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Fuzzy OpERAtIONS

Uncertainty and Imprecision
Nonrandom uncertainty Imprecision ไม่ถูกต้องแม่นยำ Vagueness ความคลุมเครือ Fuzziness ความไม่เป็นระเบียบ Ambiguity ความกำกวม

A fuzzy set is totally characterized by a membership function (MF).
Fuzzy Sets Formal definition: A fuzzy set A in X is expressed as a set of ordered pairs: Membership function (MF) Universe or universe of discourse Fuzzy set A fuzzy set is totally characterized by a membership function (MF). 04/04/60

ตัวอย่าง

ตัวอย่าง เซตของจำนวนเต็มที่มีค่าใกล้ 1

ตัวอย่าง เซตของจำนวนจริงที่มีค่าใกล้ 1

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"Jenny is very young"

สามเหลี่ยม a b c x

สี่เหลี่ยมคางหมู a b c d

S-function

Membership Functions (MFs)
Characteristics of MFs: Subjective measures Not probability functions “tall” in Asia MFs “tall” in NBA Here I like to emphasize some important properties of membership functions. First of all, it subjective measure; my membership function of all?is likely to be different from yours. Also it context sensitive. For example, I 5?1? and I considered pretty tall in Taiwan. But in the States, I only considered medium build, so may be only tall to the degree of .5. But if I an NBA player, Il be considered pretty short, cannot even do a slam dunk! So as you can see here, we have three different MFs for all?in different contexts. Although they are different, they do share some common characteristics --- for one thing, they are all monotonically increasing from 0 to 1. Because the membership function represents a subjective measure, it not probability function at all. .8 “tall” in the US .5 .1 5’10’’ Heights 04/04/60

Gaussian and Pi Functions

Definitions

If the support is finite, it is called compact support.
If the support of fuzzy set A consists of only one point, it is called a fuzzy singleton. If the membership grade of this fuzzy singleton is one, A is called a crisp singleton [Zimmermann, 1985].

Definition (Normal Fuzzy Set) A fuzzy set A is normal if ∃x ∈ X such that μA(x) = 1.
Fuzzy sets that are not normal are called subnormal. The operator norm(A) denotes normalization of a fuzzy set, i.e., A_ = norm(A) ⇔ μ A_ (x) = μ A(x) / hgt(A), ∀x.

สรุป

MF Terminology MF 1 .5 a Core X Crossover points a - cut Support

Basic Operations on Fuzzy Sets

Equivalent fuzzy sets

Fuzzy Subsets

Intersection, Union and Complementary
Definition: The membership function C (x) of the set C = A  B is defined as C (x) = min {A (x), B (x)}, x  X Definition: The membership function C(x) of C = A  B is defined as C(x) = max{A(x) , B(x)}, x  X Definition: Membership function of the complement of a fuzzy set A, A’(x) is defined as A’(x) = [1 - A(x) ], x  X

Set-Theoretic Operations
Subset: Complement: Union: Intersection:

Various fuzzy set operations
Example: Let X = { 1,2,3,4,5,6,7} A = { (3, 0.7), (5, 1), (6, 0.8) } B = {(3, 0.9), (4, 1), (6, 0.6) } A  B = { (3, 0.7), (6, 0.6) } A  B = { (3, 0.9), (4, 1), (5, 1), (6, 0.8) } A’ = { (1, 1), (2, 1), (3, 0.3), (4, 1), (6, 0.2), (7, 1) }

Operators for Fuzzy Sets
Intersection (similar to logical AND) T-norm (*) Union (similar to logical OR) S-norm or T-conorm (Å) Complement (similar to logical NOT) © 2000, 2002, 2003 Raymond P. Jefferis III 4/4/2017

Fuzzy Intersection The fuzzy intersection contains all elements that are in both A and B Mathematically, © 2000, 2002, 2003 Raymond P. Jefferis III 4/4/2017

Fuzzy Union The fuzzy intersection contains all elements that are in either A and B Mathematically, © 2000, 2002, 2003 Raymond P. Jefferis III 4/4/2017

Fuzzy Complement The fuzzy complement of A contains all elements that are not in A Mathematically, © 2000, 2002, 2003 Raymond P. Jefferis III 4/4/2017

Properties of fuzzy Sets
© 2000, 2002, 2003 Raymond P. Jefferis III 4/4/2017

Fuzzy DeMorgan Laws © 2000, 2002, 2003 Raymond P. Jefferis III
4/4/2017

Excluded middle axioms for fuzzy sets.
As enumerated before, all other operations on classical sets also hold for fuzzy sets, except for the excluded middle axioms. These two axioms do not hold for fuzzy sets since they do not form part of the basic axiomatic structure of fuzzy sets; since fuzzy sets can overlap, a set and its complement can also overlap.

The excluded middle axioms, extended for fuzzy sets, are expressed by

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