Title: Introduction to Fuzzy Set Theory
1Introduction to Fuzzy Set Theory
2Content
- Fuzzy Sets
- Set-Theoretic Operations
- MF Formulation
- Extension Principle
- Fuzzy Relations
- Approximate Reasoning
3Introduction to Fuzzy Set Theory
4Types of Uncertainty
- Stochastic uncertainty
- E.g., rolling a dice
- Linguistic uncertainty
- E.g., low price, tall people, young age
- Informational uncertainty
- E.g., credit worthiness, honesty
5Crisp or Fuzzy Logic
- Crisp Logic
- A proposition can be true or false only.
- Bob is a student (true)
- Smoking is healthy (false)
- The degree of truth is 0 or 1.
- Fuzzy Logic
- The degree of truth is between 0 and 1.
- William is young (0.3 truth)
- Ariel is smart (0.9 truth)
6Crisp Sets
- Classical sets are called crisp sets
- either an element belongs to a set or not, i.e.,
- Member Function of crisp set
or
7Crisp Sets
P the set of all people.
Y
Y the set of all young people.
8Fuzzy Sets
Crisp sets
Example
9Fuzzy Sets
Lotfi A. Zadeh, The founder of fuzzy logic.
L. A. Zadeh, Fuzzy sets, Information and
Control, vol. 8, pp. 338-353, 1965.
10DefinitionFuzzy Sets and Membership Functions
U universe of discourse.
If U is a collection of objects denoted
generically by x, then a fuzzy set A in U is
defined as a set of ordered pairs
membership function
11Example (Discrete Universe)
12Example (Discrete Universe)
Alternative Representation
13Example (Continuous Universe)
U the set of positive real numbers
about 50 years old
Alternative Representation
14Alternative Notation
U discrete universe
U continuous universe
Note that ? and integral signs stand for the
union of membership grades / stands for a
marker and does not imply division.
15Membership Functions (MFs)
- A fuzzy set is completely characterized by a
membership function. - a subjective measure.
- not a probability measure.
16Fuzzy Partition
- Fuzzy partitions formed by the linguistic values
young, middle aged, and old
17MF Terminology
18More Terminologies
- Normality
- core non-empty
- Fuzzy singleton
- support one single point
- Fuzzy numbers
- fuzzy set on real line R that satisfies convexity
and normality - Symmetricity
- Open left or right, closed
19Convexity of Fuzzy Sets
- A fuzzy set A is convex if for any ? in 0, 1.
20Introduction to Fuzzy Set Theory
21Set-Theoretic Operations
- Subset
- Complement
- Union
- Intersection
22Set-Theoretic Operations
23Properties
Involution
De Morgans laws
Commutativity
Associativity
Distributivity
Idempotence
Absorption
24Properties
- The following properties are invalid for fuzzy
sets - The laws of contradiction
- The laws of exclude middle
25Other Definitions for Set Operations
26Other Definitions for Set Operations
27Generalized Union/Intersection
- Generalized Union
- Generalized Intersection
t-norm
t-conorm
28T-Norm
Or called triangular norm.
- Symmetry
- Associativity
- Monotonicity
- Border Condition
29T-Conorm
Or called s-norm.
- Symmetry
- Associativity
- Monotonicity
- Border Condition
30Examples T-Norm T-Conorm
- Minimum/Maximum
- Lukasiewicz
- Probabilistic
31Introduction to Fuzzy Set Theory
32MF Formulation
- Triangular MF
- Trapezoidal MF
- Gaussian MF
- Generalized bell MF
33MF Formulation
34Manipulating Parameter of theGeneralized Bell
Function
35Sigmoid MF
Extensions
Abs. difference of two sig. MF
Product of two sig. MF
36L-R MF
Example
c65 ?60 ?10
c25 ?10 ?40
37Introduction to Fuzzy Set Theory
38Functions Applied to Crisp Sets
B
A
39Functions Applied to Fuzzy Sets
y
B
?B(y)
A
?A(x)
x
40Functions Applied to Fuzzy Sets
y
B
?B(y)
A
?A(x)
x
41The Extension Principle
Assume a fuzzy set A and a function f.
How does the fuzzy set f(A) look like?
y
B
?B(y)
A
?A(x)
x
42The Extension Principle
Assume a fuzzy set A and a function f.
How does the fuzzy set f(A) look like?
y
B
?B(y)
A
?A(x)
x
43The Extension Principle
The extension of f operating on A1, , An gives a
fuzzy set F with membership function
44Introduction to Fuzzy Set Theory
45Binary Relation (R)
46Binary Relation (R)
47The Real-Life Relation
- x is close to y
- x and y are numbers
- x depends on y
- x and y are events
- x and y look alike
- x and y are persons or objects
- If x is large, then y is small
- x is an observed reading and y is a corresponding
action
48Fuzzy Relations
A fuzzy relation R is a 2D MF
49Example (Approximate Equal)
50Max-Min Composition
R fuzzy relation defined on X and Y.
S fuzzy relation defined on Y and Z.
R?S the composition of R and S.
A fuzzy relation defined on X an Z.
51Example
max
52Max-Product Composition
Max-min composition is not mathematically
tractable, therefore other compositions such as
max-product composition have been suggested.
R fuzzy relation defined on X and Y.
S fuzzy relation defined on Y and Z.
R?S the composition of R and S.
A fuzzy relation defined on X an Z.
53Projection
Dimension Reduction
R
54Projection
Dimension Reduction
55Cylindrical Extension
Dimension Expansion
A a fuzzy set in X.
C(A) A?X?Y cylindrical extension of A.
56Introduction to Fuzzy Set Theory
57Linguistic Variables
- Linguistic variable is a variable whose values
are words or sentences in a natural or artificial
language. - Each linguistic variable may be assigned one or
more linguistic values, which are in turn
connected to a numeric value through the
mechanism of membership functions.
58Motivation
- Conventional techniques for system analysis are
intrinsically unsuited for dealing with systems
based on human judgment, perception emotion.
59Example
if temperature is cold and oil is cheapthen
heating is high
60Example
Linguistic Variable
?cold
if temperature is cold and oil is cheapthen
heating is high
Linguistic Value
Linguistic Value
Linguistic Variable
?cheap
?high
Linguistic Variable
Linguistic Value
61Definition Zadeh 1973
A linguistic variable is characterized by a
quintuple
62Example
A linguistic variable is characterized by a
quintuple
Example semantic rule
63Example
Linguistic Variable temperature Linguistics
Terms (Fuzzy Sets) cold, warm, hot
64Fuzzy If-Than Rules
?
A ? B
If x is A then y is B.
antecedent or premise
consequence or conclusion
65Examples
A ? B
?
If x is A then y is B.
- If pressure is high, then volume is small.
- If the road is slippery, then driving is
dangerous. - If a tomato is red, then it is ripe.
- If the speed is high, then apply the brake a
little.
66Fuzzy Rules as Relations
A ? B
?
R
If x is A then y is B.
Depends on how to interpret A ? B
A fuzzy rule can be defined as a binary relation
with MF
67Interpretations of A ? B