1 07 December 2009 Cao Hoang Tru CSE Faculty - HCMUT Vagueness • The Oxford Companion to Philosophy (1995): “Words like smart, tall, and fat are vague since in most contexts of use there is no bright line separating them from not smart, not tall, and not fat respectively …”
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Vagueness - University of Technologycse.hcmut.edu.vn/~tru/KB-SYSTEMS/fuzzy-theory-new.pdf · 1 07 December 2009 Cao Hoang Tru CSE Faculty - HCMUT Vagueness • The Oxford Companion
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1
07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Vagueness
• The Oxford Companion to Philosophy (1995):“Words like smart, tall, and fat are vague since in most contexts of use there is no bright line separating them from not smart, not tall, and not fat respectively …”
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Vagueness
• Imprecision vs. Uncertainty:The bottle is about half-full.vs.It is likely to a degree of 0.5 that the bottle is full.
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Sets
• Zadeh, L.A. (1965). Fuzzy SetsJournal of Information and Control
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Sets
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Set Definition
A fuzzy set is defined by a membership function that maps elements of a given domain (a crisp set) into values in [0, 1].
• Continuous domain::::A(u) = 1 for u∈[0, 20]A(u) = (40 - u)/20 for u∈[20, 40]A(u) = 0 for u∈[40, 120]
0
1
20 40 Age30
0.5
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Set Representation
• α-cuts::::Aα = u | A(u) ≥ αAα+ = u | A(u) > α strong α-cut
A0.5 = [0, 30]
0
1
20 40 Age30
0.5
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Set Representation
• α-cuts::::Aα = u | A(u) ≥≥≥≥ αAα+ = u | A(u) >>>> α strong α-cut
A(u) = sup α | u ∈∈∈∈ Aα
A0.5 = [0, 30]
0
1
20 40 Age30
0.5
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Set Representation
• Support: : : : supp(A) = u | A(u) > 0 = A0+
• Core: : : : core(A) = u | A(u) = 1 = A1
• Height::::h(A) = supUA(u)
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Set Representation
• Normal fuzzy set:::: h(A) = 1
• Sub-normal fuzzy set:::: h(A) <<<< 1
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Membership Degrees
• Subjective definition
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Membership Degrees
• Voting model::::Each voter has a subset of U as his/her own crisp definition of the concept that A represents.A(u) is the proportion of voters whose crisp definitions include u. A defines a probability distribution on the power set of U across the voters.