---
tag: DND9VEDAXA
kind: statement
role: proposition
name: Axioms for fuzzy intersection
sources:
  - wikipedia: en:1651811@r1360435730
    locator: Fuzzy set operations, section Fuzzy intersections > Axioms for fuzzy intersection
    note: adopted
    mention: wikipedia://en/1651811#1RTL
    sections: [Statement]
inventory: [3FKEWE4G0Q]
---

## Statement

**Axiom i1. *Boundary condition**

*i*(*a*, 1) = *a*

**Axiom i2. *Monotonicity**

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

**Axiom i3. *Commutativity**

*i*(*a*, *b*) = *i*(*b*, *a*)

**Axiom i4. *Associativity**

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

**Axiom i5. *Continuity**

*i* is a continuous function

**Axiom i6. *Subidempotency**

*i*(*a*, *a*) \< *a* for all 0 \< *a* \< 1

**Axiom i7. *Strict monotonicity**

*i* (*a*<sub>1</sub>, *b*<sub>1</sub>) \< *i* (*a*<sub>2</sub>, *b*<sub>2</sub>) if *a*<sub>1</sub> \< *a*<sub>2</sub> and *b*<sub>1</sub> \< *b*<sub>2</sub>

Axioms i1 up to i4 define a **t-norm** (a.k.a. **fuzzy intersection**). The standard t-norm min is the only idempotent t-norm (that is, *i* (*a*<sub>1</sub>, *a*<sub>1</sub>) = *a* for all *a* ∈ \[0,1\]).