3
Mathematical operations for fuzzy quantities
The extension of algebraic operations to fuzzy numbers is based on the so-called extension principle from fuzzy set theory. This extension principle is a method to extend classical functions f : M → N to the situation, when the argument is a fuzzy element in M.
3.1 Functions of fuzzy variables
In order to extend statistical functions f (x1,…xn) of samples x1,…,xn the following definition is useful.
Definition 3.1:
(Extension principle): Let f : M → N be an arbitrary function from M to N. For a fuzzy element x* in M, characterized by an arbitrary membership function μ : M → [0; 1] the generalized (fuzzy) value y* = f (x*) for the fuzzy argument value x* is the fuzzy element y* in N whose membership function ψ(·) is defined by
Remark 3.1:
The extension principle is a natural way to model the propagation of imprecision. It corresponds to the engineering principle ‘to be on the safe side’.
The following statements are important for the generalization of statistical functions.
Proposition 3.1
Let f : M → N be a classical function, and x* a fuzzy element of M with membership function μ(·). Then the fuzzy element y* = f (x*) defined by the extension principle obeys the following:
Proof:
Cδ(y*) = {y ∈ N : ψ(y) ≥ δ} where ψ(·) is defined as ...
