The artefact on this page is Σm(A, B, C; 0, 1, 2, 5, 6, 7), true on six of its 8 rows. The Karnaugh map reports 6 prime implicants and derives the cover (NOT A AND NOT B) OR (A AND C) OR (B AND NOT C), three two-cell groups: cells 0 and 1, cells 5 and 7, and cells 2 and 6. Each group is an implicant, checked equivalent on every assignment. The tautology check is violated with the counterexample A = FALSE, B = TRUE, C = TRUE, row 3, one of the two false rows.

Implicants are the units of two-level circuit design, where each one becomes an AND gate, and of the Quine–McCluskey method, which enumerates them systematically. In rule writing an implicant is a sufficient condition: a case that on its own guarantees the outcome. The term is confused with prime implicant (an implicant that cannot be enlarged) and with the implication operator; an implicant is a term, not a formula with an arrow in it.

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)

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.

Minterms and maxterms

f(A, B, C) = Σm(0, 1, 2, 5, 6, 7)f(A, B, C) = ΠM(3, 4)

Variable order: A, B, C · 8 rows

The common slip

Taking any group of 1-cells as an implicant. A group must be a rectangle of 1, 2, 4, 8 or 16 cells on the Gray-code map, so that it corresponds to a single product term; three cells in an L-shape cover true rows but are not one term.

Questions

Can a function have more prime implicants than its cover uses?

Yes, and this page is an example: the map counts 6 prime implicants, and the derived expression uses 3 of them. The others are legitimate implicants that happen not to be needed once those three are chosen.

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.