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Classical (crisp) set. A collection of elements or objects x X which can be finite , countable , or overcountable . A classical set can be described in two way: Enumerating (list) the elements ； describing the set analytically Example: stating conditions for membership --- {x|x5}
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Classical (crisp) set • A collection of elements or objects xX which can be finite, countable, or overcountable. • A classical set can be described in two way: • Enumerating (list) the elements；describing the set analytically • Example: stating conditions for membership --- {x|x5} • Define the member elements by using thecharacteristic function, in which 1 indicates membership and 0 nonmembership. Fuzzy sets - Basic Definitions
is called the membership function or grade of membership of x in Fuzzy set • If X is a collection of objects denoted generically by x then a fuzzy set in X is a set of ordered pairs: Fuzzy sets - Basic Definitions
Example 11 • A realtor wants to classify the house he offers to his clients. One indicator of comfort of these houses is the number of bedrooms in it. Let X={1,2,3,…,10} be the set of available types of houses described by x=number of bedrooms in a house. Then the fuzzy set “comfortable type of house for a 4-person family “may be described as Fuzzy sets - Basic Definitions
={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)} Example 12 Fuzzy sets - Basic Definitions
Example 2 • =“real numbers considerably larger than 10” where Fuzzy sets - Basic Definitions
or Other approaches to denote fuzzy sets 1. Solely state its membership function. 2. Fuzzy sets - Basic Definitions
=0.1/7+0.5/8+0.8/9+1/10+0.8/11+0.5/12+0.2/13 Example 3 • =“integers close to 10” Fuzzy sets - Basic Definitions
Example 4 • =“real numbers close to 10” Fuzzy sets - Basic Definitions
the fuzzy set is called normal. Normal fuzzy set • If Fuzzy sets - Basic Definitions
Supremum and Infimum • For any set of real numbers R that is bounded above, a real number r is called the supremum of R iff • r is an upper bound of R • no number less than r is an upper bound of R • r=sup R • For any set of real numbers R that is bounded below, a real number s is called the infimum of R iff • s is a lower bound of R • no number greater than s is a lower bound of R • s=inf R Fuzzy sets - Basic Definitions
Maximal element Maximal element First and Minimal element a b d c Hasse diagram 範例1 • 設X={a,b,c,d}，給定一偏序集(A， ≤)，令 ≤={(a,a), (b,b), (c,c), (d,d), (a,b), (a,c), (b,d), (a,d)} ，則c,d為A的上界，但沒有上確界，a是A的下界也是A的下確界(infA=a) 。 Fuzzy sets - Basic Definitions
Hasse diagram a b c d 範例2 • 設X={a,b,c,d}，給定一偏序集(A， ≤)，設 ≤={(a,a), (b,b), (c,c), (d,d), (a,c), (a,d), (b,c), (b,d)} ，令H={c,d} ，則H沒有上界，且a,b都是H的下界。令K={a,b,d} ，則d是K的上確界(supK=d) ，但K沒有下界。 Fuzzy sets - Basic Definitions
Support • The support of a fuzzy set ,S(), is the crisp set of all xX such that Fuzzy sets - Basic Definitions
={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)} Example 1 • “comfortable type of house for a 4-person family “may be described as Fuzzy sets - Basic Definitions
Support of Example 1 • S( )={1,2,3,4,5,6} Fuzzy sets - Basic Definitions
α - level set(α- cut) • The crisp set of elements that belong to fuzzy set at least to the degree α. is called “strong α-level set” or “strong α-cut”. Fuzzy sets - Basic Definitions
={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)} Example 1 • “comfortable type of house for a 4-person family “may be described as Fuzzy sets - Basic Definitions
={(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} A0.8={3,4} A1={4} A’0.8={4} α cut of Example 1 Fuzzy sets - Basic Definitions
Convex crisp set A in n • For every pair of points r=(ri|iNn) and s=(si|iNn) in A and every real number λ[0,1], the point t=(λri+(1-λ)si|iNn) is also in A. Fuzzy sets - Basic Definitions
Convex fuzzy set • A fuzzy set is convex if Fuzzy sets - Basic Definitions
For a finite fuzzy set Is called the relative cardinality of Cardinality | | Fuzzy sets - Basic Definitions
={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)} Example 1 • “comfortable type of house for a 4-person family “may be described as Fuzzy sets - Basic Definitions
={(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} | |=0.2+0.5+0.8+1+0.7+0.3=3.5 || ||=3.5/10=0.35 Cardinality of example 1 Fuzzy sets - Basic Definitions
Basic set-Theoretic operations (standard fuzzy set operations) • Standard complement • Standard intersection • Standard union Fuzzy sets - Basic Definitions
Standard complement • The membership function of the complement of a fuzzy set Fuzzy sets - Basic Definitions
Standard intersection • The membership function of the intersection Fuzzy sets - Basic Definitions
Standard union • The membership function of the union Fuzzy sets - Basic Definitions
=“comfortable type of house for a 4-person-family” ={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)} =“large type of house” ={(3,0.2), (4,0.4), (5,0.6), (6,0.8), (7,1), (8,1),(9,1),(10,1)} Standard fuzzy set operationsof example 11 Fuzzy sets - Basic Definitions
={(1,1),(2,1),(3,0.8),(4,0.6),(5,0.4),(6,0.2)} ={(3,0.2),(4,0.4),(5,0.6),(6,0.3)} ={(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)} Standard fuzzy set operationsof example 12 Fuzzy sets - Basic Definitions
