FUZZY SETS
AND
FUZZY LOGIC
Theory and
Applications
PART 1
From classical
sets to fuzzy sets
1. Introduction
2. Crisp sets: an overview
3. Fuzzy sets: basic types
4. Fuzzy sets: basic concepts
Introduction
• Crisp set:
to dichotomize the individuals in some
given universe of discourse into two
groups: members and nonmembers. A
sharp, unambiguous distinction exists
between the members and nonmembers
of the set.
Introduction
• Fuzzy set:
to assign each individual in the universe of
discourse a value representing its grade of
membership in the fuzzy set. This grade
corresponds to the degree to which that
individual is similar to or compatible with the
concept represented by the fuzzy set. We
perceive fuzzy sets as having imprecise
boundaries that facilitate gradual transition from
membership to nonmembership and vice versa.
Crisp sets: an overview
• Universal set: The universe of discourse,
containing all the possible elements of
concern in each particular context from
which sets can be formed, X
• Empty set: The set containing no
members, 
• Member (element): x  A,
x A
Crisp sets: an overview
• Family of sets: Ai i I 
• Three methods of defining sets:
list: A = {x,y,z}
rule: A = {x|P(x)}
characteristic function: A : X  {0,1}
1 for x  A
A( x)  
0 for x  A
Crisp sets: an overview
•
•
•
•
•
Subset: A  B
Proper subset: A  B
Equal sets: A  B  A  B and B  A
Power set: P( A)  x x  A
Cardinality: The number of members of a
finite set, |A|
• Relative complement: B  A  x x  B and x  A
Crisp sets: an overview
• Complement: A  X  A
X is the universal set
• Union:  Ai  {x x  Ai ,
i  I }
iI
A B
• Intersection:  Ai  {x x  Ai ,
iI
A B
• Fundamental properties
i  I }
Crisp sets: an overview
Crisp sets: an overview
Crisp sets: an overview
( A)  { A A  A,

• Partition:
i
i
i I },
Ai  Aj   for i  j,
 A  A.
i
iI
• Nested family: A  { A1 , A2 ,  , An },
Ai  Ai 1
• Cartesian product:
A  B  { a, b  a  A, b  B}
 Ai  { a1 , a2 ,  , an  ai  Ai , i  N n }
1i  n
• Relations: Subsets of Cartesian products
Crisp sets: an overview
• Convexity: r , s  A,
r  (1   ) s  A
• Upper bound, lower bound: x  u , x  A
y  l , y  A
Crisp sets: an overview
• Supremum, infimum:
sup A: upper bound of A
no smaller is an upper bound.
inf A: lower bound of A
no greater is an lower bound.
Crisp sets: an overview
Fuzzy sets: basic types
• Membership function: A : X  [0, 1]
• Fuzzy set
Fuzzy sets: basic types
Fuzzy sets: basic types
Fuzzy sets: basic types
Fuzzy sets: basic types
Fuzzy sets: basic types
• Fuzzy variables: Several fuzzy sets
representing linguistic concepts, such as
low, medium, high, are often employed to
define states of a fuzzy variable.
• States of fuzzy variable: Temperature
within a range [a,b] is characterized as a
fuzzy variable, with states being fuzzy sets
very low, low, medium, etc.
Fuzzy sets: basic types
Fuzzy sets: basic types
• Interval-valued fuzzy sets
Fuzzy sets: basic types
• Type 2 fuzzy sets
Fuzzy sets: basic types
• Fuzzy power set: F(A), the set of all fuzzy sets
that can be defined within A
• Type 2: A:X→F([0, 1])
• L fuzzy sets: A:X →L
where L is a set of symbols that are partially
ordered.
• Level 2 fuzzy sets: A: F(X) →[0, 1]
Fuzzy sets: basic concepts
• Concepts of young, middle-aged, and old
Fuzzy sets: basic concepts
• Discrete approximation
Fuzzy sets: basic concepts
Fuzzy sets: basic concepts
• α-cut :

A  {x A( x)  }
Fuzzy sets: basic concepts
• Strong α-cut:

A  {x A( x)  }
Fuzzy sets: basic concepts
• Level set:  ( A)  { A( x)   , x  X }
• Support: sup( A) 0 A
1
core
(
A
)

A
• Core:
Fuzzy sets: basic concepts
• Height: h( A)  sup A( x)
xX
• Normal, subnormal: h( A)  1
h( A)  1
• Convex fuzzy sets:

A is convex for all
α  [0, 1].
Fuzzy sets: basic concepts
Fuzzy sets: basic concepts
Fuzzy sets: basic concepts
Fuzzy sets: basic concepts
Fuzzy sets: basic concepts
• Standard fuzzy set operations
• Standard complement: A  1  A( x)
• Equilibrium points: {x | A( x)  A( x)}
• Standard intersection: ( Ai )(x)  inf{Ai ( x) | i  I }
iI
( A  B )(x)  min[A( x), B ( x)]
• Standard union: ( Ai )(x)  sup{Ai ( x) | i  I }
iI
( A  B)(x)  max[A( x), B( x)]
Fuzzy sets: basic concepts
Fuzzy sets: basic concepts
Fuzzy sets: basic concepts
Fuzzy sets: basic concepts
Fuzzy sets: basic concepts
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•
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Fundamental properties
Laws of contradiction and excluded middle
Fuzzy set inclusion: A  B  A( x)  B( x), x  X
Scalar cardinality: | A |  A( x)
xX
• Degree of subsethood:
| A B |
S ( A, B) 
| A|
1

(| A |   max[0, A( x)  B( x)])
| A|
xX
Fuzzy sets: basic concepts
• Notation of fuzzy sets:
A  a1 x1  a2 x2      an xn
n
A
i 1
ai
xi
A   A( x ) x
x
• Distance:
d ( A, B)  | A( x)  B( x) |
xX
Exercise 1
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1.7
1.8
1.9
1.10
1.11