WEEK-1.pdf

Course Name: Fuzzy Logic and Neural Networks
Faculty Name: Prof. Dilip Kumar Pratihar
Department : Mechanical Engineering
Week 1

Course Name: Fuzzy Logic and Neural Networks
Department : Mechanical Engineering
Topic
Lecture 01: Introduction to Fuzzy Sets

Concepts Covered:
 Classical Set/Crisp Set
 Properties of Classical Set/Crisp Set
 Fuzzy Set
 Representation of Fuzzy Set

• Universal Set/Universe of Discourse (X): A set consisting of all
possible elements
Ex: All technical universities in the world
• Classical or Crisp Set is a set with fixed and well-defined
boundary
• Example: A set of technical universities having at least five
departments each
Classical Set/Crisp Set (A)

Representation of Crisp Sets
• A={a1,a2,……,an}
• A={x|P(x)}, P: property
• Using characteristic function
μA(x)=
1, if x belongs to A,
0, if x does not belong to A.

Notations Used in Set Theory
• : Empty/Null set
• : Element x of the Universal set X belongs to set A
• : x does not belong to set A
• : set A is a subset of set B
• : set A is a superset of set B
• : A and B are equal
• : A and B are not equal
Φ
A
x ∈
x A
∉
A B
⊆
A B
⊇
A B
=
A B
≠

• : A is a proper subset of B
• : A is a proper superset of B
• : Cardinality of set A is defined as the total number of
elements present in that set
• p(A) : Power set of A is the maximum number of subsets
including the null that can be constructed from a set A
Note: ( )
p A A
= 2
A B
⊂
A B
⊃
A

Fuzzy Sets
• Sets with imprecise/vague boundaries
• Introduced by Prof. L.A. Zadeh, University of California, USA, in 1965
• Potential tool for handling imprecision and uncertainties
• Fuzzy set is a more general concept of the classical set

Representation of a Fuzzy Set
( ) ( )
( )
{ }
X
x
x
x
x
A A ∈
µ
= ,
,
Note:
Probability: Frequency of likelihood that an element is in a class
Membership: Similarity of an element to a class

Types of Fuzzy sets
1. Discrete Fuzzy set
n: Number of elements present in the set
( ) ( ) ,
/
1
∑
=
µ
=
n
i
i
i
A x
x
x
A
( ) ( )
∫µ
=
X
A x
x
x
A /
2. Continuous Fuzzy set

Convex vs. Non-Convex Membership Function Distribution
A fuzzy set A(x) will be convex, if
Where 0.0 ≤ λ ≤ 1.0
( )
{ } ( ) ( )
{ }
2
1
2
1 ,
min
1 x
x
x
x A
A
A µ
µ
≥
λ
−
+
λ
µ

Various Types of Membership Function Distributions
1. Triangular Membership














−
−
−
−
=
µ 0
,
b
c
x
c
,
a
b
a
x
min
max
triangle

2. Trapezoidal Membership
max min ,1, ,0
trapezoidal
x a d x
b a d c
µ
 
− −
 
=  
 
− −
 
 

3. Gaussian Membership
2
2
1
1






σ
−
=
µ
m
x
Gaussian
e

4. Bell-shaped Membership Function
b
shaped
Bell
a
c
x
2
1
1
−
+
=
µ −

5. Sigmoid Membership
( )
b
x
a
Sigmoid
e −
−
+
=
µ
1
1

Reference:
Pratihar D.K.: Soft Computing: Fundamentals and
Applications, Narosa Publishing House, New-Delhi,
2014

Conclusion:
Classical Set/Crisp Set has been defined
Properties of Classical Set/Crisp Set has been explained
Fuzzy Set has been defined
Deals with representation of Fuzzy Set

Course Name: FUZZY LOGIC AND NUERAL NETWORKS
Department: Mechanical Engineering, IIT Kharagpur
Topic
Lecture 02: Introduction to Fuzzy Sets (contd.)

Concepts Covered:
 A few terms of Fuzzy Sets
 Standard Operations in Fuzzy Sets
 Properties of Fuzzy Sets
 Fuzziness and Inaccuracy of Fuzzy Sets

Numerical Example
Triangular Membership: Determine μ, corresponding
to x=8.0
1.0
μ
0.0
a=2 b=6 c=10
x
8

Trapezoidal Membership
•Determine μ corresponding to x = 3.5













−
−
−
−
= 0
,
8
10
10
,
1
,
2
4
2
min
max
x
x











 −
−
= 0
,
2
10
,
1
,
2
2
min
max
x
x












−
−
−
−
= 0
,
,
1
,
min
max
c
d
x
d
a
b
a
x

•We put x = 3.5












= 0
,
2
5
.
6
,
1
,
2
5
.
1
min
max
[ ]
0
,
75
.
0
max
=
75
.
0
=

Take c=10.0, a=2.0, b=3.0
We put x=8.0

Sigmoid Membership Function:
Determine µ corresponding to x = 8.0
2 6 0
2 2 0 4
1
1
1
1
1 1
0 98
1 1
− −
− −
− × −
+
+
= =
+ +
µ =
µ =
µ =
Take b = 6.0; a = 2
we put x = 8.0
sigmoid a(x b)
sigmoid (x . )
sigmoid .
e
e
.
e e

Difference Between Crisp and Fuzzy Sets

A Few Definitions in Fuzzy Sets

• Strong α-cut of a Fuzzy Set

The membership function distribution of a fuzzy set is assumed to follow a
Gaussian distribution with mean m = 100 and standard deviation σ =20 .
Determine 0.6 – cut of this distribution.
Solution:
Gaussian distribution :
where m : Mean ; σ : Standard deviation
By substituting the values of µ = 0.6, m = 100, σ =20 and
taking log (ln) on both sides, we get
Numerical Example
2
1
2
1
 
−
 
σ
 
µ = x m
e

( )
2
2
2
1 100
2 20
1 100
2 20
1 100
2 20
1
1
0 6
1 6667
79 7846 120 2153
0 6
By taking ln
x
x
x
e
e
.
ln e ln .
x ( . , . )
.  
−
 
 
 
−
 
 
 
−
 
 
⇒ =
 
  =
 
 
⇒ =
=
Figure : 0.6-cut of a fuzzy set.

• Support of a Fuzzy Set A(x)
• Scalar Cardinality of a Fuzzy Set A(x)

( )
( ) ( ) ( ) ( ) ( )
{ }
( )
1 2 3 4
0 1 0 2 0 3 0 4
0 1 0 2 0 3 0 4 1 0
=
= + + + =
Let us consider a fu
S
zzy set A x as follow
calar Cardinality
s:
A x x , . , x , . , x , . , x , .
A x . . . . .
Numerical Example

• Core of a Fuzzy Set A(x)
It is nothing but its 1-cut
• Height of a Fuzzy Set A(x)
It is defined as the largest of membership values of the elements
contained in that set.

• Normal Fuzzy Set
For a normal fuzzy set, h(A) = 1.0
• Sub-normal Fuzzy Set
For a sub-normal fuzzy set, h(A) < 1.0

Some Standard Operations in Fuzzy Sets
• Proper Subset of a Fuzzy Set

Numerical Example
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( ) ( )
{ }
( ) ( )
( ) ( ) ( ) ( )
1 2 3 4
1 2 3 4
0 1 0 2 0 3 0 4
0 5 0 7 0 8 0 9
µ µ
=
=
∈ <
⊂
Let us consider the two fuzzy
As for al
sets, as
l
that is , is the prop
follows:
er subset of
A B
A x x , . , x , . , x , . , x , .
B x x , . , x , . , x , . , x , .
x X, x x ,
A x B x , A x B x

Some Standard Operations in Fuzzy Sets
(contd.)
• Equal fuzzy sets

Numerical Example
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( )
1 2 3 4
1 2 3 4
0 1 0 2 0 3 0 4
0 5 0 7 0 8 0 9
µ µ
=
=
∈ ≠ ≠
As for a
Let us consider the two fuzzy sets, as follows
ll
:
A B
A x x , . , x , . , x , . , x , .
B x x , . , x , . , x , . , x , .
x X, x x , A x B x

Numerical Example
( )
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( ) ( )
{ }
1 2 3 4
1 2 3 4
0 1 0 2 0 3 0 4
0 9 0 8 0 7 0 6
=
=
Let us consider a fuzzy set A x as follo
Complement
ws:
A x x , . , x , . , x , . , x , .
A x x , . , x , . , x , . , x , .

• Intersection of Fuzzy Sets

Note: Intersection is analogous to logical AND operation
x

Numerical Example
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( )
{ } { }
( ) ( ) { }
1 2 3 4
1 2 3 4
1 1 1
2
0 1 0 2 0 3 0 4
0 5 0 7 0 8 0 9
0 1 0 5 0 1
0 2 0 7 0 2
µ µ µ
µ
µ
=
=
= = =
= =


Now,
Let us consider the two fuzzy s
Similarly,
ets as follo s
w :
A B
A B
A B
A
A x x , . , x , . , x , . , x , .
B x x , . , x , . , x , . , x , .
x min x , x min . , . .
x min . , . .
( ) ( ) { }
( ) ( ) { }
3
4
0 3 0 8 0 3
0 4 0 9 0 4
µ
= =
= =


B
A B
x min . , . .
x min . , . .

Note: Union is analogous to logical OR operation

Numerical Example
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( )
{ } { }
( ) ( )
1 2 3 4
1 2 3 4
1 1 1
2
0 1 0 2 0 3 0 4
0 5 0 7 0 8 0 9
0 1 0 5 0 5
µ µ µ
µ
=
=
= = =
=


Let us consider the following two fuzzy s
Now,
Similarly,
ets:
A B
A B
A B
A x x , . , x , . , x , . , x , .
B x x , . , x , . , x , . , x , .
x max x , x max . , . .
x max{ }
( ) ( ) { }
( ) ( ) { }
3
4
0 2 0 7 0 7
0 3 0 8 0 8
0 4 0 9 0 9
µ
µ
=
= =
= =


A B
A B
. , . .
x max . , . .
x max . , . .

• Algebraic product of Fuzzy Sets

Numerical Example
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( ) ( ) ( )
{ }
1 2 3 4
1 2 3 4
1 2 3 4
0 1 0 2 0 3 0 4
0 5 0 7 0 8 0 9
0 05 0 14 0 24 0 36
=
=
=
Let us consider the following two fuzzy sets:
A x x , . , x , . , x , . , x , .
B x x , . , x , . , x , . , x , .
A x B x x , . , x , . , x , . , x , .
.

• Multiplication of a Fuzzy Set by a Crisp Number

Numerical Example
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( ) ( )
{ }
1 2 3 4
1 2 3 4
0 1 0 2 0 3 0 4 0 2
0 02 0 04 0 06 0 08
=
=
and a crisp number
Let us consider a fuzzy set
A x x , . , x , . , x , . , x , . d .
d A x x , . , x , . , x , . , x , .
.

• Power of a Fuzzy Set
AP(x): p-th power of a fuzzy set A(x) such that
Concentration: p=2
Dilation: p=1/2

Numerical Example
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( ) ( )
{ }
1 2 3 4
2
1 2 3 4
0 1 0 2 0 3 0 4 2
0 01 0 04 0 09 0 16
and power
Let us consider a fuzzy set
A x x , . , x , . , x , . , x , . p
A x x , . , x , . , x , . , x , .
=
=

• Algebraic Sum of two Fuzzy Sets A(x) and B(x)
where

Numerical Example
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( ) ( ) ( )
{ }
1 2 3 4
1 2 3 4
1 2 3 4
0 1 0 2 0 3 0 4
0 5 0 7 0 8 0 9
0 55 0 76 0 86 0 94
=
=
∴ + =
A x x , . , x , . , x , . , x , .
B x x , . , x , . , x , . , x , .
A x B x x , . , x , . , x , . , x , .

• Bounded Sum of two Fuzzy Sets
where

Numerical Example
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( ) ( )
{ }
( ) ( ) ( ) ( ) ( ) ( )
{ }
1 2 3 4
1 2 3 4
1 2 3 4
0 1 0 2 0 3 0 4
0 5 0 7 0 8 0 9
0 6 0 9 1 0 1 0
=
=
∴ ⊕ =
A x x , . , x , . , x , . , x , .
B x x , . , x , . , x , . , x , .
A x B x x , . , x , . , x , . , x , .

• Algebraic Difference of two Fuzzy Sets
where

Numerical Example
•Let us consider the following two fuzzy sets:
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
( ) {( ,0.1),( ,0.2),( ,0.3),( ,0.4)}
( ) {( ,0.5),( ,0.7),( ,0.8),( ,0.9)}
, ( ) {( ,0.5),( ,0.3),( ,0.2),( ,0.1)}
( ) ( ) {( ,0.1),( ,0.2),( ,0.2),( ,0.1)}
A x x x x x
B x x x x x
Now B x x x x x
A x B x x x x x
=
=
=
∴ − =

• Bounded Difference of two Fuzzy Sets
where

Numerical Example
1 2 3 4
1 2 3 4
1 2 3 4
( ) {( ,0.1),( ,0.2),( ,0.3),( ,0.4)}
( ) {( ,0.5),( ,0.7),( ,0.8),( ,0.9)}
( ) ( ) {( ,0.0),( ,0.0),( ,0.1),( ,0.3)}
A x x x x x
B x x x x x
A x B x x x x x
=
=
Θ =

• Cartesian product of two Fuzzy Sets
Two fuzzy sets A(x) defined in X
and B(y) defined in Y
Cartesian product of two fuzzy sets is denoted by A(x)×B(y),
such that

Numerical Example
1 2 3 4
1 2 3
1 1
1 2
( ) {( ,0.2),( ,0.3),( ,0.5),( ,0.6)}
( ) {( ,0.8),( ,0.6),( ,0.3)}
min( ( ), ( )) min(0.2,0.8) 0.2
min( ( ), ( )) min(0.2,0.6) 0.2
A B
A B
A x x x x x
B y y y y
x y
x y
µ µ
µ µ
=
=
= =
= =

1 3
2 1
2 2
2 3
min( ( ), ( )) min(0.2,0.3) 0.2
min( ( ), ( )) min(0.3,0.8) 0.3
min( ( ), ( )) min(0.3,0.6) 0.3
min( ( ), ( )) min(0.3,0.3) 0.3
A B
A B
A B
A B
x y
x y
x y
x y
µ µ
µ µ
µ µ
µ µ
= =
= =
= =
= =

3 1
3 2
3 3
4 1
min( ( ), ( )) min(0.5,0.8) 0.5
min( ( ), ( )) min(0.5,0.6) 0.5
min( ( ), ( )) min(0.5,0.3) 0.3
min( ( ), ( )) min(0.6,0.8) 0.6
A B
A B
A B
A B
x y
x y
x y
x y
µ µ
µ µ
µ µ
µ µ
= =
= =
= =
= =

4 2
4 3
min( ( ), ( )) min(0.6,0.6) 0.6
min( ( ), ( )) min(0.6,0.3) 0.3
0.2 0.2 0.2
0.3 0.3 0.3
0.5 0.5 0.3
0.6 0.6 0.3
A B
A B
x y
x y
A B
µ µ
µ µ
= =
= =
 
 
 
∴ × =
 
 
 

Composition of fuzzy relations
Let A = [aij] and B = [bjk] be two fuzzy relations expressed in the matrix form.
Composition of these two fuzzy relations, that is, C is represented as follows:
C=A о B
In matrix form
[cik] = [aij] о [bjk]
Where
cik =max[min(aij, bjk)]

Numerical Example
•Let us consider the following two Fuzzy relations:
[ ]
[ ]
[ ]
ik
jk
ij
c
b
B
a
A






=
=






=
=
6
.
0
7
.
0
8
.
0
6
.
0
1
.
0
3
.
0
7
.
0
5
.
0
3
.
0
2
.
0
•Elements of matrix can be determined as follows:

[ ]
[ ]
[ ]
2
.
0
1
.
0
,
2
.
0
max
)
1
.
0
,
3
.
0
min(
),
3
.
0
,
2
.
0
min(
max
)
,
min(
),
,
min(
max 21
12
11
11
11
=
=
=
= b
a
b
a
c

[ ]
[ ]
[ ]
3
.
0
3
.
0
,
2
.
0
max
)
8
.
0
,
3
.
0
min(
),
6
.
0
,
2
.
0
min(
max
)
,
min(
),
,
min(
max 22
12
12
11
12
=
=
=
= b
a
b
a
c

[ ]
[ ]
[ ]
3
.
0
3
.
0
,
2
.
0
max
)
6
.
0
,
3
.
0
min(
),
7
.
0
,
2
.
0
min(
max
)
,
min(
),
,
min(
max 23
12
13
11
13
=
=
=
= b
a
b
a
c

[ ]
[ ]
[ ]
3
.
0
1
.
0
,
3
.
0
max
)
1
.
0
,
7
.
0
min(
),
3
.
0
,
5
.
0
min(
max
)
,
min(
),
,
min(
max 21
22
11
21
21
=
=
=
= b
a
b
a
c

[ ]
[ ]
[ ]
7
.
0
7
.
0
,
5
.
0
max
)
8
.
0
,
7
.
0
min(
),
6
.
0
,
5
.
0
min(
max
)
,
min(
),
,
min(
max 22
22
12
21
22
=
=
=
= b
a
b
a
c

[ ]
[ ]
[ ]
6
.
0
6
.
0
,
5
.
0
max
)
6
.
0
,
7
.
0
min(
),
7
.
0
,
5
.
0
min(
max
)
,
min(
),
,
min(
max 23
22
13
21
23
=
=
=
= b
a
b
a
c







=
∴
6
.
0
3
.
0
7
.
0
3
.
0
3
.
0
2
.
0
C

Properties of Fuzzy Set
Fuzzy sets follow the properties of crisp sets except the following two:
• Law of excluded middle

Measure of Fuzziness of Fuzzy Set

Measure of Inaccuracy of Fuzzy Set

References:
 Soft Computing: Fundamentals and Applications by D.K. Pratihar,
Narosa Publishing House, New-Delhi, 2014
 Fuzzy Sets and Fuzzy Logic: Theory and Applications by G.J. Klir,
B. Yuan, Prentice Hall, 1995

Conclusion:
• A few terms related to Fuzzy Sets have been defined
• Some standard Operations in Fuzzy Sets have been
explained
• Properties of Fuzzy Sets have been explained
• Fuzziness and Inaccuracy of Fuzzy Sets are
determined

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