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2.3 Triangular norms

A vector (x1*,…, xk*) of fuzzy numbers xi* is not a fuzzy vector. For the generalization of statistical inference, functions defined on sample spaces are essential. Therefore it is basic to form fuzzy elements in the sample space M × … × M, where M denotes the observation space of a random quantity. These fuzzy elements are fuzzy vectors. By this it is necessary to form fuzzy vectors from fuzzy samples. This is done by applying so-called triangular norms, also called t-norms.

Definition 2.5:

A function T : [0; 1] × [0; 1] → [0; 1] is called a t-norm, if for all x, y, z, ∈ [0; 1] the following conditions are fulfilled:

1. T (x, y) = T (y, x) commutativity.

2. T (T (x, y), z) = T (x, T (y, z)) associativity.

3. T (x, 1) = x.

4. x ≤ y ⇒ T (x, z) ≤ T (y, z).

Examples of t-norms are:

a. Minimum t-norm

b. Product t-norm

c. Limited sum t-norm

Remark 2.5:

For statistical and algebraic calculations with fuzzy data the minimum t-norm is optimal. For the combination of fuzzy components of vector data in some examples the product t-norm is more suitable.

For more details on mathematical aspects of t-norms see Klement et al. (2000).Based on t-norms the combination of fuzzy ...

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