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Chap3 Mathematical basics of fuzzy logic

Abstract

  • Fuzzy information and fuzzy sets
  • Operations with fuzzy sets
  • Fuzzy relations
  • Linguistic variables

Given μ is a fuzzy set in G, then {x| x∈G,μ(x)>0} is called support of fuzzy set μ {x| x∈G,μ(x)=1} is called kernel of fuzzy set μ

fuzzy-similar: have the same kernal strongly fuzzy-similar: have the same kernal and the same support

And link(intersection)

\[ \mu_1 \cap \mu_2:G \rightarrow[0,1], with (\mu_1 \cap \mu_2)(x) = MIN\{\mu_1 (x) + \mu_2(x)\} \]

Or link(union)

\[ \mu_1 \cup \mu_2 : G\rightarrow[0,1] mit (\mu_1 \cap \mu_2)(x) = MAX \{\mu_1 (x), \mu_2(x)\} \]

Special operators: Modifiers - Concentration operator CON - Dilatation operator DIL

fuzzy relations $$ R: G_1 \times G_2 \rightarrow [0,1] $$

\[ (\mu_1 \times \mu_2)(x,y)= MIN\{ \mu_1(x), \mu_2(y)\}, with (x,y) \in G_1 \times G_2 \]
\[ \mu_1 \times \mu_2 =\mu_1^T \circ \mu_2 \]

product calculation is replaced by the MIN opertor and the addition is replaced by the MAX operatior

\[ \mu_{R \circ S}(x, z)= MAX_{y \in G_2} {MIN \{\mu_R(x,y), \mu_S(y,z)}\}, with (x,z) \in G_1 \times G_3 \]

fuzzy inference

\[ \mu_2(y) = MAX_{x \in G_1} \{ MIN{\mu_1 (x), \mu_R(x,y)}\} = \mu_1 \circ R, y\in G_2 \]

inference with several rules

  1. Construct implication relation for each rule i
\[ \mu_{R_i}(x,y) = I(\mu_{A_i}(x), \mu_{B_i}(y)) \]
  1. Aggregate relations \(R_i\) into one
\[ \mu_R(x,y) = aggr(\mu_{R_i}(x,y)) \]

The aggr operator is the minimum for implications and the maximum for junctions

  1. Use relational composition to derive B' from A'
\[ B^\prime = A^\prime \circ R \]

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