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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 sets - Basic DefinitionsFuzzy sets - Basic Definitions 22
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~
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 33
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
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 44
Example 1Example 122
={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)}
A~
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 55
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
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 66
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 /~
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 77
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
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 88
Example 4Example 4
=“real numbers close to 10”=“real numbers close to 10”A~
x
xA /
101
1~2
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 99
Normal fuzzy setNormal fuzzy set
If If 1sup ~ xAX
the fuzzy set is called normal.A~
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 1010
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
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 1111
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
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 1212
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
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 1313
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~
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 1414
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~
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 1515
Support of Example 1Support of Example 1
S( )={1,2,3,4,5,6}S( )={1,2,3,4,5,6}A~
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 1616
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”.
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 1717
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~
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 1818
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}
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 1919
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..
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 2020
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
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 2121
Cardinality Cardinality | || |A~
Xx
x AAdxxxA ~~
~
A~For a finite fuzzy set
X
AA
~~ Is called the relative cardinality of A~
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 2222
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~
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 2323
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~
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 2424
Basic set-Theoretic operations Basic set-Theoretic operations (standard fuzzy set operations)(standard fuzzy set operations)
Standard Standard complementcomplementStandard Standard intersectionintersectionStandard Standard unionunion
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 2525
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
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 2626
Standard intersectionStandard intersection
The membership function of the intersection The membership function of the intersection xC~BAC ~~~
Xxxxx BAC ,, min ~~~
A~ B
C
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 2727
Standard unionStandard union
The membership function of the union The membership function of the union xD~BAD ~~~
Xxxxx BAD ,, max ~~~
A~ BC
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 2828
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”
Fuzzy sets - Basic DefinitionsFuzzy sets - Basic Definitions 2929
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)}