By Barnabas Bede (auth.)
ISBN-10: 3642352200
ISBN-13: 9783642352201
ISBN-10: 3642352219
ISBN-13: 9783642352218
This e-book provides a mathematically-based advent into the attention-grabbing subject of Fuzzy units and Fuzzy good judgment and can be used as textbook at either undergraduate and graduate degrees and in addition as reference advisor for mathematician, scientists or engineers who want to get an perception into Fuzzy Logic.
Fuzzy units were brought via Lotfi Zadeh in 1965 and because then, they've been utilized in many functions. in this case, there's a tremendous literature at the sensible purposes of fuzzy units, whereas thought has a extra modest assurance. the most objective of the current publication is to minimize this hole through delivering a theoretical advent into Fuzzy units in line with Mathematical research and Approximation concept. famous purposes, as for instance fuzzy keep watch over, also are mentioned during this e-book and put on new flooring, a theoretical beginning. additionally, a number of complex chapters and a number of other new effects are incorporated. those contain, between others, a brand new systematic and positive strategy for fuzzy inference platforms of Mamdani and Takagi-Sugeno forms, that investigates their approximation strength by way of supplying new blunders estimates.
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Extra info for Mathematics of Fuzzy Sets and Fuzzy Logic
Sample text
9. For any R, S ∈ F (X × Y ) and Q ∈ F (Y × Z) we have (i) (R ∨ S) ◦ Q = (R ◦ Q) ∨ (S ◦ Q) (ii) (R ∧ S) ◦ Q ≤ (R ◦ Q) ∧ (S ◦ Q). Proof. (i) We prove the equality by double inclusion. , (R ∨ S) ◦ Q ≤ (R ◦ Q) ∨ (S ◦ Q). On the other hand, from the previous proposition we have R◦Q ≤ (R∨S)◦Q and S ◦ Q ≤ (R ∨ S) ◦ Q, and then (R ◦ Q) ∨ (S ◦ Q) ≤ (R ∨ S) ◦ Q. Combining the two inequalities, the required conclusion follows. (ii) We have (R ∧ S) ◦ Q(x, z) = (R ∧ S)(x, y) ∧ Q(y, z) y∈Y (R(x, y) ∧ Q(y, z)) ∧ (S(x, y) ∧ Q(y, z)) = y∈Y ≤ (R(x, y) ∧ Q(y, z)) ∧ y∈Y (S(x, y) ∧ Q(y, z)) y∈Y = (R ◦ Q) ∧ (S ◦ Q)(x, z).
6. 23, (iv) 7. 24, (ii) 8. 26, (ii), (iii), (iv). 9. 31. 10. 34. 11. Consider the fuzzy relational equation R • P = Q. Find a solvability condition and find a solution of this equation, supposing that it is solvable. 12. Let R ∈ F (X × X) be a fuzzy relation. Let us consider the eigen fuzzy sets equation R ◦ A = A, with A ∈ F (X). Prove that the equation has a least and a greatest solution and these can be calculated as Al = lim Rn ◦ ∅. n→∞ and Ag = lim Rn ◦ X. n→∞ Hint: Use Tarski-Knaster and Kleene’s fixed point theorems (see appendix).
Proof. Let Mr fulfill the properties (i)-(iv). , u is a normal, fuzzy convex, upper semicontinuous and compactly supported fuzzy set). The level sets of u are ur = Mr , r ∈ (0, 1] and u0 ⊆ M0 . Now we prove these statements step by step. “Normal:” Since M1 is nonempty, for x ∈ M1 we have u(x0 ) = 1, for some x0 ∈ M0 , so u is normal. “Fuzzy Convex:” In order to prove fuzzy convexity we consider now a fixed element in a fixed interval x ∈ [a, b] ⊆ M0 . Suppose that u(a) = ra = sup{r|a ∈ Mr } and u(b) = rb = sup{r|b ∈ Mr }.