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Classical (crisp) set

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- 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.

- Enumerating (list) the elements；describing the set analytically

Fuzzy sets - Basic Definitions

is called the membership function or grade of membership of x in

- 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

- 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)}

Fuzzy sets - Basic Definitions

- =“real numbers considerably larger than 10”

where

Fuzzy sets - Basic Definitions

or

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

- =“integers close to 10”

Fuzzy sets - Basic Definitions

- =“real numbers close to 10”

Fuzzy sets - Basic Definitions

the fuzzy set is called normal.

- If

Fuzzy sets - Basic Definitions

- 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

- 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

- 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

- 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)}

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

Fuzzy sets - Basic Definitions

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

Fuzzy sets - Basic Definitions

- 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)}

- “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}

Fuzzy sets - Basic Definitions

- 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

- A fuzzy set is convex if

Fuzzy sets - Basic Definitions

For a finite fuzzy set

Is called the relative cardinality of

Fuzzy sets - Basic Definitions

={(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 “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

Fuzzy sets - Basic Definitions

- Standard complement
- Standard intersection
- Standard union

Fuzzy sets - Basic Definitions

- The membership function of the complement of a fuzzy set

Fuzzy sets - Basic Definitions

- The membership function of the intersection

Fuzzy sets - Basic Definitions

- 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)}

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)}

Fuzzy sets - Basic Definitions