Fuzzy Relations and the Extension Principle

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Chapter 4

• Fuzzy Relations and the Extension Principle

2
Introduction

• In mathematics courses you talked about relations
  and functions.
• In fuzzy mathematics relations are more
  appropriate.
• Some words you will see include, projection,
  cylindric extension, and the extension principle.
• The extension principle describes how to extend a
  classical function to a fuzzy relation.

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Cartesian Products
Let U and V be two arbitrary classical (non
fuzzy, crisp) sets. The Cartesian product of U
and V, denoted by
Is the non fuzzy set of all ordered pairs (u,v)
such that
Example U1,2,apple,orange, V2,4,pear
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More Cartesian Products
Cartesian Products of more than two sets.
Consider 3 crisp sets, U1, U2, and U3. Let
The Cartesian product of the three sets is the
ordered triple
Examples
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Relations
A (non-fuzzy) relation among (non-fuzzy) sets U1,
Un is a subset of the Cartesian Product of the
sets. That is, if Q(U1,,U2) denotes the
relation then
A binary relation between (non-fuzzy) sets U and
V is a subset of the Cartesian product
Example
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More (Crisp) Relations
Relations are sets.
With set operations.
With membership functions.
Relational Matrices for binary relations.
For Previous Example

• In groups of two
• Construct an example of a binary
• relation.
• Include a relational matrix.

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From Crisp to Fuzzy Relations
Consider the 2 sets consisting of cities in
Georgia.
UAtlanta, Savannah, Statesboro
VMacon, Columbus
(Draw Map)
Define the concept of very far between these two
sets of cities. Use a fuzzy (why?) relational
matrix to define the relation that models the
concept very far. (Quantify relative farness in
the interval 0,1.)
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Fuzzy Relations
A fuzzy relation is a fuzzy set defined in the
Cartesian product of crisp sets.
More precisely The fuzzy relation Q in
Is defined as the fuzzy set
What property must the fuzzy membership function?
A binary fuzzy relation ??
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Fuzzy Relation Examples
Let U1U2
Almost equal
Let UV
Much larger than
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Some special relations Projections and
Cylindric Extensions
Remember that a fuzzy relation is a fuzzy
membership function defined on a crisp set.
Sometimes we need a fuzzy set that is smaller
than Q. Sometimes we need a fuzzy set that is
larger than Q.
Recall the projection of a vector onto one of
the coordinate axes. (Draw picture.)
Let Q be a fuzzy relation in the CP of U1 and U2.
The projection of Q onto U1 is
What is the projection of Q onto U2?
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Projections
Lets now consider a relation on 3 three Fuzzy
sets U1, U2, and U3 and look at various
projections.
Let Q be a relation on the CP of U1, U2, and U3.
It is defined by the membership function
It can be projected onto any of the three sets
U1, U2, U3.
It can be projected onto any of the pairwise CPs
The six projections are
Do you see the pattern?
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Projection Examples
Example Define the relation u2 is between u1 and
u3 as
Compute the 6 projections.
Project onto U3
Project onto CP of U1 and U2
Read definition of projection (Def. 4.2 pg. 50)
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Another Projection Example
Consider the relation defined by the relational
matrix
Projection of very far onto U
Projection of very far onto V
We will refer back to this example.
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Projections constrain a fuzzy relation (i.e. a
fuzzy set) to a subspace of the original domain.
Projections produce a smaller fuzzy relation by
limiting the number of independent variables.
Cylindric extensions produce a larger fuzzy
relation by Increasing the number of independent
variables.
Cylindric extensions extend the fuzzy relation to
a larger set, (I.e. to a superset) of the
original domain.
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Cylindric Extensions
Consider a fuzzy set defined on U1 with
membership function
Its cylindric extension to the CP of U1 and U2 is
Suppose you have a fuzzy set defined on the CP of
U1 and U2 with membership function
What is the cylindric extension to the CP of U1,
U2, and U3?
Read definition 4.3, pg. 52.
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Cylindric Extension Example
Consider the fuzzy sets defined as
U1?
U2?
Now find the cylindric extension of Q1 to the CP
of U1 and U2.
Now find the cylindric extension of Q2 to the CP
of U1 and U2.
Now compare with the original relational matrix.
Is this always true?
Which is bigger?
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Cartesian Product of Fuzzy Sets
New concept required to answer the question.
Let A1,An be fuzzy sets in U1Un respectively.
The Cartesian product of A1,,An denoted
is a fuzz relation in
whose membership function is defined as
where
denotes any t-norm operator.
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Answer to Question Lemma 4.1
If Q is a fuzzy relation in the CP of U1 to Un,
and Q1,Qn are its projections onto U1, , Un,
respectively, then
Where we use the min for the t-norm in the fuzzy
CP.
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Compositions of Fuzzy Relations
Picture of Compositions
Let P(u,V) and Q(v,W) be two crisp binary
relations that share a common set V. The
composition of P and Q, denoted by
Is defined as a relation in the CP of U and W
such that
If and only if there exists at least one y in V
such that (x,y) is in P and (y,z) is in Q.
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Fuzzy Compositions in terms of membership
functions(Lemma 4.2)
t(a,b)min(a,b)
t(a,b)ab
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Examples
Continue the cities-of-the- world example.
U and V are defined as before
Add the new fuzzy set WNYC, Beijing with
relational matrix
Compute the max-min and max-product compositions.
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Extension Principle
How to convert a crisp function into a fuzzy
relation.
Given y f(x), x in U, y in V.
A is a fuzzy set in U with its membership function
The membership function of the fuzzy set in V
that is the image of all xs in U is
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Homework/Quiz
Homework 1) Exercise 4.2 2) Exercise 4.3 3)
Exercise 4.4 4) Exercise 4.5
Quiz Either Exercise 4.3 or Exercise 4.4