Fuzzy Logic and Its Applications
Textbook:
Fuzzy Logic with Engineering
Applications,
2
nd
Ed. John

Wiley,
2004
, T.J.Ross,
References
:
Fuzzy
Logic
with
Engineering
Applications,
1995
T
.
J
.
Ross,
McGraw

Hill
Fuzzy
Set
Teory,
1997
G
.
Klir
et
al
.
Prentice
Hall
Fuzzy
Sets
and
Fuzzy
Logic
1995
G
Klir
et
al
.
Prentice
Hall
Introduction
In
1965
,
Prof
.
Lofti
Zadeh
published
the
first
article
“Fuzzy
Sets”
.
It
becomes
billions
of
dollars
business
.
America
Europe
Asia
Thousands
of
patents
Uncertainty
:
Incomplete
Ambiguity
:
Imprecise
Applications
:
Air
Conditioner
Washing
Machine
Subway
System
Camera
Aerospace
Nuclear
Submarine
Pattern
Recognition
Control
Image
Processing
Computer
Vision
They reflect a recent trend to view fuzzy logic (FL),
neurocomputing (NC), genetic computing (GC), Rough
Sets (RS) and probabilistic computing (PC) as an
association of computing methodologies falling under
the rubric of so

called soft computing.
Among the basic concepts that underlie human
cognition, three stand out in importance: granulation,
organization, and causation.
Granulation involves a partitioning of a whole into parts;
organization involves an integration of parts into a
whole; and causation relates to an association of causes
with effects
A granule may be viewed as a clump of points (objects)
drawn together by indistinguishability, similarity, or
functionality. Modes of information granulation (IG) in
which granules are crisp play an important role in many
theories, methods and techniques, among them interval
analysis, quantization, rough set theory, qualitative
process theory, and chunking.
In fuzzy logic, fuzzy IG underlies the basic concepts of
linguistic variables, fuzzy if

then rules, and fuzzy graphs
This perception is reinforced by viewing it in the context
of generalization. More specifically, any theory, method,
technique, or problem may be fuzzified (or f

generalized)
by replacing the concept of a crisp set with that of a
fuzzy set.
Similarly, any theory, method, technique, or problem can
be granulated (g

generalized) by partitioning variables,
functions, and relations into granules.
Furthermore, we can combine fuzzification with
granulation, which gives rise to fuzzy granulation (f

granulation). Fuzzy granulation, then, provides a basis
for what might be called f.g

generalization.
The generalization of two

valued logic leads to
multivalued logic and parts of fuzzy logic. But fuzzy logic
in its wide sense
—
which is the sense in which it is used
today
—
results from f.g

generalization. This crucial
difference between multivalued logic and fuzzy logic
explains why fuzzy logic has so many applications,
whereas multivalued logic does not.
Introduction
Fuzzy
set
theory
provides
a
means
for
representing
uncertainties
.
Probablity
–
random uncertainty
But
some
uncertainty
is
non

random
In
fact,
a
huge
amount!
Natural
Language
is
vague
and
imprecise
.
Fuzzy
set
theory
uses
Linguistic
variables,
rather
than
quantitative
variables
to
represent
imprecise
concepts
.
Fuzzy Logic
Fuzzy
Logic
is
suitable
to
Very
complex
models
Judgemental
Reasoning
Perception
Decision
making
Requiring
precision
–
high
cost,
long
time
Statistics
and
random
processes
Based
on
Randomness
.
Fuzziness
Example
.
Random
Errors
generally
average
out
over
time
or
space
Non

random
errors
will
not
generally
average
out
and
likely
to
grow
with
time
.
Information World
Information World
Crisp
set
has
a
unique
membership
function
A
(x)
=
1
x
A
0
x
A
A
(x)
{
0
,
1
}
Fuzzy
Set
can
have
an
infinite
number
of
membership
functions
A
[
0
,
1
]
Fuzziness
Examples
:
A
number
is
close
to
5
Fuzziness
Examples
:
He/she
is
tall
Fuzziness
Randomness
versus
Fuzziness
Drinking
Water
Problem
Classical
Sets
Fuzzy
Sets
Operations on Classical Sets
Union
:
A
B
=
{x

x
A
or
x
B}
Intersection
:
A
B
=
{x

x
A
and
x
B}
Complement
:
A’
=
{x

x
A,
x
X}
X
–
Universal
Set
Set
Difference
:
A

B
=
{x

x
A
and
x
B}
Set
difference
is
also
denoted
by
A

B
Properties of Classical Sets
A
B
=
B
A
A
B
=
B
A
A
(B
C)
=
(A
B)
C
A
(B
C)
=
(A
B)
C
A
(B
C)
=
(A
B)
(A
C)
A
(B
C)
=
(A
B)
(A
C)
A
A
=
A
A
A
=
A
A
X
=
X
A
X
=
A
A
=
A
A
=
Properties of Classical Sets
If
A
B
C,
then
A
C
De Morgan
’
s Law:
(A
B)
’
= A
’
B
’
(A
B)
’
= A
’
B
’
Proof:
LHS=
{x  x
(A and B)}=
{x  x
A or
x
B)}=
A
’
B
’
= RHS
Can be extended to
n sets
Generalized De Morgan Law:
A
A
’
X
X
Using ( ) to keep original processing order
Generalized Duality Law:
X
X
Using ( ) to keep original processing order
Law of the excluded middle:
A
A
’
= X
Law of the Contradiction:
A
A
’
=
These laws are not true for Fuzzy
Sets!
Fuzzy Sets
Characteristic
function
X,
indicating
the
belongingness
of
x
to
the
set
A
X(x)
=
1
x
A
0
x
A
or
called
membership
Hence,
A
B
X
A
B
(x)
=
X
A
(x)
X
B
(x)
=
max(X
A
(x),X
B
(x))
Note
:
Some
books
use
+
for
,
but
still
it
is
not
ordinary
addition!
Some
more
explanations
follow
…
Fuzzy Sets
A
B
X
A
B
(x)
=
X
A
(x)
X
B
(x)
=
min(
X
A
(x),
X
B
(x))
A’
X
A’
(x)
=
1
–
X
A
(x)
A
B
X
A
(x)
X
B
(x)
A’’
=
A
Fuzzy Sets
Note
(x)
[
0
,
1
]
not
{
0
,
1
}
like
Crisp
set
A
=
{
A
(x
1
)
/
x
1
+
A
(x
2
)
/
x
2
+
…
}
=
{
A
(xi)
/
xi
}
Note
:
‘+’
add
‘/
’
divide
Only
for
representing
element
and
its
membership
.
Also
some
books
use
(x)
for
Crisp
Sets
too
.
Fuzzy Set Operations
A
B
(x)
=
A
(x)
B
(x)
=
max(
A
(x),
B
(x))
A
B
(x)
=
A
(x)
B
(x)
=
min(
A
(x),
B
(x))
A’
(x)
=
1

A
(x)
De
Morgan’s
Law
also
holds
:
(A
B)’
=
A’
B’
(A
B)’
=
A’
B’
But,
in
general
A
A’
A
A’
Properties of Fuzzy Sets
A
B
=
B
A
A
B
=
B
A
A
(B
C)
=
(A
B)
C
A
(B
C)
=
(A
B)
C
A
(B
C)
=
(A
B)
(A
C)
A
(B
C)
=
(A
B)
(A
C)
A
A
=
A
A
A
=
A
A
X
=
X
A
X
=
A
A
=
A
A
=
If
A
B
C,
then
A
C
A’’
=
A
Sets as Points in Hypercubes
Explore
to
n

dimension
Classical
Relations
Fuzzy
relations
Logic,
Approximate
reasoning,
Rule

based
learning
systems,
Nonlinear
Simulation,
Classification,
Pattern
Recognition,
etc
.
Cartesian Product
A
=
{a,b}
B
=
{
0
,
1
}
A
x
B
=
{
(a,
0
)
(a,
1
)
(b,
0
)
(b,
1
)
}
Ordered
Pairs
Consider
A
x
A
or
A
x
B
x
C
if
C
is
given
Based
on
the
above,
Crisp
Relations
are
discussed
next
…
Crisp Relations
A
subset
of
a
Cartesian
Product
A
1
x
A
2
x
…
x
Ar
is
called
an
r

ary
relation
over
A
1
,A
2
,
…
,Ar
If
r
=
2
,
the
relation
is
a
subset
of
A
1
x
A
2
Binary
relation
from
A
1
into
A
2
The
strength
of
a
relation
:
Characteristic
Function
X(x,y)
=
1
(x,y)
X
x
Y
0
(x,y)
X
x
Y
For
Classical
relations,
the
value
is
1
or
0
If
the
universes
or
sets
are
finite,
we
can
use
relational
matrix
to
represent
it
.
Crisp Relations
Example
:
If
X
=
{
1
,
2
,
3
}
Y
=
{a,b,c}
R
=
{
(
1
a),(
1
c),(
2
a),(
2
b),(
3
b),(
3
c)
}
a b c
1 1 0 1
R =
2 1 1 0
3 0 1 1
Using
a
diagram
to
represent
the
relation
Crisp Relations
Relations
can
also
be
defined
for
continuous
universes
R
=
{
(x,y)

y
2
x,
x
X,
y
Y
}
X
=
1
y
2
x
0
otherwise
Crisp Relations
Cardinality
:
N
:
#
of
elements
in
X
M
:
#
of
elements
in
y
Cardinality
of
R
n
X
x
Y
=
n
X
•
n
Y
=
M
•
N
Cardinality
of
the
Power
set
of
this
relation
n
P(X
x
Y)
=
2
M
N
Operations on Crisp Relations
Complete
Relation
Matrix
1 1 1
1
1 1
1 1 1
Null
relation
Matrix
0 0 0
0 0 0
0 0 0
Operations on Crisp Relations
Union
R
S
X
R
S
(x,y)
X
R
S
(x,y)
=
max{
X
R
(x,y),X
S
(x,y)
}
Intersection
R
S
X
R
S
(x,y)
X
R
S
(x,y)
=
min{
X
R
(x,y),X
S
(x,y)
}
Complement
R’
X
R’
(x,y)
X
R’
(x,y)
=
1
–
X
R
(x,y)
Containment
R
S
X
R
(x,y)
X
S
(x,y)
Identity
0
X
E
Properties of Crisp Relations
Commutativity
Associativity
Distributivity
Idempotency
All
hold
De
Morgan
Law
Excluded
middle
Law
Etc
.
Properties of Crisp Relations
Composition
Let
R
be
a
relation
representing
a
mapping
from
X
to
Y
X
Y
University
sets
Let
S
be
a
relation,
a
mapping
from
Y
to
Z
Can
we
find
T
from
R
to
S?
Properties of Crisp Relations
T
:
mapping
from
X
to
Z
T
=
R
S
Two
ways
to
compute
X
T
(xz)
1.
X
T
(xz)
=
(X
R
(xy)
X
s
(yz))
=
max(min{X
R
(xy),X
S
(yz)})
Max

min composition
2.
X
T
(xz)
=
(X
R
(xy)
X
s
(yz))
Max

product composition
multiplication
y
Y
y
Y
y
Y
Properties of Crisp Relations
Using
Matrix
representation
:
y
1
y
2
y
3
y
4
x
1
1
0
1
0
R
=
x
2
0
0
0
1
x
3
0
0
0
0
z
1
z
2
y
1
0
1
z
1
z
2
y
2
0
0
x
1
0
0
S
=
y
3
0
1
T
=
x
2
0
0
y
4
0
0
x
3
0
0
T
(x
1
,z
1
)
=
max[min(
1
,
0
)
min(
0
,
0
)
min(
1
,
0
)
min(
0
,
0
)]
=
max[
0
,
0
,
0
,
0
]
=
0
Similar,
but
not
the
same
as
matrix
multiplication!
Fuzzy Relations
Cardinality
of
Fuzzy
Relations
Since
the
cardinality
of
fuzzy
sets
on
any
universe
is
infinity,
the
cardinality
of
a
fuzzy
relation
is
also
infinity
.
Note
:
other
books
have
different
discussions!
Operations on Fuzzy Relations
Union
:
R
S
=
max{
R
(x,y),
S
(x,y)
}
Intersection
:
R
S
=
min{
R
(x,y),
S
(x,y)
}
Complement
:
R
’
(x,y)
=
1

R
(x,y)
Containment
:
R
S
R
(x,y)
S
(x,y)
Properties of Fuzzy Relations
Commutativity
Associativity
Distributivity
Idempotency
All
hold
De
Morgan
Law
Excluded
middle
Law
Etc
.
Note
:
R
R’
E
R
R’
0
In
general
.
Properties of Fuzzy Relations
Fuzzy
Cartesian
Product
and
Composition
R
(x
y)
=
A
x
B
(x
y)
=
min(
A
(x),
B
(y))
Example
:
A
=
0
.
2
/x
1
+
0
.
5
/x
2
+
1
/x
3
B
=
0
.
3
/y
1
+
0
.
9
/y
2
y
1
y
2
0
.
2
x
1
0
.
2
0
.
2
A
x
B
=
0
.
5
0
.
3
0
.
9
=
x
2
0
.
3
0
.
5
1
x
3
0
.
3
0
.
9
Properties of Fuzzy Relations
Vector
Outer
Product
If
R
is
a
fuzzy
relation
on
the
space
X
x
Y
S
is
a
fuzzy
relation
on
the
space
Y
x
Z
Then,
fuzzy
composition
is
T
=
R
S
1.
Fuzzy
max

min
composition
T
(xz)
=
(
R
(xy)
s
(yz))
2
.
Fuzzy
max

production
composition
T
(xz)
=
(
R
(xy)
s
(yz))
Note
:
R
S
S
R
y
Y
y
Y
Properties of Fuzzy Relations
Example
:
y
1
y
2
z
1
z
2
z
3
R
=
x
1
0
.
7
0
.
5
S
=
y
1
0
.
9
0
.
6
0
.
2
x
2
0
.
8
0
.
4
y
2
0
.
1
0
.
7
0
.
5
z
1
z
2
z
3
Using
max

min,
T
=
x
1
0
.
7
0
.
6
0
.
5
x
2
0
.
8
0
.
6
0
.
4
z
1
z
2
z
3
Using
max

product,
T
=
x
1
0
.
63
0
.
42
0
.
25
x
2
0
.
72
0
.
48
0
.
20
Note
:
Set,
Relation,
Composition
How
to
find
new
membership
from
the
given
ones!
Tolerance and Equivalence Relation
Crisp
Equivalence
Relation
R
X
x
X
Relation
has
the
following
properties
:
Reflexivity
(xi
xi)
R
or
X
R
(xi
xi)
=
1
Symmetry
(xi
xj)
R
(xj
xi)
R
or
X
R
(xi
xj)
=
X
R
(xj
xi)
Transitivity
(xi
xj)
R
and
(xj
xk)
R
(xi
xk)
R
or
X
R
(xi
xj)
=
1
and
X
R
(xj
xk)
=
1
X
R
(xi
xk)
=
1
Tolerance and Equivalence Relation
Graph
representation
:
Crisp Tolerance Relation
(or proximity relation)
Only
has
reflexivity
and
symmetry
A
tolerance
relation,
R
1
can
become
an
Equivalence
Relation
by
at
most
(n

1
)
compositions
(
n

1
),
n
is
the
cardinal
member
of
X
.
R
1
n

1
=
R
1
R
1
…
R
1
=
R
Crisp Tolerance Relation
(or proximity relation)
Example
:
1
1
0
0
0
1
1
0
0
1
R
1
=
0
0
1
0
0
0
0
0
1
0
0
1
0
0
1
1
1
0
0
1
1
1
0
0
1
Try
R
1
2
=
0
0
1
0
0
0
0
0
1
0
1
1
0
0
1
Note
:
symmetric,
reflexive,
but
not
transitive,
why?
X(x
1
x
2
)
=
1
X(x
2
x
5
)
=
1
but
X(x
1
x
5
)
1
(=
0
)
Now,
it
is
transitive!
Fuzzy Tolerance and Equivalence Relation
A
fuzzy
relation
R
has
:
1
.
Reflexivity
R
(xi
xi)
=
1
2
.
Symmetry
R
(xi
xj)
=
R
(xj
xi)
3
.
Transitivity
R
(xi
xj)
=
1
R
(xj
xk)
=
2
R
(xi
xk)
=
where
min{
1
,
2
}
Fuzzy
tolerance
relation
R
1
has
reflexivity,
symmetry
.
It
can
be
transformed
into
a
fuzzy
equivalence
relation
by
at
most
(n

1
)
(
n

1
)
compositions
.
R
1
n

1
=
R
1
R
1
…
R
1
=
R
Fuzzy Tolerance and Equivalence Relation
Example
:
1
0
.
8
0
0
.
1
0
.
2
0
.
8
1
0
.
4
0
0
.
9
R
1
=
0
0
.
4
1
0
0
0
.
1
0
0
1
0
.
5
0
.
2
0
.
9
0
0
.
5
1
R
1
(x
1
x
2
)
=
0
.
8
R
1
(x
2
x
5
)
=
0
.
9
But
R
1
(x
1
x
5
)
=
0
.
2
min(
0
.
8
,
0
.
9
)
not
transitive
Fuzzy Tolerance and Equivalence Relation
Value
Assignment
How
to
find
the
membership
values
for
the
relation?
1.
Cartesian
Production
Note
:
you
have
to
know
the
membership
value
for
the
sets!
Will
discuss
in
chapter
4
.
2
.
Y
=
f(x)
X
–
input
vector
Y
–
output
vector
3
.
Look
up
table
y
1
y
2
y
3
x
1
x
2
x
3
Fuzzy Tolerance and Equivalence Relation
Value
Assignment
4
.
Linguistic
rule
of
knowledge
–
chapters
7
–
9
5
.
Classification
–
chapter
11
6
.
Similarity
methods
in
data
manipulation
The
more
robust
a
data
set,
the
more
accurate
the
relation
entries!
Cosine Amplitude
X
=
{x
1
,x
2
,
…
,x
n
}
each
element
is
also
a
vector
X
i
=
{x
i
1
,x
i
2
,
…
,x
im
}
ij
=
R
(x
i
,x
j
)
It
will
be
n
x
n
symmetric,reflexive
…
i
.
e
.
a
tolerance
relation!
Note
:
this
relates
to
the
vector
dot
product
for
cosine
function
Cosine Amplitude
Example:
x
1
x
2
x
3
x
4
x
5
xi
1
0.3 0.2 0.1 0.7 0.4
xi
2
0.6 0.4 0.6 0.2 0.6
xi
3
0.1 0.4 0.3 0.1 0
Using the above formula:
1
symm
0.836 1
R
1
=
0.914 0.934 1
0.682 0.6 0.441 1
0.982 0.74 0.818 0.7741 1
Cosine Amplitude
1
symm
0.538 1
R1 =
0.667 0.667 1
0.429 0.333 0.25 1
0.818 0.429 0.538 0.429 1
Computationally simple!
Max

min Method:
Other Similarity Methods
Absolute Exponential:
Exponential Similarity Coefficient:
Where, S
k
= any general measure for all the data i.e.
(S
k
)
2
≥
0
Other Similarity Methods
Other methods produce scalar quantities which are similar
to the cosine amplitude, such as the following:
Geometric average minimum:
Scalar Product:
Where:
Other Similarity Methods
Some methods are analogous to popular statistical
quantities, such as:
Correlation Coefficient:
Where:
and
Arithmetic Average Minimum:
Other Similarity Methods
Some methods are based on the inverse relationships,
for example:
Absolute Reciprocal:
Where M is selected to make 0 ≤ r
ij
≤ 1
Absolute subtrahend:
Where c is selected to make 0 ≤ r
ij
≤ 1
Other Similarity Methods
Other
methods
are
nonparametric,
such
as
:
Nonparametric
:
where
x
’
ik
=
x
ik
–
x
i
and
x
’
jk
–
x
j
n
+
= number of elements > 0 in
{x
’
i1
x
’
j1
,x
’
i2
x
’
j2
,
…
,x
’
im
x
’
jm
}
n

= number of elements < 0 in
{x
’
i1
x
’
j1
,x
’
i2
,x
’
j2
,
…
,x
’
im
,x
’
jm
}
In the above equations, terms such as x
’
i1
x
’
j1
are
products of data elements.
Membership Function
Membership Functions characterize the fuzziness of
fuzzy sets. There are an infinite # of ways to
characterize fuzzy
infinite ways to define fuzzy
membership functions.
Membership function essentially embodies all fuzziness
for a particular fuzzy set, its description is essential to
fuzzy property or operation.
Features of Membership Function
Core
:
comprises
of
elements
x
of
the
universe,
such
that
A
(x)
=
1
Support
:
comprises
of
elements
x
of
universe,
such
that
A
(x)
>
0
Boundaries
:
comprise
the
elements
x
of
the
universe
0
<
A
(x)
<
1
A
normal
fuzzy
set
has
at
least
one
element
with
membership
1
For
fuzzy
set,
if
one
and
only
one
element
has
a
membership
=
1
,
this
element
is
called
as
the
prototype
of
set
.
A
subnormal
fuzzy
set
has
no
element
with
membership=
1
.
Features of Membership Function
Graphically,
Features of Membership Function
A convex fuzzy set has a membership whose value is:
1. strictly monotonically increasing, or
2. strictly monotonically decreasing, or
3. strictly monotonically increasing, then strictly
monotonically decreasing
Or another way to describe:
(y) ≥ min[
(x),
(z)], if x < y < z
If A and B are convex sets, then A
B is also a convex set
Crossover points have membership 0.5
Height of a Fuzzy set is the maximum value of the
membership: max{
A
(x)}
Features of Membership Function
If height <
1
, the fuzzy set is subnormal.
Fuzzy number: like a number is close to
5
. It has to have
the properties:
1
. A must be a normal fuzzy set.
2
.
A must be closed for all
(
0
,
1
].
3
. The support,
0
A must be bounded.
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