Fuzzy Sets and Basic Operations on Fuzzy Sets

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Fuzzy Sets and Basic Operations on Fuzzy Sets

• Ch. 2

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Example Cars in Berkeley

• U all the cars in Berkeley, CA
• some sets
• X cylinder cars (X 4, 6, 8, ??, etc.)
• foreign cars
• American cars
• Write in set notation
• Write in discrimination function notation.

• Questions
• What car manufacturers are American?
• how many US-made parts are required for a car to
  be classified as Made-in-America?
• Are there clear set boundaries?

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Classical vs. Fuzzy Sets

• Classical (or crisp) set theory is used when the
  set boundaries are clear and easy to distinguish.
  (black and white)
• Fuzzy set theory is used when the set boundaries
  are not clear and easy to distinguish. (shades of
  gray)

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Classical Sets Fuzzy Sets

• U is the
• universal set or
• universe of discourse
• Recall that a classical set A or a (crisp set) in
  the universe of discourse can be defined by
• listing all of its members
• specifying the properties that members must have

• Membership, Characteristic (or discrimination)
  function

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Fuzzy Sets Representation

• A fuzzy set in a universe of discourse U is
  characterized by a membership function that takes
  values in the interval 0, 1.
• Representations

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Cars in Berkeley (cont.)

• D is the set of domestic (I.e. American) cars in
  Berkeley.
• How do you quantify the domesticity of a car?
• p(x)

• F is the set of foreign cars in Berkeley. F is
  the set of non domestic cars in Berkeley.

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Numbers close to ZERO
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Numbers Close to ZERO
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Comments

• Fuzzy sets characterize fuzzy properties
• Close to zero, hot, cold, fast, slow
• How to determine membership functions?
• Use human experts
• Use sensor data
• Are humans sensors?
• The Gaussian and Triangular membership functions
  that described close to zero defined different
  fuzzy sets even though they tried to describe the
  same concept.

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Basic Concepts of Fuzzy Sets

• Consider a fuzzy set A in the universe of
  discourse U
• The support is the crisp set that contains all of
  the elements in U that have non zero membership
  values

• If the support of a fuzzy set is empty, it is
  called the empty fuzzy set.
• A fuzzy singleton is a fuzzy set whose support is
  a single point in U.

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Activity Define hot warm comfortable
cool cold
Hot Warm Comfortable Cool cold
High
Typical
Low
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Basic Concepts of Fuzzy Sets

• The following fuzzy sets help motivate the
  concept of the center of a fuzzy set.

• For A2 the center is clear it is where the
  function peaks
• For A3 it is the center of the plateau.
• For A1, and A4 it is where the function becomes a
  max that is closest to the origin.

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Basic Concepts of Fuzzy Sets

• The crossover point(s) of a fuzzy set is(are) the
  point(s) in U whose membership value equals .5.
• The height of a fuzzy set is the largest
  membership value attained by any point.
• A normal fuzzy set has height of 1.
• An alpha-cut of a fuzzy set is a crisp set that
  contains all the elements in U that have
  membership values greater than alpha

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Multi-Dimensional Fuzzy Sets

• IF Temperature is HOT and Humidity is LOW THEN
  set cooling-level ______ and fan-speed ________

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Operations on Fuzzy Sets

• Assume that A and B are fuzzy sets in the same
  Universe of Discourse, U.

• Equality
• containment
• complement
• union
• intersection

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Example

• Given fuzzy sets Close-to-zero and Close-to-one,
  determine the union and intersection.

• Is one contained in the other?
• Draw a fuzzy set contained in another fuzzy set.

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Example

• Given fuzzy sets, close-to-0 and determine its
  complement

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Crisp vs. Fuzzy Set Theory

• The basic definitions and operations of set
  theory determine what Theorems are true.
• Question Are any of the Theorems of Crisp set
  Theory true in Fuzzy Set Theory?
• Ans Yes
• Question Are all of the Theorems of Crisp Theory
  true in Fuzzy Set Theory?
• Ans No
• How do we Tell?
• Try to Prove them.
• Is this good?
• Yes. Different Theorems allow us to solve
  different problems.

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DeMorgans Laws

• Very useful theorems in Crisp set Theory and
  Logic. (Remember from Boolean Algebra?)
• Do they hold in Fuzzy Set Theory?
• Translate into Fuzzy Set Theory
• Express in terms of membership functions

• DeMorgans Law is valid provided the following
  equality is valid

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Proof of DeMorgans LAW

• We break this up into 2 simpler cases that
  describe all possible situations
• The left is easy once we know which membership
  function is largest

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Homework/Quiz

• Read
• Exercises 2.1, 2.2, 2.3, 2.5, 2.6, 2.7
• Quiz One of the parts in exercise 2.3

• Comments
• Read about alpha-cuts, projections (and
  hyperplanes)
• Failure of Law of Excluded Middle in Fuzzy Set
  Theory (Exercise 2.6) is a major distinction
  between Crisp and Fuzzy Set Theory
• These differences allow the theories to solve
  different problems