Formulation of the . . . Formulation of the . . . Let Us First Consider . . . How to Detect the . . . Can We Detect Crisp Sets Based Only on How to Detect 1- . . . the Subsethood Ordering of Fuzzy Sets? How to Detect 1- . . . Fuzzy Sets And/Or Crisp Sets Based on What About Interval- . . . Subsethood of Interval-Valued Fuzzy Interval-Valued Fuzzy . . . Sets? Open Question Home Page Christian Servin 1 , Gerardo Muela 2 , and Vladik Kreinovich 2 Title Page 1 Computer Science and Information Technology Systems Department ◭◭ ◮◮ El Paso Community College, 919 Hunter, El Paso, TX 79915, USA cservin@gmail.com ◭ ◮ Page 1 of 12 2 Department of Computer Science, University of Texas at El Paso El Paso, TX 79968, USA. gdmuela@miners.utep.edu, vladik@utep.edu Go Back Full Screen Close Quit
Formulation of the . . . Formulation of the . . . 1. Formulation of the Problem Let Us First Consider . . . • A fuzzy set is a function µ : U → [0 , 1] from some set How to Detect the . . . U ( Universe of discourse ) to the interval [0 , 1]. How to Detect 1- . . . How to Detect 1- . . . • This function is also known as a membership function . What About Interval- . . . • A fuzzy set A with a memb. f-n µ A ( x ) is a subset of a Interval-Valued Fuzzy . . . fuzzy set B if µ A ( x ) ≤ µ B ( x ) for all x . Open Question • ⊆ is an order in the sense that it is: Home Page Title Page – reflexive ( A ⊆ A ), ◭◭ ◮◮ – asymmetric ( A ⊆ B and B ⊆ A imply A = B ), – transitive ( A ⊆ B and B ⊆ C imply A ⊆ C ). ◭ ◮ Page 2 of 12 • Traditional ( crisp ) sets S are particular cases of fuzzy sets: µ S ( x ) = 1 if x ∈ S and µ S ( x ) = 0 if x �∈ S . Go Back • A natural question: can we detect crisp sets based only Full Screen on the subsethood ordering of fuzzy sets? Close Quit
Formulation of the . . . Formulation of the . . . 2. Formulation of the Problem in Precise Terms Let Us First Consider . . . • Suppose now that we have a class F of all fuzzy sets How to Detect the . . . with the subsethood ordering A ⊆ B . How to Detect 1- . . . How to Detect 1- . . . • However, we have no access to the actual values of the What About Interval- . . . corresponding membership functions. Interval-Valued Fuzzy . . . • Based only on this ordering relation A ⊆ B , can we Open Question then detect crisp sets? Home Page • In this talk, we show that this is indeed possible. Title Page • We also show that based on ⊆ for interval-valued fuzzy ◭◭ ◮◮ sets, we can: ◭ ◮ – detect crisp sets, and Page 3 of 12 – type-1 fuzzy sets. Go Back Full Screen Close Quit
Formulation of the . . . Formulation of the . . . 3. Let Us First Consider [0 , 1] -Based Fuzzy Sets Let Us First Consider . . . • First, we will prove that the empty set ∅ can be How to Detect the . . . uniquely determined based on the subsethood relation. How to Detect 1- . . . How to Detect 1- . . . • Second, we will show that 1-element crisp sets, i.e., sets What About Interval- . . . of the type { x 0 } , can be thus determined. Interval-Valued Fuzzy . . . • Then, we consider 1-element fuzzy sets, for which for Open Question some x 0 ∈ U , µ A ( x 0 ) > 0 and µ A ( x ) = 0 for all x � = x 0 . Home Page • Then, we will prove that 1-element fuzzy sets can be Title Page determined based on the subsethood relation. ◭◭ ◮◮ • Finally, we prove that crisp sets can be uniquely deter- ◭ ◮ mined based on the subsethood relation. Page 4 of 12 Go Back Full Screen Close Quit
Formulation of the . . . Formulation of the . . . 4. How to Detect the Empty Set and 1-Element Let Us First Consider . . . Crisp Sets How to Detect the . . . • An empty set ∅ is a fuzzy set for which µ ∅ ( x ) = 0 for How to Detect 1- . . . all x ∈ U . How to Detect 1- . . . What About Interval- . . . • A fuzzy set A is an empty set if and only if A ⊆ B for Interval-Valued Fuzzy . . . all fuzzy sets B . Open Question • A non-empty fuzzy set A is a one-element crisp set if Home Page and only if the following two conditions are satisfied: Title Page – the class { B : B ⊆ A } is linearly ordered and ◭◭ ◮◮ – for no proper superset A ′ of A , the class ◭ ◮ { B : B ⊆ A ′ } is linearly ordered. Page 5 of 12 • Indeed, for A = { x 0 } , all subsets B are fuzzy 1-element sets, so { B : B ⊆ A ′ } is linearly ordered. Go Back Full Screen Close Quit
Formulation of the . . . Formulation of the . . . 5. How to Detect 1-Element Crisp Sets (cont-d) Let Us First Consider . . . • If µ A ( x 1 ) > 0 and µ A ( x 2 ) > 0, then we can take: How to Detect the . . . How to Detect 1- . . . – µ B 1 ( x 1 ) = µ A ( x 1 ) and µ B 1 ( x ) = 0 for all other x , How to Detect 1- . . . and What About Interval- . . . – µ B 2 ( x 2 ) = µ A ( x 2 ) and µ B 2 ( x ) = 0 for all other x . Interval-Valued Fuzzy . . . • Then, B 1 �⊆ B 2 and B 2 �⊆ B 1 . Open Question Home Page • So, µ A ( x ) < 0 for only one x . Title Page • If µ A ( x ) < 1, then A ⊂ A ′ = { x } and { B : B ⊆ A ′ } is ◭◭ ◮◮ linearly ordered. ◭ ◮ • Thus, µ A ( x ) = 1, so A is a 1-element crisp set. Page 6 of 12 • So, we can indeed detect 1-element crisp sets based on the subsethood relation ⊆ . Go Back Full Screen Close Quit
Formulation of the . . . Formulation of the . . . 6. How to Detect 1-Element Fuzzy Sets and Crisp Let Us First Consider . . . Sets How to Detect the . . . • A 1-element fuzzy set A can be detected as a non- How to Detect 1- . . . empty subset of a 1-element crisp set. How to Detect 1- . . . What About Interval- . . . • Crisp sets can be detected as follows: Interval-Valued Fuzzy . . . A is crisp ⇔ ∀ A ( B is a one-element fuzzy subset of A ⇒ Open Question Home Page ∃ C (( B ⊆ C ⊆ A ) & ( C is a 1-element crisp set))) . Title Page ⇒ This is clearly true for crisp sets. ◭◭ ◮◮ ⇐ Vice versa, if 0 < µ A ( x 0 ) < 1 for some x 0 , then: ◭ ◮ def – the set B = A ∩ { x 0 } is a 1-element fuzzy subset Page 7 of 12 of A , and Go Back – the set B cannot be embedded in a 1-element crisp Full Screen subset of A . Close Quit
Formulation of the . . . Formulation of the . . . 7. What About Interval-Valued Fuzzy Sets? Let Us First Consider . . . • An empty set ∅ is an interval-valued fuzzy set for which How to Detect the . . . µ ∅ ( x ) = [0 , 0] for all x ∈ U . How to Detect 1- . . . How to Detect 1- . . . • An interval-valued fuzzy set A is an empty set if and What About Interval- . . . only if A ⊆ B for all interval-valued fuzzy sets B . Interval-Valued Fuzzy . . . • We say that A is special if for some x 0 and a > 0, Open Question µ A ( x 0 ) = [0 , a ] and µ A ( x ) = 0 for all x � = x 0 . Home Page • A non-empty interval-valued fuzzy set A is special if Title Page and only if the class { B : B ⊆ A } is linearly ordered. ◭◭ ◮◮ • Indeed, if µ A ( x 1 ) � = [0 , 0] and µ A ( x 2 ) � = [0 , 1], then for ◭ ◮ B 1 = A ∩ { x 1 } and B 2 = A ∩ { x 2 } , we have Page 8 of 12 B 1 �⊂ B 2 and B 2 �⊂ B 1 . Go Back • If µ A ( x ) = [ a, a ] with a > 0, then for B 1 = [0 , a ] and Full Screen B 2 = [0 . 5 · a, 0 . 5 · a ], we have B 1 �⊂ B 2 and B 2 �⊂ B 1 . Close Quit
Formulation of the . . . Formulation of the . . . 8. Interval-Valued Fuzzy Sets (cont-d) Let Us First Consider . . . • We say that A is a 1-element type-1 fuzzy set if for How to Detect the . . . some x 0 and a > 0: How to Detect 1- . . . How to Detect 1- . . . µ A ( x 0 ) = [ a, a ] and µ A ( x ) = [0 , 0] for all x � = x 0 . What About Interval- . . . • A non-empty A is a 1-element type-1 set if and only if Interval-Valued Fuzzy . . . it is satisfies the following three properties: Open Question Home Page – the set A is not special (in the sense of the above Title Page definition), ◭◭ ◮◮ – there exists a special set B ⊆ A for which the class { C : B ⊆ C ⊆ A } is linearly ordered, and ◭ ◮ – for no proper superset A ′ of A , the class Page 9 of 12 { C : B ⊆ C ⊆ A ′ } is linearly ordered. Go Back Full Screen Close Quit
Formulation of the . . . Formulation of the . . . 9. Interval-Valued Fuzzy Sets: Final Result Let Us First Consider . . . • Since we have subsethood, we also have union: the How to Detect the . . . union of A β is the ⊆ -smallest set that contains all A β . How to Detect 1- . . . How to Detect 1- . . . • We can thus define type-1 fuzzy sets as unions of What About Interval- . . . 1-element type-1 fuzzy sets. Interval-Valued Fuzzy . . . • Once we can detect type-1 fuzzy sets, we can use tech- Open Question niques from the previous sections to detect crisp sets. Home Page • Thus: Title Page – we can indeed detect type-1 fuzzy sets and crisp sets ◭◭ ◮◮ – based only on subsethood relation between interval- ◭ ◮ valued fuzzy sets. Page 10 of 12 Go Back Full Screen Close Quit
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