Classical (crisp) set

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# Classical (crisp) set - PowerPoint PPT Presentation

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|x5}

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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|x5}
• Define the member elements by using thecharacteristic function, in which 1 indicates membership and 0 nonmembership.

Fuzzy sets - Basic Definitions

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

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

• 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

Support
• The support of a fuzzy set ,S(), is the crisp set of all xX 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|iNn) and s=(si|iNn) in A and every real number λ[0,1], the point t=(λri+(1-λ)si|iNn) 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

• 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