---
tag: DAFFVP23YY
kind: statement
role: proposition
name: Axioms for fuzzy complements
sources:
  - wikipedia: en:1651811@r1360435730
    locator: Fuzzy set operations, section Fuzzy complements > Axioms for fuzzy complements
    note: adopted
    mention: wikipedia://en/1651811#XT1L
    sections: [Statement]
inventory: [4K621NZZJX]
---

## Statement

**Axiom c1. *Boundary condition**

*c*(0) = 1 and *c*(1) = 0

**Axiom c2. *Monotonicity**

For all *a*, *b* ∈ \[0, 1\], if *a* \< *b*, then *c*(*a*) > *c*(*b*)

**Axiom c3. *Continuity**

*c* is continuous function.

**Axiom c4. *Involutions**

*c* is an involution, which means that *c*(*c*(*a*)) = *a* for each *a* ∈ \[0,1\]

*c* is a *strong negator* (a.k.a. *fuzzy complement*).

A function c satisfying axioms c1 and c3 has at least one fixpoint a<sup>\*</sup> with c(a<sup>\*</sup>) = a<sup>\*</sup>, and if axiom c2 is fulfilled as well there is exactly one such fixpoint. For the standard negator c(x) = 1-x the unique fixpoint is a<sup>\*</sup> = 0.5 .