# 2 Classical and Fuzzy Sets by sherif.abas

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```									       2. Classical and Fuzzy Sets

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3.1. Crisp vs. Fuzzy Sets:
The concept of a set is fundamental to mathematics. How-ever,
our own language uses sets extensively. For example, car
indicates the set of cars. When we say a car, we mean one out of
the set of cars.

The classical example in fuzzy sets is tall men. The elements
of the fuzzy set “tall persons” are all persons, but their degrees of
membership depend on their height.

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Example: Crisp vs. Fuzzy Sets: TALL-Men set

Person      Height    Degree of Membership
Crisp         Fuzzy
Person 1       205          1           1.00
Person 2       182          1           0.81
Person 3       175          0           0.38
Person 4       167          0           0.10
Person 5       155          0           0.04
Person 6       152          0           0.00

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Crisp Set
(clearly defined Set)

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Unlike two-valued Boolean logic, fuzzy logic is multi-valued. It
deals with degrees of membership and degrees of truth.

Fuzzy logic uses the continuum of logical values between 0
(completely false) and 1 (completely true), accepting that things
can be partly true and partly false at the same time.

0    0     0 1     1     1   0 0   0.2   0.4   0.6   0.8   1 1
(a) Boolean Logic.         (b) Multi-valued Logic.

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Example: Fuzzy set of Tall Men

The x-axis represents the Universe of Discourse (universe of all
available information on a given problem/ the space where the fuzzy
variables are defined). According to this representation, the universe
of person’s heights consists of all ’tall’ men.
The y-axis represents the Membership Value of the fuzzy set. In
our case, the fuzzy set of “tall men” maps height values into
corresponding membership values.
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3.2. Crisp Sets:
Let X be the universe of discourse and its elements be denoted as x.

In the classical set theory, a crisp set A of X is defined by a
function fA(x) called the characteristic function of A:
fA(x) : X  {0/1}, where
This set maps universe X to a set of two elements. For any element
x of universe X, characteristic function fA(x) is equal to 1 if x is an
element of set A, and is equal to 0 if x is not an element of A.

x∈X            ⇒ x belongs to X
Notations:

A⊂B            ⇒ A is fully contained in B (if x ∈ A, then x ∈ B)
A⊆B            ⇒ A is contained in or is equivalent to B
⇒ A ⊆ B and B ⊆ A (A is equivalent to B)
The null set, ∅, is the set containing no elements
(A ↔ B)

Conventional or crisp sets are binary. An element either
belongs to the set (True/1) or doesn’t (False/0) .
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3.3. Fuzzy Sets:
In the fuzzy theory, a fuzzy set A of universe X is defined by function
µA(x) called the membership function of set A
µA(x) : X  {0, 1}, where          µA(x) = 1 if x is totally in A;
µA(x) = 0 if x is not in A;
0 < µA(x) < 1 if x is partly in A.
This set allows a continuum of possible choices. For any element x of
universe X, membership function µA(x) equals the degree to which x
is an element of set A. This degree, a value between 0 and 1,
represents the degree of membership, also called membership value,
of element x in set A.

Thus
Membership functions of fuzzy sets are continuous. An element
can belong to the set to a certain degree {0  1} .

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First determine the member-
3.4. Fuzzy Set Representation:    ship functions:
In the “tall men” example, we
can obtain Crisp set of tall,
short and average men, and
also of Fuzzy set.
The universe of discourse –
the men’s heights – consists
of three sets: short, average
and tall men. A man who is
184 cm tall is:
Crisp: a member of the tall
set but not of the others
Fuzzy: he is a member of
average men set with a
degree of membership of
0.1, and at the same time,
he is also a member of the
tall men set with a degree
of 0.4, and is not a member
of short men set
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Fuzzy Set Representation (Contd.)
There are Continuous universes
comprising of an infinite collection of
elements (see fuzzy set of Tall Men on
slide 6) , and Discrete universes, that
are composed of a finite collection of
elements. In the last case when
elements are discrete the grade of
membership can be represented by:
A = {(1,0.7), (2, 0.5),(3,1.0)} Or
A = 0.7/1 + 0.5/2+1.0/3
be sure to notice that the symbol ‘+’ implies not addition but
union.
More Generally, we use                      Or

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Example: Discrete vs. Continuous Membership functions

A discrete membership function for ”x is close to 1”

A Continuous membership function for ”x is close to 1”

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3.5. Standard Operations on Fuzzy Sets:
Consider a universal set X which is defined on the age domain.
X = {5, 15, 25, 35, 45, 55, 65, 75, 85}

5          0        0         0         0
15          0       0.2       0.1        0
25          0        1        0,9        0
35          0       0.8        1         0
45          0       0.4        1        0.1
55          0       0.1        1        0.2
65          0        0         1        0.6
75          0        0         1         1
85          0        0         1         1

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1) Complement:
Definition:
Crisp Sets: Who does not belong to the set?
Fuzzy Sets: How much do elements not belong to the set?

The complement of a set is an opposite of this set. For
example, if we have the set of tall men, its complement is
the set of NOT tall men. When we remove the tall men set
from the universe of discourse, we obtain the complement.

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The complement set    of a fuzzy set A is found as
follows:
The degreee of Membership is given by:

In our example A is the “adult” set and is given by:

A = {(5, 0), (15, 0.1), (25, 0.9),(35,1)}.
Then:

= {(5, 1), (15, 0.9), (25, 0.1),(35,0}.

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2. Union ( )
Definition
Crisp Sets: Which element belongs to either set?
Fuzzy Sets: How much of the element is in either set?

The union of two crisp sets consists of every element that
falls into either set. For example, the union of tall men
and fat men contains all men who are tall OR fat.

In fuzzy sets, the union is the largest membership value of
the element in either set.

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Membership value of member x in the union takes the greater
value of membership between A and B

Of course, A and B are subsets of A B, now if B represents
“young” and is given by:
B= {(5, 0), (15, 0.2), (25, 1),(35,0.8 ), (45, 0.4),(55,0.2)}
and as before adult A={(5, 0), (15, 0.1), (25, 0.9),(35,1)}

Then the union of “young” and “adult” is
A    B = {(15,0.2), (25,1), (35,1), (45,1), (55,1),
(65,1),(75,1), (85,1)}

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3. Intersection ( ( )
Definition
Crisp Sets: Which element belongs to both sets?
Fuzzy Sets: How much of the element is in both sets?

In classical set theory, an intersection between two sets
contains the elements shared by these sets. For
example, the intersection of the set of tall men and the
set of fat men is the area where these sets overlap. In
fuzzy sets, an element may partly belong to both sets
with different memberships.

A fuzzy intersection is the lower membership in both sets of
each element.
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Intersection of fuzzy sets A and B takes the smaller value of
membership function between A and B.

Intersection A     B is a subset of A or B. For instance, the
intersection of “young” and “adult” is

A    B= {(15, 0.1), (25, 0.9), (35, 0.8), (45, 0.4), (55, 0.1)}.

Note that complement, union and intersection sets are
applicable even if membership function is restricted to 0 or 1
(i.e. crisp set)

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Another Example:
Let A be a fuzzy interval “between 5 and 8” and B be a fuzzy

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The following figure shows the fuzzy set “between 5 and 8” AND

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The Fuzzy set “between 5 and 8” OR “about 4” is shown in the
next figure (Union)

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This figure gives an example for a negation. The blue line is the
NEGATION (Complement) of the fuzzy set A. i.e. “not a fuzzy
interval between 5 and 8”

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Example:
Consider the fuzzy set of the points
included in a circle

Distance membership function

membership of the inclusion within the
circle with black representing the
points not in the set i.e. μP1=0

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2010                        Hany Selim                             27
membership of the inclusion within
a shifted circle with black
representing the points not in the
set i.e. μP2

Union of the 2 sets
μP1 μP2

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The intersection of the two sets
μP1     μP2

The complement of the first
set above:
μP1

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The intersection of the first set
and its complement is not empty
(remember by Crisp A.A=0)

The union of the first set and its
complement is not 1 (remember
by Crisp A+A=1)

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3.6. Properties of Fuzzy Sets:
Equality of two fuzzy sets

Inclusion of one set into another fuzzy set

Cardinality of a fuzzy set

An empty fuzzy set

-cuts (alpha-cuts)

Fuzzy set Normality

Height of the Fuzzy set

Fuzzy set Core

Fuzzy set Support

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1. Equality:
Fuzzy set A is considered equal to a fuzzy set B, IF AND
ONLY IF (iff):

A(x) = B(x), xX

A = 0.3/1 + 0.5/2 + 1/3
B = 0.3/1 + 0.5/2 + 1/3

therefore A = B

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2. Inclusion:
Inclusion of one fuzzy set into another fuzzy set. Fuzzy set
A  X is included in (is a subset of) another fuzzy set,
B  X:
A(x)  B(x), xX

Consider X = {1, 2, 3} and sets A and B

A = 0.3/1 + 0.5/2 + 1/3;
B = 0.5/1 + 0.55/2 + 1/3

then A is a subset of B, or A  B

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3. Cardinality:
Cardinality of a non-fuzzy set, Z, is the number of elements
in Z. BUT the cardinality of a fuzzy set A, the so-called
SIGMA COUNT, is expressed as a SUM of the values of the
membership function of A, A(x):

cardA = A(x1) + A(x2) + … A(xn) = ΣA(xi),
for i=1..n

Consider X = {1, 2, 3} and sets A and B

A = 0.3/1 + 0.5/2 + 1/3;
B = 0.5/1 + 0.55/2 + 1/3
then
cardA = 1.8
cardB = 2.05
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4. Empty Fuzzy Set:
A fuzzy set A is empty, IF AND ONLY IF:

A(x) = 0, xX

Consider X = {1, 2, 3} and set A

A = 0/1 + 0/2 + 0/3

then A is empty

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5. Alpha-cut:
An -cut or -level set of a fuzzy set A  X is a Crisp set
A  X, such that:

A={A(x), xX}

Consider X = {15,25,35,45,55} representing age
and set A representing young
A = 0.2/15 + 1/25 + 0.8/35+0.3/45+0/55

then     A0.5 = {25, 35},
A0.1 = {15, 25, 35, 45},
A1 = {25}
A0.5 means “the age that we can say of being young with possibility
not less than 0.5”.

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6. Fuzzy set Normality:
A fuzzy subset of X is called normal if there exists at least

one element xX such that A(x) = 1

• A fuzzy subset that is not normal is called subnormal.

• All crisp subsets except for the null set are normal.

•Example:
A = 0.2/15 + 1/25 + 0.8/35+0.3/45+0/55
is a Normal set.
But
A = 0.2/15 + 0.9/25 + 0.8/35+0.3/45+0/55
Is a Subnormal set.
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7. Height of the Fuzzy set:
The height of a fuzzy subset A is the large membership
grade of an element in A

height(A) = maxx (A(x))

Example:
If    A = 0.2/15 + 0.6/25 + 0.8/35+0.3/45+0/55
then  height(A)=0.8

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8. Fuzzy set Core:
the core of A is the crisp subset of X consisting of all
elements with membership grade = 1:

core(A) = {x A(x) = 1 and xX}

Example:
If    A = 0.2/15 + 1/25 + 0.8/35+1/45+0/55
then  core(A)={25,45}

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9. Fuzzy set Support:
the support of A is the crisp subset of X consisting of all
elements with a degree of membership differing from zero:

supp(A) = {x A(x)  0 and xX}

Example:
If    A = 0.2/15 + 0/25 + 0.8/35+1/45+0/55
then  supp(A)={15, 35,45}

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3.7. Fuzzy Set Math Operations:

kA = {kA(x), xX}
Let k =0.5, and
A = {0.5/a, 0.3/b, 0.2/c, 1/d}
then
kA = {0.25/a, 0.15/b, 0.1/c, 0.5/d}

Ak = {A(x)k, xX}
Let k =2, and
A = {0.5/a, 0.3/b, 0.2/c, 1/d}
then
Ak = {0.25/a, 0.09/b, 0.04/c, 1/d}

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3.8. Properties of Fuzzy:

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Quiz:

Consider two fuzzy subsets of the set X,
X = {a, b, c, d, e }

referred to as A and B

A = {1/a, 0.3/b, 0.3/c, 0.8/d, 0/e}
and
B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e}

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1. Cardinality:
card(A) =                 A = {1/a, 0.3/b, 0.3/c ,0.8/d, 0/e}
B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e}

card(B) =

2. Complement of A and of B
A=

B=

3. Union:
AB=

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4. Intersection:   A = {1/a, 0.3/b, 0.3/c, 0.8/d, 0/e}
AB=               B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e}

5. a-cut:
A0.2 =

A0.3 =

A0.8 =

A1 =

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6. Support:
supp(A) =
A = {1/a, 0.3/b, 0.3/c, 0.8/d, 0/e}
B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e
supp(B) =

7. Core:
core(A) =

core(B) =

8. aA    for a=0.5

9. Aa:   for a=2

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Example:

Where the possibility for X=3 is 1, the probability is only 0.1

The example shows, that a possible event does not imply that it is
probable. However, if it is probable it must also be possible. You
might view a fuzzy membership function as your personal
distribution, in contrast with a statistical distribution based on
observations

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