C C h a k r a b o r t y , w w w .m y r e a d e r s .i n f o R Fuzzy Set Theory : Soft Computing Cour Lecture 29 – 34, notes, slides
Fuzzy Set Theory
Soft Computing
Introduction to fuzzy t, topics : classical t theory, fuzzy t
theory, crisp and non-crisp Sets reprentation, capturing uncertainty, examples. Fuzzy membership and graphic interpretation
of fuzzy ts - small, prime numbers, universal, finite, infinite,
empty space; Fuzzy Operations -inclusion, comparability, equality,
complement, union, interction, difference; Fuzzy properties
related to union, interction, distributivity, law of excluded middle,
law of contradiction, and cartesian product. Fuzzy relations :
definition, examples, forming fuzzy relations, projections of fuzzy
中秋节的句子relations, max-min and min-max compositions.
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R Fuzzy Set Theory Soft Computing
Topics (Lectures 29, 30, 31, 32, 33, 34 6 hours)
Slides
1. Introduction to fuzzy Set
What is Fuzzy t? Classical t theory; Fuzzy t theory; Crisp and Non-crisp Sets : Reprentation; Capturing uncertainty, Examples
03-102. Fuzzy t Fuzzy Membership; Graphic interpretation of fuzzy ts : small, prime numbers, universal, finite, infinite, empty space;
淘宝网首页打不开Fuzzy Operations : Inclusion, Comparability, Equality, Complement, Union, Interction, Difference;
Fuzzy Properties : Related to union – Identity, Idempotence, Associativity, Commutativity ; Related to Interction – Absorption, Identity, Idempotence, Commutativity, Associativity; Additional properties - Distributivity, Law of excluded middle, Law of contradiction; Cartesian product .
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3. Fuzzy Relations Definition of Fuzzy Relation, examples;
手机内存卡怎么用Forming Fuzzy Relations – Membership matrix, Graphical form; Projections of Fuzzy Relations – first, cond and global; Max-Min and Min-Max compositions.
茭瓜33-41
4. References
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Fuzzy Set Theory
What is Fuzzy Set ?
• The word "fuzzy" means "vagueness ". Fuzziness occurs when the boundary of a piece of information is not clear-cut.
宽敞反义词• Fuzzy ts have been introduced by Lotfi A. Zadeh (1965) as an extension of the classical notion of t. •
Classical t theory allows the membership of the elements in the t in binary terms, a bivalent condition - an element either belongs or does not belong to the t. Fuzzy t theory permits the gradual asssment of the membership of elements in a t, described with the aid of a membership function valued in the real unit interval [0, 1]. •
Example: Words like young, tall, good , or high are fuzzy. − There is no single quantitative value which defines the term young. − For some people, age 25 is young, and for others, age 35 is young. − The concept young has no clean boundary. − Age 1 is definitely young and age 100 is definitely not young; − Age 35 has some possibility of being young and usually depends on the context in which it is being considered. 03
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SC - Fuzzy t theory - Introduction
1. Introduction In real world, there exists much fuzzy knowledge; Knowledge that is vague, impreci, uncertain, ambiguous, inexact , or probabilistic in nature. Human thinking and reasoning frequently involve fuzzy information, originating from inherently inexact human concepts. Humans, can give satisfactory answers, which are probably true. However, our systems are unable to answer many questions. The reason is, most systems are designed bad upon classical t theory and two-valued logic which is unable to cope with unreliable and incomplete information and give expert opinions. We want, our systems should also be able to cope with unreliable and incomplete information and give expert opinions. Fuzzy ts have been able provide solutions to many real world problems. Fuzzy Set theory is an extension of classical t theory where elements have degrees of membership. 04
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R SC - Fuzzy t theory - Introduction • Classical Set Theory A Set is any well defined collection of objects. An object in a t is
called an element or member of that t.
− Sets are defined by a simple statement describing whether a particular element having a certain property belongs to that particular t.
− Classical t theory enumerates all its elements using
A = { a 1 , a 2 , a 3 , a 4 , . . . . a n }
If the elements a i (i = 1, 2, 3, . . . n ) of a t A are subt of universal t X , then t A can be reprented for all elements x ∈ X by its characteristic function
1 if x ∈ X
µA (x) =
0 otherwi
− A t A is well described by a function called characteristic
function .
This function, defined on the universal space X , assumes :
a value of 1 for tho elements x that belong to t A , and a value of 0 for tho elements x that do not belong to t A . The notations ud to express the mathematically are
Α : Χ → [0, 1] A(x) = 1 , x is a member of A Eq.(1)
A(x) = 0 , x is not a member of A
Alternatively, the t A can be reprented for all elements x ∈ X by its characteristic function µA (x) defined as
1 if x ∈ X
µA (x) = Eq.(2)
0 otherwi
− Thus in classical t theory µA (x) has only the values 0('fal') and 1 ('true''). Such ts are called crisp ts.
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