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Fuzzy Set
This document introduces fuzzy sets. It defines a fuzzy set as a set where elements have gradual membership rather than binary membership. Fuzzy sets allow membership values between 0 and 1. Operations on fuzzy sets like union, intersection, and complement are defined. An example fuzzy set distinguishes young and adult ages on a scale rather than a binary classification. Fuzzy sets permit ambiguous or uncertain boundaries unlike classical sets.
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Fuzzy Set
- 1. Fuzzy Set BY :EHSAN HAMZEI - 810392121
- 2. Sets? Sets Fuzzy Classic 1 A ={(u, μA(u))|u ∈ X} A = {u|u ∈ X}
- 3. Introduction 2 Fuzzysets have been introduced by Zadeh 1965. Fuzziness occurs when the boundary of a piece of information is not clear-cut. Classical set theory allows membership in binary terms while, Fuzzy set permits gradual assessment of membership.
- 4. Fuzzy Set 3 Definition: A = {(u, μA(u))|u ∈ X} 0<= μA(u) <=1 Ex: {(1,0.1), (2,0.3), (5,1), (6,0.7)}
- 5. Fuzzy Set 4 Equality: μA(x) = μB(x), ∀x∈X Subset: (A ⊆ B) μA(x) ≤ μB(x), ∀x∈X
- 6. Fuzzy Set 5 Empty Fuzzy Set: μA(x) = 0, ∀x∈X kA: kA = {kμA(x), ∀x∈X} EX: k =0,5 , A={(a,0.5),(b,0.3),(c,0.2),(d,1)} kA = {(a,0.25),(b,0.15),(c,0.1),(d,0.5)}
- 7. Fuzzy Set Operations6 Complement: μ(-A) = 1- μ(A) Union: μ(AUB) = Max(μ(A), μ(B))
- 8. Fuzzy Set Operations7 Intersect: μ(A∩B) = Min(μ(A), μ(B)) A|B: A ∩ B’ = Min(μ(A), μ(-B))
- 9. Fuzzy Set (Example)8 Age ? (Young, Adult): X = {15, 25, 35, 45, 55} Young = {(15,0.9), (25,0.8), (35,0.5),(45,0.1),(55,0)} Adult = {(15,0), (25,0.5), (35,0.8), (45,1), (55,1)}
- 10. Fuzzy Set (Example)8 15 25 35 45 55 Young 0.9 0.8 0.5 0.1 0 Adult 0 0.5 0.8 1 1 C(Adult) 1 0.5 0.2 0 0 0 0.2 0.4 0.6 0.8 1 1.2 Age?? Young Adult C(Adult)
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