Introduction Results Second Auxiliary Result Third Auxiliary Result How to Detect Crisp Sets Based on Main Result Subsethood Ordering of Normalized Interval-Valued Case Fuzzy Sets? How to Detect Type-1 Sets First Conclusion Based on Subsethood Ordering of Second Conclusion Normalized Interval-Valued Fuzzy Sets? Possible Future Work Home Page Christian Servin 1 , Olga Kosheleva 2 , Vladik Kreinovich 2 Title Page 1 Computer Science and Information Technology Systems Department ◭◭ ◮◮ El Paso Community College, El Paso, Texas 79915, USA cservin@gmail.com ◭ ◮ 2 University of Texas at El Paso, El Paso, Texas 79968, USA olgak@utep.edu, vladik@utep.edu Page 1 of 83 Go Back Full Screen Close Quit
Introduction Results 1. Introduction Second Auxiliary Result • A fuzzy set is usually defined as function A from a Third Auxiliary Result certain set U ( Universe of discourse ) to [0 , 1]. Main Result Interval-Valued Case • Traditional – “crisp” – sets can be viewed as particular First Conclusion cases of fuzzy sets, for which A ( a ) ∈ { 0 , 1 } for all x . Second Conclusion • In most applications, we consider normalized fuzzy sets, Possible Future Work i.e., fuzzy sets for which A ( x ) = 1 for some x ∈ U . Home Page • For crisp sets, this corresponds to considering non- Title Page empty sets. ◭◭ ◮◮ • For two crisp sets, A is a subset or B if and only if ◭ ◮ A ( x ) ≤ B ( x ) for all x . Page 2 of 83 • The same condition is used as a definition of the sub- sethood ordering between fuzzy sets: Go Back • a fuzzy set A is a subset of a fuzzy set B Full Screen • if A ( x ) ≤ B ( x ) for all x . Close Quit
Introduction Results 2. Introduction (cont-d) Second Auxiliary Result • Subsets B ⊆ A which are different from the set A are Third Auxiliary Result called proper subsets of A . Main Result Interval-Valued Case • A natural question is: First Conclusion • if we have a class of all normalized fuzzy sets with Second Conclusion the subsethood relation, Possible Future Work • can we detect which of these fuzzy sets are crisp? Home Page • It is known that: Title Page ◭◭ ◮◮ • if we alow all possible fuzzy sets – even non-normalized ones, ◭ ◮ • then we can detect crisp sets. Page 3 of 83 • In this talk, we show that such a detection is possible Go Back even if we restrict ourselves only to normalized sets. Full Screen Close Quit
Introduction Results 3. Results Second Auxiliary Result • We want to describe general crisp sets in terms of sub- Third Auxiliary Result sethood relation ⊆ between fuzzy sets. Main Result Interval-Valued Case • For this purpose, let us first describe some auxiliary First Conclusion notions in these terms. Second Conclusion • In this part of the talk, we only consider normalized Possible Future Work fuzzy sets. Home Page • Proposition. Title Page • A normalized fuzzy set is a 1-element crisp set ◭◭ ◮◮ • if and only if it has no proper normalized fuzzy sub- ◭ ◮ sets, i.e., if and only if B ⊆ A implies B = A . Page 4 of 83 • Let us first prove that: Go Back • a 1-element crisp set A = { x 0 } (i.e., a set for which A ( x 0 ) = 1 and A ( x ) = 0 for all x � = x 0 ) Full Screen • has the desired property. Close Quit
Introduction Results 4. Proof of the First Auxiliary Result (cont-d) Second Auxiliary Result • Indeed, if B ⊆ A , then B ( x ) ≤ A ( x ) for all x . Third Auxiliary Result Main Result • For x � = x 0 , we have A ( x ) = 0, so we have B ( x ) = 0 as Interval-Valued Case well. First Conclusion • Since B is a normalized fuzzy set, it has to attain value Second Conclusion 1 somewhere. Possible Future Work Home Page • We have B ( x ) = 0 for all x � = x 0 . Title Page • So, the only point x ∈ U at which B ( x ) = 1 is the point x 0 . ◭◭ ◮◮ • Thus, we have B ( x 0 ) = 1. ◭ ◮ Page 5 of 83 • So, indeed, we have B ( x ) = A ( x ) for all x , i.e., B = A . Go Back Full Screen Close Quit
Introduction Results 5. Proof of the First Auxiliary Result (cont-d) Second Auxiliary Result • Vice versa, let us prove that: Third Auxiliary Result Main Result • each normalized fuzzy set A which is different from Interval-Valued Case a 1-element crisp set First Conclusion • has a proper normalized fuzzy subset. Second Conclusion • Indeed, since A is normalized, we have A ( x 0 ) = 1 for Possible Future Work some x 0 . Home Page • Then, we can take B = { x 0 } . Title Page ◭◭ ◮◮ • Clearly, B ⊆ A , and, since A is not a 1-element crisp set, B � = A . ◭ ◮ • The proposition is proven. Page 6 of 83 Go Back Full Screen Close Quit
Introduction Results 6. Second Auxiliary Result Second Auxiliary Result • Definition. By a 2-element set , we mean a normalized Third Auxiliary Result fuzzy set A for which A ( x ) > 0 for exactly two x ∈ U . Main Result Interval-Valued Case • Proposition. First Conclusion • Let A be a normalized fuzzy set A which is not a Second Conclusion 1-element crisp set. Possible Future Work • Then, the following two conditions are equivalent Home Page to each other: Title Page • A is a non-crisp 2-element set, and ◭◭ ◮◮ • the class { B : B ⊆ A } is linearly ordered, i.e.: ◭ ◮ if B 1 , B 2 ⊆ A then B 1 ⊆ B 2 or B 2 ⊆ B 1 . Page 7 of 83 Go Back Full Screen Close Quit
Introduction Results 7. Third Auxiliary Result Second Auxiliary Result • Proposition. A normalized fuzzy set A is a crisp 2- Third Auxiliary Result element set ⇔ the following 2 conditions hold: Main Result Interval-Valued Case • the set A itself is not a 1-element crisp set and not First Conclusion a 2-element non-crisp set, but Second Conclusion • each proper norm. fuzzy subset B ⊆ A is either a Possible Future Work crisp 1-element sets or a non-crisp 2-element set. Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 83 Go Back Full Screen Close Quit
Introduction Results 8. Main Result Second Auxiliary Result • Proposition. A normalized fuzzy set is crisp if and Third Auxiliary Result only if we have one of the following two cases: Main Result Interval-Valued Case • A is a 1-element fuzzy set, or First Conclusion • for every subset B ⊆ A which is a non-crisp 2- Second Conclusion element set, ∃ a crisp 2-element set C for which Possible Future Work B ⊆ C ⊆ A. Home Page Title Page • Previous propositions show that the following proper- ties can be described in terms of subsethood: ◭◭ ◮◮ ◭ ◮ • of being a crisp 1-element set, • of being a crisp 2-element set, and Page 9 of 83 • of being a non-crisp 2-element set. Go Back • Thus, this Proposition shows that crispness can indeed Full Screen be described in terms of subsethood. Close Quit
Introduction Results 9. Interval-Valued Case Second Auxiliary Result • The traditional fuzzy logic assumes that: Third Auxiliary Result Main Result • experts can meaningfully describe their degrees of Interval-Valued Case certainty First Conclusion • by numbers from the interval [0 , 1]. Second Conclusion • In practice, however, experts cannot meaningfully se- Possible Future Work lect a single number describing their certainty. Home Page • Indeed, it is not possible to distinguish between, say, Title Page degrees 0.80 and 0.81. ◭◭ ◮◮ • A more adequate description of the expert’s uncer- ◭ ◮ tainty is: Page 10 of 83 • when we allow to characterize the uncertainty Go Back • by a whole range of possible numbers, i.e., by an Full Screen � � interval A ( x ) , A ( x ) . Close Quit
Introduction Results 10. Interval-Valued Case (cont-d) Second Auxiliary Result • This idea leads to interval-valued fuzzy numbers, i.e., Third Auxiliary Result mappings that assign, Main Result Interval-Valued Case • to each element x from the Universe of discourse, First Conclusion � � • an interval A ( x ) = A ( x ) , A ( x ) . Second Conclusion � � • For two interval-valued degrees A = A, A and B = Possible Future Work � � B, B , it is reasonable to say that A ≤ B if Home Page A ≤ B and A ≤ B. Title Page • Thus, we can define a subsethood relation between two ◭◭ ◮◮ interval-valued fuzzy sets A and B as ◭ ◮ A ( x ) ≤ B ( x ) for all x. Page 11 of 83 • An interval-valued fuzzy set is normalized if A ( x 0 ) = 1 Go Back for some x 0 . Full Screen • Traditional ( type-1 ) fuzzy sets can be viewed as partic- Close ular cases of interval-valued fuzzy sets. Quit
Introduction Results 11. Interval-Valued Case (cont-d) Second Auxiliary Result • Namely, they correspond to “degenerate” intervals Third Auxiliary Result Main Result [ A ( x ) , A ( x )] . Interval-Valued Case First Conclusion • Here, we have a similar problem: Second Conclusion • can we detect traditional fuzzy sets Possible Future Work • based only on the subsethood relation between interval- Home Page valued fuzzy sets? Title Page • Let us show that this is indeed possible. ◭◭ ◮◮ ◭ ◮ Page 12 of 83 Go Back Full Screen Close Quit
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