2. Classical and Fuzzy Sets

12012

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, carindicates the set of cars. When we say a car, we mean one out ofthe set of cars.

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

22012

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

Person Height Degree of MembershipCrisp Fuzzy

Person 1 205 1 1.00Person 2 182 1 0.81Person 3 175 0 0.38Person 4 167 0 0.10Person 5 155 0 0.04Person 6 152 0 0.00

32012

Crisp Set

(clearly defined Set)

42012

Unlike two-valued Boolean logic, fuzzy logic is multi-valued. Itdeals 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 thingscan be partly true and partly false at the same time.

(a) Boolean Logic. (b) Multi-valued Logic.0 1 10 0.2 0.4 0.6 0.8 100 1 10

52012

The x-axis represents the Universe of Discourse (universe of allavailable information on a given problem/ the space where the fuzzyvariables are defined). According to this representation, the universeof person’s heights consists of all ’tall’ men.

The y-axis represents the Membership Value of the fuzzy set. Inour case, the fuzzy set of “tall men” maps height values intocorresponding membership values.

62012

Example: Fuzzy set of Tall Men

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 afunction fA(x) called the characteristic function of A:

fA(x) : X {0/1}, whereThis set maps universe X to a set of two elements. For any elementx of universe X, characteristic function fA(x) is equal to 1 if x is anelement of set A, and is equal to 0 if x is not an element of A.Notations:x ∈ X ⇒ x belongs to XA ⊂ 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) ⇒ A ⊆ B and B ⊆ A (A is equivalent to B)The null set, ∅, is the set containing no elements

Conventional or crisp sets are binary. An element eitherbelongs to the set (True/1) or doesn’t (False/0) .

72012

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 AµA(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 ofuniverse X, membership function µA(x) equals the degree to which xis 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.

ThusMembership functions of fuzzy sets are continuous. An elementcan belong to the set to a certain degree {0 1} .

82012

3.4. Fuzzy Set Representation:First determine the member-ship functions:In the “tall men” example, wecan obtain Crisp set of tall,short and average men, andalso of Fuzzy set.The universe of discourse –

the men’s heights – consistsof three sets: short, averageand tall men. A man who is184 cm tall is:Crisp: a member of the tallset but not of the othersFuzzy: he is a member ofaverage men set with adegree of membership of0.1, and at the same time,he is also a member of thetall men set with a degreeof 0.4, and is not a memberof short men set

92012

Fuzzy Set Representation (Contd.)There are Continuous universescomprising of an infinite collection ofelements (see fuzzy set of Tall Men onslide 6) , and Discrete universes, thatare composed of a finite collection ofelements. In the last case whenelements are discrete the grade ofmembership can be represented by:

A = {(1,0.7), (2, 0.5),(3,1.0)} OrA = 0.7/1 + 0.5/2+1.0/3

10

be sure to notice that the symbol ‘+’ implies not addition butunion.More Generally, we use Or

2012

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”

112012

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}

12

age(element) infant young adult senior5 0 0 0 0

15 0 0.2 0.1 025 0 1 0,9 035 0 0.8 1 045 0 0.4 1 0.155 0 0.1 1 0.265 0 0 1 0.675 0 0 1 185 0 0 1 1

2012

132012

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. Forexample, if we have the set of tall men, its complement isthe set of NOT tall men. When we remove the tall men setfrom the universe of discourse, we obtain the complement.

142012

The complement set of a fuzzy set A is found asfollows: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}.

152012

16

Not Adult

2012

2. Union ( )DefinitionCrisp 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 thatfalls into either set. For example, the union of tall menand fat men contains all men who are tall OR fat.

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

172012

Membership value of member x in the union takes the greatervalue 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” isA B = {(15,0.2), (25,1), (35,1), (45,1), (55,1),

(65,1),(75,1), (85,1)}

182012

19

Young Adult

2012

3. Intersection ( ( )DefinitionCrisp 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 setscontains the elements shared by these sets. Forexample, the intersection of the set of tall men and theset of fat men is the area where these sets overlap. Infuzzy sets, an element may partly belong to both setswith different memberships.

A fuzzy intersection is the lower membership in both sets ofeach element.

202012

Intersection of fuzzy sets A and B takes the smaller value ofmembership function between A and B.

Intersection A B is a subset of A or B. For instance, theintersection 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 areapplicable even if membership function is restricted to 0 or 1(i.e. crisp set)

212012

22

Young Adult

2012

Another Example:Let A be a fuzzy interval “between 5 and 8” and B be a fuzzynumber “about 4”.

232012

The following figure shows the fuzzy set “between 5 and 8” AND“about 4” (Intersection)

242012

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

252012

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

262012

Example:Consider the fuzzy set of the pointsincluded in a circle

2010 27Hany Selim

Distance membership function

Representation of the grade ofmembership of the inclusion within thecircle with black representing thepoints not in the set i.e. μP1=0

272012

Representation of the grade ofmembership of the inclusion withina shifted circle with blackrepresenting the points not in theset i.e. μP2

2010 28

Union of the 2 setsμP1 μP2

282012

The intersection of the two sets

μP1 μP2

2010 29

The complement of the firstset above:

μP1

292012

2010 30

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

The union of the first set and itscomplement is not 1 (rememberby Crisp A+A=1)

302012

312012

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

322012

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

A(x) = B(x), xX

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

therefore A = B

332012

2. Inclusion:Inclusion of one fuzzy set into another fuzzy set. Fuzzy setA X is included in (is a subset of) another fuzzy set,B X:

A(x) B(x), xX

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

342012

3. Cardinality:Cardinality of a non-fuzzy set, Z, is the number of elementsin Z. BUT the cardinality of a fuzzy set A, the so-calledSIGMA COUNT, is expressed as a SUM of the values of themembership function of A, A(x):

cardA = A(x1) + A(x2) + … A(xn) = ΣA(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

thencardA = 1.8cardB = 2.05

352012

4. Empty Fuzzy Set:A fuzzy set A is empty, IF AND ONLY IF:

A(x) = 0, xX

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

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

then A is empty

362012

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

A={A(x), xX}

Consider X = {15,25,35,45,55} representing ageand 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 possibilitynot less than 0.5”.

372012

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

one element xX 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+0/55is a Normal set.ButA = 0.2/15 + 0.9/25 + 0.8/35+0.3/45+0/55Is a Subnormal set.

382012

7. Height of the Fuzzy set:The height of a fuzzy subset A is the large membershipgrade 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+0/55then height(A)=0.8

392012

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

core(A) = {x A(x) = 1 and xX}

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

402012

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

supp(A) = {x A(x) 0 and xX}

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

412012

3.7. Fuzzy Set Math Operations:

kA = {kA(x), xX}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, xX}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}

422012

3.8. Properties of Fuzzy:

432012

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}

442012

1. Cardinality:card(A) =

card(B) =

2. Complement of A and of BA =

B =

3. Union:A B =

45

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}

2012

4. Intersection:A B =

5. a-cut:A0.2 =

A0.3 =

A0.8 =

A1 =

46

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}

2012

6. Support:supp(A) =

supp(B) =

7. Core:core(A) =

core(B) =

8. aA for a=0.5

9. Aa: for a=2

47

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

2012

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 isprobable. However, if it is probable it must also be possible. Youmight view a fuzzy membership function as your personaldistribution, in contrast with a statistical distribution based onobservations

482012