AE4M33RZN, Fuzzy logic: Fuzzy relations

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Properties of fuzzy <strong>relations</strong> Then the relation R ⊆ Δ × Δ is called property using set axioms reflexive E ⊆ R symmetric R = R -1 ∘-anti-symmetric R ∩ ∘ R -1 ⊆ E ∘-transitive R ○ ∘ R ⊆ R ∘-partial order reflexive, ∘-transitive and ∘-anti-symmetric ∘-equivalence reflexive, ∘-transitive and ∘-symmetric
Properties in a finite domain If the universe Δ is finite, the relation can be written as a matrix. Their properties are reflected in the relation's matrix: • Reflexivity: Cells on the main diagonalare 1 • Symmetricity: Cells symmetric over the main diagonalare equal • Anti-symmetricity: Cells symmetric over the main diagonal . have conjunction equal to zero • For S- and A-anti-symmetricity,one of the elements must be zero • For L-anti-symmetricity,their sum must be less or equal to 1 • Transitivity: More difficult (see example on the next slide). . . . .
Page 1 and 2: AE4M33RZN, Fuzzy logic: Fuzzy relat
Page 3: Fuzzy implication We already know f
Page 13 and 14: Generalized fuzzy inclusion: Exampl
Page 18 and 19: Cutworhiness: Examples • The theo
Page 20 and 21: Google: “probability” Sources:
Page 22 and 23: Crisp relations Definition A binary
Page 24 and 25: Crisp relations: Properties The equ
Page 26 and 27: Projection Defintion Let R ∈
Page 28 and 29: Cylindrical extension Can we recons
Page 30 and 31: Composition of fuzzy relations Defi
Page 34 and 35: Example on fuzzy relation propertie
Page 36 and 37: Proof of 234. Proving (21) and (22)
Page 38: Extensions: Sometimes it is useful

AE4M33RZN, Fuzzy logic: Fuzzy relations

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