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.

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
 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.

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 Operations 6
 Complement:
 μ(-A) = 1- μ(A)
 Union:
 μ(AUB) = Max(μ(A), μ(B))

8. Fuzzy Set Operations 7
 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)

11. Thanks 26