Fuzzy Relation

Definition

An n-ary fuzzy relation R is a fuzzy set (see “Fuzzy Set”) in the universe U₁ × … × U_n, i.e., R ⫇ U₁ × … × U_n. A special case is a binary fuzzy relation in the universe U × V, i.e., R ⫇ U × V.

Key Points

Mathematically, an n-ary fuzzy relation is a function R: U₁ × … × U_n → L where L is a residuated lattice.

Let R ⫇ U × V and S ⫇ V × W be two binary fuzzy relations. The composition of R and S is a fuzzy relation R ∘ S determined by the membership function

$$ \begin{array}{c}\big(R\circ S\big)\left(x,z\right)=\underset{y\in V}{\vee}\left(R\left(x,y\right)\wedge S\left(y,z\right)\right),\hfill \\ {}\hfill \kern11.5em x\in U,z\in W.\hfill \end{array} $$

(1)

The operation ∧ in (1) can be replaced by arbitrary t-norm.

Cross-References

• Fuzzy Set

• Residuated Lattice

• Triangular Norms