fuzzy sets - basic definitions1 classical (crisp) set a collection of elements or objects x x which...

Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 11

Classical (crisp) setClassical (crisp) setA collection of elements or objects A collection of elements or objects xxXX

which can be which can be finitefinite, , countablecountable, or , or overcoovercountableuntable..

A A classical setclassical set can be described in two can be described in two way:way:Enumerating (list) the elementsEnumerating (list) the elements ；； describindescribin

g the set analyticallyg the set analyticallyExample: stating conditions for membership --- Example: stating conditions for membership ---

{x|x{x|x5}5}Define the member elements by using theDefine the member elements by using the cc

haracteristic functionharacteristic function, in which 1 indicates , in which 1 indicates membership and 0 nonmembership.membership and 0 nonmembership.


Fuzzy setFuzzy set

If If XX is a collection of objects denoted is a collection of objects denoted generically by generically by xx then a fuzzy set in then a fuzzy set in XX is a is a set of ordered pairs: set of ordered pairs:

A~

XxxxA A ~,~

xA~ is called the membership function or grade of membership of x in A~


Example 1Example 111

A realtor wants to classify the house he A realtor wants to classify the house he offers to his clients. One indicator of offers to his clients. One indicator of comfortcomfort of these houses is the of these houses is the number number of bedroomsof bedrooms in it. Let in it. Let X={1,2,3,…,10}X={1,2,3,…,10} be be the set of available types of houses the set of available types of houses described by described by x=number of bedrooms in x=number of bedrooms in a housea house. Then the fuzzy set . Then the fuzzy set ““comfortable type of house for a 4-comfortable type of house for a 4-person familyperson family “may be described as “may be described as


Example 1Example 122

={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)}

A~


Example 2Example 2

=“real numbers considerably larger than 1=“real numbers considerably larger than 10”0”A~

XxxxAA

~,~

where

10 ,101

10 ,012~xx

xx

A


Other approaches to denote Other approaches to denote fuzzy setsfuzzy sets

1. Solely state its membership function.1. Solely state its membership function.

...//~22~11~ xxxxA AA 2.

n

iiiAxx

1

~ / or x Axx /~


Example 3Example 3

=“integers close to 10”=“integers close to 10”A~

A~ =0.1/7+0.5/8+0.8/9+1/10+0.8/11+0.5/12+0.2/13


Example 4Example 4

=“real numbers close to 10”=“real numbers close to 10”A~

x

xA /

101

1~2


Normal fuzzy setNormal fuzzy set

If If 1sup ~ xAX

the fuzzy set is called normal.A~


Supremum and InfimumSupremum and Infimum For any set of real numbers R that is For any set of real numbers R that is bounded abounded a

bovebove, a real number r is called the , a real number r is called the supremumsupremum of of R iffR iff r is an upper bound of Rr is an upper bound of R no number less than r is an upper bound of Rno number less than r is an upper bound of R r=sup Rr=sup R

For any set of real numbers R that is For any set of real numbers R that is bounded bbounded belowelow, a real number s is called the , a real number s is called the infimuminfimum of R of R iffiff s is a lower bound of Rs is a lower bound of R no number greater than s is a lower bound of Rno number greater than s is a lower bound of R s=inf Rs=inf R


範例範例 11

設設 X={a,b,c,d}X={a,b,c,d} ，給定一，給定一偏序集偏序集 (A(A ， ≤， ≤ )) ，令，令 ≤≤={(a,a), (b,b), (c,c), (d,d),={(a,a), (b,b), (c,c), (d,d), (a,b), (a,c), (b,d), (a,d)} (a,b), (a,c), (b,d), (a,d)} ，，則則 c,dc,d 為為 AA 的上界，但沒的上界，但沒有上確界，有上確界， aa 是是 AA 的下的下界也是界也是 AA 的下確界的下確界 (infA(infA=a) =a) 。。

a

b

d

c

Hasse diagram

Maximal element

Maximal element

First and Minimal element


範例範例 22

設設 X={a,b,c,d}X={a,b,c,d} ，給定一偏序，給定一偏序集集 (A(A ， ≤， ≤ )) ，設，設 ≤≤ ={(a,a), ={(a,a), (b,b), (c,c), (d,d), (a,c), (a,(b,b), (c,c), (d,d), (a,c), (a,d), (b,c), (b,d)} d), (b,c), (b,d)} ，令，令 H={c,H={c,d} d} ，則，則 HH 沒有上界，且沒有上界，且a,ba,b 都是都是 HH 的下界。令的下界。令 KK={a,b,d} ={a,b,d} ，則，則 dd 是是 KK 的的上確界上確界 (supK=d) (supK=d) ，但，但 KK沒有下界。沒有下界。

a

b

c d

Hasse diagram


SupportSupport

The The supportsupport of a fuzzy set , of a fuzzy set ,S(S( )), is the cri, is the crisp set of all xsp set of all xX such that X such that 0~ x

AA~A~


Example 1Example 1

““comfortable type of house for a 4-comfortable type of house for a 4-person familyperson family “may be described as “may be described as

={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)}

A~


Support of Example 1Support of Example 1

S( )={1,2,3,4,5,6}S( )={1,2,3,4,5,6}A~


αα - level set - level set((αα- cut)- cut)

The crisp set of elements that belong to fuzThe crisp set of elements that belong to fuzzy set at least to the degree zy set at least to the degree αα..A~

xXxA A~

xXxAA~

'

is called “strong strong αα-level set-level set” or “strstrong ong αα-cut-cut”.


Example 1Example 1


={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)}

A~


αα cut cut of Example 1 of Example 1

A~ ={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)}

A0.2={1,2,3,4,5,6}

A0.5={2,3,4,5}

A’0.8={4}

A0.8={3,4}

A1={4}


Convex crisp set Convex crisp set AA in in nn

For every pair of points For every pair of points r=(rr=(rii|i|iNNnn)) and and ss

=(s=(sii|i|iNNnn)) in in AA and every real number and every real number

λλ[0,1],[0,1], the point the point t=(t=(λλrrii+(1-+(1-λλ)s)sii|i|iNNnn)) is a is a

lso in lso in AA..


Convex fuzzy setConvex fuzzy set

A fuzzy set is convex if A fuzzy set is convex if A~

1,0,,,, min1 212~1~21~ XxxxxxxAAA


Cardinality Cardinality | || |A~

Xx

x AAdxxxA ~~

~

A~For a finite fuzzy set

X

AA

~~ Is called the relative cardinality of A~


Example 1Example 1


={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)}

A~


Cardinality of example 1Cardinality of example 1

A~ ={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)}

X={1,2,3,4,5,6,7,8,9,10}

A~| |=0.2+0.5+0.8+1+0.7+0.3=3.5

|| ||=3.5/10=0.35A~


Basic set-Theoretic operations Basic set-Theoretic operations (standard fuzzy set operations)(standard fuzzy set operations)

Standard Standard complementcomplementStandard Standard intersectionintersectionStandard Standard unionunion


Standard complementStandard complement

The membership function of the complement The membership function of the complement of a fuzzy set of a fuzzy set A~

XxxxAAc

,1 ~~

A~ cA


Standard intersectionStandard intersection

The membership function of the intersection The membership function of the intersection xC~BAC ~~~

Xxxxx BAC ,, min ~~~

A~ B

C


Standard unionStandard union

The membership function of the union The membership function of the union xD~BAD ~~~

Xxxxx BAD ,, max ~~~

A~ BC


Standard fuzzy set operationsStandard fuzzy set operationsof example 1of example 111

A~

={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)}

=“comfortable type of house for a 4-person-family”

B~

={(3,0.2), (4,0.4), (5,0.6), (6,0.8), (7,1), (8,1),(9,1),(10,1)}

=“large type of house”


Standard fuzzy set operationsStandard fuzzy set operationsof example 1of example 122

cB~ ={(1,1),(2,1),(3,0.8),(4,0.6),(5,0.4),(6,0.2)}

BAC ~~~ ={(3,0.2),(4,0.4),(5,0.6),(6,0.3)}

BAD ~~~ ={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.8),(7,1),(8,1),(9,1),(10,1)}

fuzzy sets - basic definitions1 classical (crisp) set a collection of elements or objects x x which...

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