AI FuzzySets

8/20/2019 AI FuzzySets

1/21

Conventional Sets andFuzzy Sets

2

Conventional Sets

A set is a collection of things, for example the room

temperature, the set of all real numbers, etc….


2/21

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


3/21

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 Sets

Operations 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 ]


4/21

7

Conventional Sets

Operations 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 Sets

Operations 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


5/21

9

Conventional Sets

Properties 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 Sets

Properties 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

A

B


6/21

11

Conventional Sets

Properties 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 Sets

Mapping

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

µ A(x) =1, x Є A

0, x AЄ

X

5 8


7/21

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, µ A(x ) |x Є X}

A discrete and finite fuzzy set is represented as follow

A =

When x is continuous A = ∫ µ A(x ) /x

µ A(x1 ) /x1 + µ A(x2 ) /x2 +………


8/21

15

Fuzzy Sets

0 0 0 .1 .3 .5 .7 .9 1 1 Low

0

1

100

0

1

90

.8

.5

70

1

.3

60

.5

0

40

.1

0

30

0

0

10

.5 .8 0 Medium

.8 .1 0 High

80 50 20 Score

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 .8 0 Medium

80 50 20 Score

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


9/21

17

Fuzzy Sets

Fuzzy Sets Operations

Union: the membership functions of the union of the two

fuzzy sets A and B is defined as the maximum of both

µ A U B (x) = µ A (x) V µ B (x)

18

Fuzzy Sets


Intersection: the membership functions of the intersection

of the two fuzzy sets A and B is defined as the minimum of

both

µ A Π B (x) = µ A (x) ^ µ B (x)


10/21

19

Fuzzy Sets


Complement: the membership functions of the complement

of fuzzy set A is defined as

20

Fuzzy Sets


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

21

Fuzzy Sets

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

x Є X.

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


12/21

23

Fuzzy Sets

Example

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 Sets

Normal 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 µ A (x)

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

maximum degree of membership less than one Degree of

Membership

Universe of Discourse0

1


13/21

25

Fuzzy Sets

Support of a Fuzzy Set

Support of a fuzzy set A

(written as supp(A)) is a

(crisp) set of points in X for

which µ A is positive supp(A)

= { x Є X | µ A (x)>0}

0

100

0

90

.8

70

1

60

.5

40

.1

30

0

10

.5 .8 0 Medium

80 50 20 Score

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

26

Fuzzy Sets

Convex Fuzzy Set

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

following µ A ( λ x1 + (1 – λ ) x2 ) ≥ min ( µ A ( x1) , µ A ( x2 )),

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


14/21

27

Fuzzy Sets

α -cut of a Fuzzy Set

α -cut is defined as a crisp set A α (or a crisp interval) for a

particular degree of membership, α : A α = [a α , b α ] , where α can take on values between [0,1]

28

Fuzzy Sets

α -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 .8 0 Medium

80 50 20 Score

B 0.8 = Medium score 0.8 = { 50, 60, 70}


15/21

29

Fuzzy Sets

Fuzzy 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 Sets

Fuzzy Numbers


16/21

31

Fuzzy Sets

Resolution Principle

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

µ A (x) = α ^ µ A α (x); x Є X

This means that a fuzzy set can be decomposed into α A α , α

Є [0, 1].

X

µ A(x)

1

α2

α1

α2 Aα2

α1 Aα1

Aα2 Aα1

32

Fuzzy Sets

Resolution 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


17/21

33

Fuzzy Sets

Representation 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 α A α .

A = U α A α

X

µ A(x)

1

α2

α

1

Aα2 Aα1

34

Fuzzy Sets

Representation Theorem: Example

If we are given: A 0.1 = {1, 2, 3, 4, 5}, A 0.4 = {2, 3, 5}, A 0.8 =

{2, 3}, and A 1 = {3}

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

1].

A = 0.1 A 0.1 + 0.4 A 0.4 + 0.8 A 0.8 + 1 A 1

= 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


18/21

35

Fuzzy Sets

Extension 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 Sets

Extension 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)

x B (y) = x f(A) (y)

µ B (y) = V µ f(A) (y); y=f(x)


19/21

37

Fuzzy Sets

Extension 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

µ B (2) = V [ µ A (0) ] = 1

µ B

(6) = V [ µ A

(-1), µ A

(1) ] = 1

µ B (10) = V [ µ A (-2), µ A (2) ] = 0

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

38

Fuzzy Sets

Extension 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 = µ 1 /x 1 + µ 2 /x 2 + µ 3 /x 3 + ……., the extension principle

states: ƒ(A) = ƒ(µ 1 /x 1 + µ 2 /x 2 + µ 3 /x 3 + …….) = µ 1 /ƒ (x 1 )+

µ 2 /ƒ(x 2 ) + µ 3 /ƒ(x 3 )+ …….

Or the resulting set B = µ A (x 1 )/y 1 + µ A (x 2 )/y 2 + ……

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

element y of V, then the max membership is taken


20/21

39

Fuzzy Sets


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) = x 2 , 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 Sets


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}


21/21

41

Fuzzy Sets


0.1

-3 -2 -1 0 1 2 3

0.4

0.9

0.3

0.8

0.9

-3 -2 -1 0 1 2 3

0.3

0.8

ƒ (x) = x2 – 3

Documents

AI FuzzySets