Fuzzy Sets and
Applications
Introduction
Fuzzy Sets and Operations
Why fuzzy sets?
Types of Uncertainty
1. Randomness : Probability
Knowledge about the relative frequency of each event in some
domain
Lack of knowledge which event will be in next time
2. Incompleteness : Imputation by EM
Lack of knowledge or insufficient data
3. Ambiguity : Dempster-Shafer’s Belief Theory
=> Evidential Reasoning
Uncertainty due to the lack of evidence
ex) “The criminal is left-handed or not”
Why fuzzy sets?
Types of Uncertainty (continued)
4. Imprecision :
Ambiguity due to the lack of accuracy of observed data
ex) Character Recognition
5. Fuzziness (vagueness) : Uncertainty due to the vagueness
of boundary
ex) Beautiful woman, Tall man
Why fuzzy sets?
Powerful tool for vagueness
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Description of vague linguistic terms and
algorithms
Operation on vague linguistic terms
Reasoning with vague linguistic rules
Representation of clusters with vague
boundaries
History of Fuzzy Sets
History of Fuzzy Sets and Applications
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1965 Zadeh
Fuzzy Sets
1972 Sugeno
Fuzzy Integrals
1975 Zadeh
Fuzzy Algorithm & Approximate Reasoning
1974 Mamdani
Fuzzy Control
1978 North Holland Fuzzy Sets and Systems
1982 Bezdek
Fuzzy C-Mean
1987 Korea
Fuzzy Temperature Control
Current Scope of Fuzzy Society
Foundation
Methods
Applications
Fuzzy Sets
Fuzzy Relation
Fuzzy Numbers
Extension Principle
Fuzzy Optimization
Clustering
Statistics
Pattern Recognition
Data Processing
Fuzzy Measure
Fuzzy Integrals
Fuzzy Measure
Decision Making
Evaluation
Estimation
Expert Systems
Fuzzy Logic
Linguistic Variable
Fuzzy Algorithm
Approximate Reasoning
Fuzzy Computer
Fuzzy Control
Applications
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지식
자동기능
모델
분석
평가
시스템
진단/결정
인식추론
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로봇
인공지능
인공생명
인간신뢰도 모델
사고/행동 모델
수요경향 모델
대중인식분석
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보험시스템
CAD/CAI
의료진단
장비진단
경영결정
Topic in the Class
• Theory on fuzzy sets
1) fuzzy set
2) fuzzy number
3) fuzzy logic
4) fuzzy relation
• Applications
1) fuzzy database
2) fuzzy control and expert system
3) robot
4) fuzzy computer
5) pattern recognition
• Rough Sets & Applications
Fuzzy Sets
Definition) Fuzzy sunset F on U, the universe of
discourse can be represented with the membership
grade, F(u) for all u U, which is defined by
F : U  [0,1].
Note:
1) The membership function F(u) represents the
degree of belongedness of u to the set F.
2) A crisp set is a special case of a fuzzy set, where
F : U  {0,1}.
Fuzzy Sets
F = {(ui, F(ui) |u iU }
= {F(ui) / ui |u iU }
=  F(ui) / ui
F =  F(u) / u
if U is discrete
if U is continuous
ex) F = {(a, 0.5), (b, 0.7), (c, 0.1)}
ex) F = Real numbers close to 0
F =  F(x) / x
where F(x) = 1/(1+x2)
ex) F = Real numbers very close to 0
F =  F(x) / x
where F(x) = {1/(1+x2)}2
Fuzzy Sets
Definition) Support of set F is defined by
supp(F) = { u U| F(u) 0}
Definition) Height of set F
h(F) = Max{ F(u),  u U}
Definition) Normalized fuzzy set is the fuzzy set with
h(F) = 1
Definition)  - level set,  - cut of F
F = {u U| F(u)  }
Fuzzy Sets
Definition) Convex fuzzy set F: The fuzzy set that satisfies
F(u)  F(u1)  F(u2)
(u1 < u < u2)
uF

u1
u2
u
Operations
Suppose U is the universe of discourse and
F, and G are fuzzy sets defined on U.
Definition) F = G (Identity)  F(u) = G(u)
Definition) F  G (Subset)  F(u) < G(u)
Definition) Fuzzy union: F  G
F G(u) = Max[F(u), G(u)]
= F(u)  G(u)
Definition) Fuzzy intersection: F  G
F G(u) = Min[F(u), G(u)]
= F(u)  G(u)
Definition) Fuzzy complement) FC(~F)
Fc(u) = 1- F(u)
 u U
 u U
 u U
Operations
Properties of Standard Fuzzy Operators
1) Involution
: (Fc)c = F
2) Commutative : F  G = G  F
FG=GF
3) Associativity : F  (G  H) = (F  G)  H
F  (G  H) = (F  G)  H
4) Distributivity : F  (G  H) = (F  G)  (F H)
F  (G  H) = (F  G)  (F  H)
5) Idempotency : F  F = F
FF=F
Operations
6) Absorption :
F  (F  G) = F
F  (F  G ) = F
7) Absorption by  and U :
F   = , F  U = U
8) Identity :
F =F F U=F
9) DeMorgan’s Law:
(F  G) C= FC  GC
(F  G) C= FC  GC
10) Equivalence
:
(FC  G)  (F  GC) = (FC  GC)  (F  G)
11) Symmetrical difference:
(FC  G)  (F  GC) = (FC  GC)  (F  G)
Operations
Note: The two conventional identity do not satisfy in standard
operation;
Law of contradiction
: F  FC = 
Law of excluded middle : F  FC = U
Other fuzzy operations
(1) Disjunctive Sum: F G = (F  GC)  (FC  G)
(2) Set Difference:
Simple Difference : F-G = F  GC
F -G(u) = Min[F(u), 1-G(u)]
 u U
Bounded Difference: F G
FG(u) = Max[0, F(u)-G(u)]
 u U
Operations
(3) Bounded Sum: F  G
F  G(u) = Min[1, F(u) + G(u)]
(4) Bounded Product: F  G
F  G(u) = Max[0, F(u) + G(u)-1]
 u U
 u U
(5) Product of Fuzzy Set for Hedge
F2 : F2 (u) = [F (u)]2
Fm : Fm (u) = [F (u)]m
(6) Cartesian Product of Fuzzy Sets F1 F2    Fn
F1  F2   Fn(u1 , u2,  ,un) = Min[F1(u1),  ,,Fn(un) ]
 ui  Fi
Generalized Fuzzy Sets
Interval-Valued Fuzzy Set
A : X   ([0,1])
 ([0,1]) denotestheset of all intervalsin [0,1]
Fuzzy Set of Type 2
A : X  F ([0,1])
F ([0,1]) denotesthefamilyof all fuzzy sets definedin [0,1]
L-Fuzzy Set
A: X  L
L denotesa lattice,theset at least partiallyordered.
Generalized Fuzzy Sets
Level-2 Fuzzy Set
A : F ( X )  [0,1]
F ( X ) denotestheall fuzzy sets defined on X
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Ex: “x is close to r”
If r is precisely specified, then it can be
represented by an ordinary fuzzy set
If r is approximately specified, A(B), the
fuzzy set A of a fuzzy set B can be used.
Additional Definitions
Cardinality of A (Sigma Count of A)
A 
 A( x)
xX
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Ex: A = .1/1 + .5/2 + 1./3 + .5/4 + .1/6
S ( A, B)
|A| = 2.2
Degree of Subsethood S(A,B)
S ( A, B ) 
S ( A, B ) 
1
( A   max[0, A( x)  B ( x)])
A
x X
A B
A
Hamming Distance
d ( A, B)   A( x)  B( x)
xX
Decomposition of Fuzzy Sets
Decomposition using  - level set
For every A  F ( X ),
A   A
 [ 0,1]
where


A    A      level set of A.
Ex:
A  .2 / x1  .4 / x2  .6 / x3  .8 / x4  1/ x5
Additional Notions of Operators
Axiomatic Definition of Complement C
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Boundary Condition
C (0)  1 and C (1)  0.
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Monotonicity
For all a, b[0,1], if a  b thenC (a)  C (b).
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Continuity
C is a continuous function.
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Involutive
C (C (a))  a for all a [0,1]
Additional Notions of Operators
Some complement operators
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Sugeno Class
c (a) 
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1 a
1  a
  (1, )
Yager Class
cw (a)  (1  a w )1/ w
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w  (0, )
Note: Parameters can be adjusted to obtain
some desired behavior.
Additional Notions of Operators
Characterization Theorem of Complement
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By strictly increasing function
C (a)  g 1 ( g (1)  g (a))
where g is a strictlyincreasingcontiuousfunction
from[0,1]to R such thatg (0)  0.
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By strictly decreasing function
C (a)  f 1 ( f (0)  f (a))
where f is a strictlydecreasingcontiuousfunction
from[0,1]to R such that f (1)  0.
Additional Notions of Operators
Axiomatic Definition of t-norm i
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Boundary Condition i(a,1)  a
Monotonicity b  d  i(a, b)  i(a, d )
Commutative i(a, b)  i(b, a)
Associative i(a, i(b, d ))  i(i(a, b), d )
Continuous i is a continuous function.
Subidempotecy i(a, a)  a
Strict Monotonicity a1  a2 and b1  b2  i(a1, b1 )  i(a2 , b2 )
Additional Notions of Operators
Some intersection operators
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Algebraic Product
Bounded Difference
Drastic Intersection
i ( a, b)  a  b
i(a, b)  max(0, a  b  1)
a when b  1

imin ( a, b)   b when a  1
0 ot herwise.
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Yager’s t-norm
iw (a, b)  1  min(1,[(1  a)w  (1  b)w ]1/ w )
where w  (0, ).
Additional Notions of Operators
Notes:
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Boundary of t-norm
imin (a, b)  i(a, b)  min(a, b)
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Characterization Theorem
t-norm can be generated by a generating
function.
Additional Notions of Operators
Axiomatic Definition of co-norm u
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Boundary Condition u(a,0)  a
Monotonicity b  d  u(a, b)  u(a, d )
Commutative u(a, b)  u(b, a)
Associative u(a, u(b, d ))  u(u(a, b), d )
Continuous u is a continuous function.
Subidempotecy u(a, a)  a
Strict Monotonicity a1  a2 and b1  b2  u(a1, b1 )  u(a2 , b2 )
Additional Notions of Operators
Some union operators
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Algebraic Sum
Bounded Sum
Drastic Union
u(a, b)  a  b  a  b
i(a, b)  min(1, a  b)
a when b  0

u max (a, b)  b when a  0
 1 ot herwise.
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Yager’s conorm
uw (a, b)  min(1, (a w  bw )1/ w )
where w  (0, ).
Additional Notions of Operators
Notes:
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Boundary of co-norm
max(a, b)  u(a, b)  umax (a, b)
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Characterization Theorem
Co-norm can be generated by a generating
function.
Additional Notions of Operators
Dual Triples <intersec, union, comp>
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Generalized DeMorgan’s Law
c(i(a, b))  u (c(a), c(b))
c(u (a, b))  i(c(a), c(b))

Examples
 min(a, b), max(a, b),1  a 
 ab, a  b  ab,1  a 
 max(0, a  b  1), min(1, a  b),1  a 
 imin (a, b), umax (a, b),1  a 
Additional Notions of Operators
Aggregation Operators for IF
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Compensation (Averaging) Operators h
min(a, b)  h(a, b)  max(a, b)
i ( a, b)  h ( a, b)  u ( a, b)

Example:

Gamma Model
n
i 1
n
where

i 1

n
 (a1 , a2 ,...,an )  ( ai ) (1   (1  ai ) )
 i 1
i
 n and   [0,1].
Averaging Operators:
Mean, Geometric Mean, Harmonic Mean
i
i 1
Additional Notions of Operators
Ordered Weighted Averaging
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Definition
n
OWA(a1 , a2 ,...,an )   wi bi
i 1
n
where
w
i 1
i
 1 and bi are thei - th componentof sorteddata a j s
in ascendingorder..
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Note: Different operations
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Min -> [1,0,…,0], Max -> [0, 0, …, 1]
Median -> [0, 0, ..1, 0, ..0]
Mean -> [1/n, …. 1/n]