Fuzzy SetBY : EHSAN HAMZEI - 810392121

Sets?

Sets

Fuzzy Classic

1

A = {(u, μA(u))|u ∈ X} A = {u|u ∈ X}

Introduction 2

Fuzzy sets 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.

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)}

Fuzzy Set 4

Equality: μA(x) = μB(x), ∀x∈X

Subset: (A ⊆ B) μA(x) ≤ μB(x), ∀x∈X

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)}

Fuzzy Set Operations 6

Complement: μ(-A) = 1- μ(A)

Union: μ(AUB) = Max(μ(A), μ(B))

Fuzzy Set Operations 7

Intersect: μ(A∩B) = Min(μ(A), μ(B))

A|B: A ∩ B’ = Min(μ(A), μ(-B))

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)}

Fuzzy Set (Example) 8

15 25 35 45 550

0.2

0.4

0.6

0.8

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1.2 Age??

Young Adult C(Adult)

Thanks 26