1

Conventional Sets and Fuzzy Sets

2

Conventional Sets

A set is a collection of things, for example the room temperature, the set of all real numbers, etc….

2

3

Conventional Sets

Such collection of things are called the Universe of Discourse, X, and is defined as the range of all possible values for a variable.

Universe of Discourse can be divided into sets or subsets. For Example, consider a set A of the real numbers between 5 and 8 from the universe of discourse X.

Conventional sets called crisp sets

X

5 8

4

Conventional Sets

If we have two sets A and B consisting of a collection of elements in X universe of discourse.

x Є X (x belongs to X)

x Є A (x belongs to A)

x Є X (x does not belong to A)

A B (A is fully contained in B; if x Є A, then x Є B)

A B (A is contained in or is equivalent to B)

A = B (A B and B A)

The null set Ø is the set with no elements, and the whole set , X, is the set of elements in the universe.

Operations on Classical Sets

3

5

Conventional Sets

A U B ; the union represent all elements that reside in both sets A and B. This is called the logic or.

Operations on Classical Sets: Union

A

B

A U B = [x | x Є A or x Є B ]

6

Conventional SetsOperations on Classical Sets: Intersection

A Π B ; the intersection represent all elements that

simultaneously reside in both sets A and B. This is called the

logic and.

A

B

A Π B = [x | x Є A and x Є B ]

4

7

Conventional SetsOperations on Classical Sets: Complement

Є

Ā ; the complement of set A is the collection of all

elements on the universe that do not reside in set A.

Ā = [x | x A and x Є X ]

A

8

Conventional SetsOperations on Classical Sets: Difference

Є

A | B ; the collection of all elements on the universe

that reside in A and do not reside in B at the same time.

A | B = [x | x Є A and x B]

B

A

5

9

Conventional SetsProperties of Classical Sets

Commutativity: A U B = B U A; also for the intersection

Associativity: A U (B U C) = (A U B) U C

Distributivity: A U (B Π C) = (A U B) Π (A U C)

Transitivity: if A B C, then A C.

Identity: A U Ø = A

A Π Ø = Ø

A U X = X

A Π X = A

10

Conventional SetsProperties of Classical Sets

Law of Excluded Middle:

A U Ā = X

A Π Ā = Ø

De Morgan’s law:

A Π B = Ā U B

A U B = Ā Π B

The complement of a union or an intersection is equal to the

intersection or union of the respective complement

AB

AB

6

11

Conventional SetsProperties of Classical Sets: Example

The survival of the arch will be represented by E1 Π E2.

The collapse is E1 Π E2. Logically collapse will occur if either

members fail, i.e., E1 U E2.

Consider an arch consists of two members,

if either members fails then the arch will

collapse. If E1 represents survival of

member 1 and E2 member 2.

Load

12

Conventional SetsMapping

If an element x is contained in X and corresponds to an element y contained in Y, it is termed a mapping from X to Y, ƒ : X Y.

This is called the characteristic function

µA(x) = 1, x Є A

0, x AЄ

X

5 8

7

13

Fuzzy Sets

In classic sets, the transition of an element in the

universe between being a member and non member in a

given set is abrupt.

In fuzzy sets, this transition occurs gradually

A fuzzy set is a set containing elements that have varying

degree of membership in the set.

Accordingly, elements in a fuzzy sets can be members of

other fuzzy set on the same universe.

Elements of fuzzy sets are mapped to a universe of

membership values using a function-theoretic form

14

Fuzzy Sets

This function maps elements of fuzzy set A to a real numbered value between 0 and 1.

A fuzzy set A in the universe X can be defined as set of ordered pairs

A = {(x, µA(x) |x Є X}

A discrete and finite fuzzy set is represented as follow

A =

When x is continuous A = ∫ µA(x) /x

µA(x1) /x1 + µA(x2) /x2 +………

8

15

Fuzzy Sets

000.1.3.5.7.911Low

0

1

100

0

1

90

.8

.5

70

1

.3

60

.5

0

40

.1

0

30

0

0

10

.5.80Medium

.8.10High

805020Score

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60 70 80 90 100 110

low medium high

Example

16

Fuzzy Sets

0

100

0

90

.8

70

1

60

.5

40

.1

30

0

10

.5.80Medium

805020Score

Example

B = Medium score = {(10, 0), (20, 0), (30, .1), (40, .5), (50, .8), (60, 1), (70, .8), (80, .5), (90, 0), (100, 0)}

Or B = (30, .1), (40, .5), (50, .8), (60, 1), (70, .8), (80, .5)}

Or B = 0.1/30 + 0.5/40 + 0.8/50 + 1/60 + 0.8/70 + 0.5/80

9

17

Fuzzy SetsFuzzy Sets Operations

Union: the membership functions of the union of the two fuzzy sets A and B is defined as the maximum of both

µA U B (x) = µA (x) V µ B (x)

18

Fuzzy SetsFuzzy Sets Operations

Intersection: the membership functions of the intersection of the two fuzzy sets A and B is defined as the minimum of both

µA Π B (x) = µA (x) ^ µ B (x)

10

19

Fuzzy SetsFuzzy Sets Operations

Complement: the membership functions of the complement of fuzzy set A is defined as

20

Fuzzy SetsFuzzy Sets Operations

The same operations of the classical sets are still valid for the fuzzy sets.

Commutativity: A U B = B U A; also for the intersection

Associativity: A U (B U C) = (A U B) U C

Distributivity: A U (B Π C) = (A U B) Π (A U C)

Transitivity: if A B and B C, then A C.

De Morgan’s law

11

21

Fuzzy Sets

Two fuzzy sets are equal if and only if µA (x) = µ B (x) for all

x Є X.

A is a sub set of B: A B, if and only if µA (x) < µ B (x) for all

x Є X.

Fuzzy Sets Operations

22

Fuzzy SetsExample

Consider the following two fuzzy sets:

A = { 1/2 + .5/3 + .3/4 + .2/5}

B = {.5/2 + .7/3 + .2/4 + .4/5}

Complement Ā = { 1/1 + 0/2 + .5/3 + .7/4 +.8/5}

Complement B = { 1/1 + .5/2 + .3/3 + .8/4 +.6/5}

Union: A U B = {1/2 + .7/3 + .3/4 +.4/5}

Intersection: A Π B = {.5/2 + .5/3 + .2/4 +.2/5}

12

23

Fuzzy SetsExample

Consider the following two fuzzy sets:

A = { 1/2 + .5/3 + .3/4 + .2/5}

B = {.5/2 + .7/3 + .2/4 + .4/5}

Difference A | B = A Π B = {.5/2 + .3/3 + .3/4 +.2/5}

De Mogan’s law = A U B = Ā Π B = { 1/1 + 0/2 + .3/3 +

.7/4 +.6/5}

24

Fuzzy SetsNormal Fuzzy Set

A fuzzy set A is normal if its maximal degree of membership

is unity (i.e., there must exist at least one x for which µA(x)

= 1. On the other hand, non-normal fuzzy sets have

maximum degree of membership less than oneDegree of Membership

Universe of Discourse0

1

13

25

Fuzzy SetsSupport of a Fuzzy Set

Support of a fuzzy set A

(written as supp(A)) is a

(crisp) set of points in X for

which µA is positive supp(A)

= { x Є X | µA(x)>0}

0

100

0

90

.8

70

1

60

.5

40

.1

30

0

10

.5.80Medium

805020Score

Support (B) = Medium score = {30, 40, 50, 60, 70, 80}

26

Fuzzy SetsConvex Fuzzy Set

A fuzzy set A is convex if and only if it satisfies the

following µA( λx1 + (1 – λ ) x2 ) ≥ min ( µA( x1) , µA( x2 )),

where λ is in the interval [0,1], and x1 < x2

14

27

Fuzzy Setsα-cut of a Fuzzy Setα-cut is defined as a crisp set Aα (or a crisp interval) for a

particular degree of membership, α: Aα= [aα , bα] , where α

can take on values between [0,1]

28

Fuzzy Setsα-cut of a Fuzzy Set: Example

Consider the score example

0

100

0

90

.8

70

1

60

.5

40

.1

30

0

10

.5.80Medium

805020Score

B0.8 = Medium score 0.8 = { 50, 60, 70}

15

29

Fuzzy SetsFuzzy Numbers

Fuzzy number is a fuzzy set which is both normal and

convex. In addition, the membership function of a fuzzy

number must be piecewise continuous .

Most common types of fuzzy numbers are triangular and

trapezoidal. Other types of fuzzy numbers are possible,

such as bell-shaped or gaussian fuzzy numbers, as well as a

variety of one sided fuzzy numbers. Triangular fuzzy

numbers are defined by three parameters, while

trapezoidal require four parameters

30

Fuzzy SetsFuzzy Numbers

16

31

Fuzzy SetsResolution Principle

A fuzzy set A can be expanded in terms of its α-cuts.

µA (x) = α ^ µA α (x); x Є X

This means that a fuzzy set can be decomposed into αAα , αЄ [0, 1].

X

µA(x)

1

α2

α1

α2Aα2

α1Aα1

Aα2Aα1

32

Fuzzy SetsResolution Principle: Example

Consider the following fuzzy set:

A = {.1/50 + .3/60 + .5/70 + .8/80 + 1/90 + 1/100}

Using the resolution principle:

A = .1 {1/50 + 1/60 + 1/70 + 1/80 +1/90 + 1/100}

+ .3 {1/60 + 1/70 + 1/80 +1/90 + 1/100}

+ .5 {1/70 + 1/80 +1/90 + 1/100}

+ .8 {1/80 +1/90 + 1/100}

+ 1 {1/90 + 1/100}

= .1 A.1 + .3 A.3 + .5 A.5 + .8 A.8 + 1A1

17

33

Fuzzy SetsRepresentation Theorem

As opposed to the resolution principle, a fuzzy set A can be

represent in terms of its α-cuts. i.e., A fuzzy set can be

retrieved as a union of its αAα.

A = U αAα

X

µA(x)

1

α2

α1

Aα2Aα1

34

Fuzzy SetsRepresentation Theorem: Example

If we are given: A0.1 = {1, 2, 3, 4, 5}, A0.4 = {2, 3, 5}, A0.8 = {2, 3}, and A1 = {3}

Then, fuzzy set A can be expressed as: A = U αAα for α Є [0, 1].

A = 0.1 A0.1 + 0.4 A0.4 + 0.8 A0.8 + 1 A1

= 0.1 {1/1 + 1/2 + 1/3 + 1/4 +1/5}

+ 0.4 {1/2 + 1/3 + 1/5}

+ 0.8 {1/2 + 1/3}

+ 1 {1/3}

= 0.1/1 + 0.8/2 + 1/3 + 0.1/4 + 0.4/5

18

35

Fuzzy SetsExtension Principle

Consider a single relationship between one independent variable x and one dependent variable y.

ƒ(x)x y

The function ƒ(x) represents the mapping of x on y.

y = ƒ(x)

The function y = ax + b, are mapping from one universe X to another universe Y and is written as:

ƒ : X Y

Sometimes it is called the image of x under ƒ for y=ƒ(x)

36

Fuzzy SetsExtension Principle

The extension principle can be also applied to fuzzy sets.

Given a function f : U V , and a set A in U for x Є U,

then its image, set B, in the universe V is found from the

mapping, B = ƒ(A)

xB (y) = xf(A) (y)

µB (y) = V µf(A) (y); y=f(x)

19

37

Fuzzy SetsExtension Principle: Example

Consider a crisp set A = [0, 1] defined in the universe X =

{-2, -1, 0, 1, 2}, where A = {0/-2 + 0/-1 + 1/0 + 1/1

+0/2} and mapping function y= |4x|+2. Find the set B on

an output universe Y using the extension principle.

The universe Y = f(x) for x Є X

Then Y = {2, 6, 10}, the mapping for membership

µB (2) = V [µA (0)] = 1

µB (6) = V [µA (-1), µA (1)] = 1

µB (10) = V [µA (-2), µA (2)] = 0

Then B = {1/2 + 1/6 + 0/10} or B = [2, 6 ]

38

Fuzzy SetsExtension Principle

The same operations of the classical sets are still valid for

the fuzzy sets.

Given a function ƒ : U V and a fuzzy set A in U, where

A = µ1/x1 + µ2/x2+ µ3/x3 + ……., the extension principle

states: ƒ(A) = ƒ(µ1/x1 + µ2/x2+ µ3/x3 + …….) = µ1/ƒ (x1)+

µ2/ƒ(x2) + µ3/ƒ(x3)+ …….

Or the resulting set B = µA(x1)/y1 + µA(x2)/y2 + ……

If more than one element of U is mapped to the same

element y of V, then the max membership is taken

20

39

Fuzzy SetsExtension Principle: Example

Consider X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}; a fuzzy set A =

“large” is given as = {0.5/6 + 0.7/7 + 0.8/8 + .9/9 +1/10}

Given a function ƒ : y=f(x) = x2, find the fuzzy set B =

“large”2

B = {0.5/36 + 0.7/49 + 0.8/64 + .9/81 + 1/100}

One to one mapping always reserve the membership values

40

Fuzzy SetsExtension Principle: Example

Consider A = {0.1/-2 + 0.4/-1 + 0.8/0 + 0.9/1 +0.3/2}

ƒ(x) = x2 – 3, using extension principle to find B = ƒ(x)

B ={0.1/(4-3)+0.4/(1-3)+0.8/(0-3)+0.9/(1-3)+0.3/(4-3)}

B = {0.1/1 + 0.4/-2 + 0.8/-3 + 0.9/-2 + 0.3/1}

B = {(0.1 V 0.3)/1 + (0.4 V 0.9)/-2 + 0.8/-3}

B = {0.3/1 + 0.9/-2 + 0.8/-3}

21

41

Fuzzy SetsExtension Principle: Example

0.1

-3 -2 -1 0 1 2 3

0.4

0.9

0.3

0.80.9

-3 -2 -1 0 1 2 3

0.3

0.8

ƒ(x) = x2 – 3