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Fuzzy Sets and Basic Operations on Fuzzy Sets

- Ch. 2

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?

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)

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

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

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.

(No Transcript)

Numbers close to ZERO

Numbers Close to ZERO

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.

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.

Activity Define hot warm comfortable

cool cold

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.

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

Multi-Dimensional Fuzzy Sets

- IF Temperature is HOT and Humidity is LOW THEN

set cooling-level ______ and fan-speed ________

Operations on Fuzzy Sets

- Assume that A and B are fuzzy sets in the same

Universe of Discourse, U.

- Equality
- containment
- complement
- union
- intersection

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.

Example

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

complement

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.

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

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

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

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