---
tag: 8QGXKM0000
kind: statement
role: proposition
name: Axioms for fuzzy union
sources:
  - wikipedia: en:1651811@r1360435730
    locator: Fuzzy set operations, section Fuzzy unions > Axioms for fuzzy union
    note: adopted
    mention: wikipedia://en/1651811#ZH7I
    sections: [Statement]
inventory: [MGNCZXP7TB]
---

## Statement

**Axiom u1. *Boundary condition**

*u*(*a*, 0) =*u*(0 ,*a*) = *a*

**Axiom u2. *Monotonicity**

*b* ≤ *d* implies *u*(*a*, *b*) ≤ *u*(*a*, *d*)

**Axiom u3. *Commutativity**

*u*(*a*, *b*) = *u*(*b*, *a*)

**Axiom u4. *Associativity**

*u*(*a*, *u*(*b*, *d*)) = *u*(*u*(*a*, *b*), *d*)

**Axiom u5. *Continuity**

*u* is a continuous function

**Axiom u6. *Superidempotency**

*u*(*a*, *a*) > *a* for all 0 \< *a* \< 1

**Axiom u7. *Strict monotonicity**

*a*<sub>1</sub> \< *a*<sub>2</sub> and *b*<sub>1</sub> \< *b*<sub>2</sub> implies *u*(*a*<sub>1</sub>, *b*<sub>1</sub>) \< *u*(*a*<sub>2</sub>, *b*<sub>2</sub>)

Axioms u1 up to u4 define a **t-conorm** (a.k.a. **s-norm** or **fuzzy union**). The standard t-conorm max is the only idempotent t-conorm (i. e. u (a1, a1) = a for all a ∈ \[0,1\]).