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Fuzzy set theory. Abby yinger. Definitions. Set – any well defined collection of objects. An object in a set is called an element or member of that set . Crisp Sets – these are sets that only have values of 0 (‘False’) and 1 (‘True’).

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fuzzy set theory

Fuzzy set theory

Abby yinger

definitions
Definitions
  • Set – any well defined collection of objects. An object in a set is called an element or member of that set.
  • Crisp Sets – these are sets that only have values of 0 (‘False’) and 1 (‘True’).
  • Classical set Theory – either an element belongs to the set or it does not. For example, for the set of integers, either an integer is even or it is not (it is odd). So a classical set allows for the membership of the elements in the set, is enumerates all its elements using; A = { , , , . . . . . . }.
    • Fuzzy set Theory is an extension of classical set theory where elements have varying degrees of membership.
what is a fuzzy set
What is a fuzzy set?
  • Definition: A fuzzy set is any set that allows its members to have different grades of membership (membership function) in the interval [0,1].
  • A fuzzy set A is written as a set of pairs {x, A(x)} as A = {{x, A(x)}}, x in the set X, where x is an element of the universal space X, and A(x) is the value of the function A for this element.
  • The proposition of Fuzzy Sets are motivated by the need to capture and represent real world data with uncertainty due to imprecise measurements or by vagueness in the language.
what is the problem
What is the problem?
  • The problem that I have chosen is one that is solved by fuzzy sets. This is simply the problem that before we had Fuzzy sets there was no way to show uncertainty.
  • In 1965 LotfiZadeh introduced the idea of fuzzy sets. He decided that we needed a way to capture the idea of uncertainty in sets instead of ignoring them. The word “Fuzzy” means “Vagueness”. Fuzzyness occurs when the boundary of a piece of information is not clear-cut.
  • Example:
  • Words like young, tall, bad, or low are fuzzy:
  • This is because there is no single quantitative value which defines the term tall.
  • For some people, 7 feet is tall, and for others, 8 feet is tall.
  • This means that the concept of tall has no clear boundary.
  • The Fuzzy Set Theory is a set that is characterized by its membership function. It’s range is contained in the unit interval.
examples of fuzzy sets
Examples of fuzzy sets

Height considered Tall

Height considered Short

Average Height

fuzzy set operations
Fuzzy set operations
  • Fuzzy set operations include:
    • Addition and subtraction
    • Unions and Intersections (in provided picture)
    • Multiplication and Division

Set B

Set A

The intersection of A and B

sources
Sources
  • http://www.myreaders.info/html/soft_computing.html
  • http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol4/sbaa/report.fuzzysets.html