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Fuzzy Set

A function UA :X-->[0,1] is called a membership function and the set A defined by it is called a Fuzzy set. The universe X is called referential set and [0,1] is called valuation set. The membership function UA of a set A can be denoted by A itself, then we get
A: X-->[0,1]
Suppose C(3,5) R2: Then the set S={P ∈ R2:Cp≤2} and the set T={P ∈ R2: P is a near point of C} is a fuzzy set.

Suppose M:X-->I defines a fuzzy set of “Middle Aged People” where X is the ages of human beings, such that
M(X)={0 when x≤20,(x-20)/15 when 20<x<35,1 when 35≤x≤45,(60-x)/15 when 45<x<60,0 when 60≤x

Suppose x ={a,b,c,d,e} and the fuzzy set f:X-->I is I defined by
f(a)=0.8,f(b)=0.4,f(c)=0,f(d)=0.1,f(e)=1
Then the fuzzy set can be written as,
f=(0.8,0.4,0,0.1,0)

Difference Between Classical Set And Fuzzy Set:

Classical Set:

• 1. A classical set is defined as collection objects, which share characteristics.
• 1. A classical set is a container that wholly includes or wholly excludes any given element. For example, the set of days of the week unquestionably includes Tuesday, Wednesday and Saturday. It just unquestionably excludes butter, liberty, shoe polish and so on.
• 1. Operation on classical sets-
     Union: AUB={x|x ∈ A or x ∈ B}
     Intersection: A∩B= {x|x ∈ A and x ∈ B}
     Complement: A= {x ∉ A,x ∈ X}
     Difference:
     A-B=A/B={x|x ∈ A and x ∉ B}
     = A-(A∩B)
• 1. Properties Of Classical Sets-
     Law of excluded middle :
     AUA ̅= X
     Law of Contradiction:
     A∩A ̅ = φ
     De Morgan's Law:
     |(A∩B) ̅ |=A ̅UB ̅, |(AUB) ̅ |= A ̅∩B ̅
• 1. Union, Intersection and complement with the figure:
     

Fuzzy Set

• 1. A function UA :X[0,1] is called a membership function and the set A defined by it is called a Fuzzy set. The membership function UA of a set A can be denoted by A itself, then we get
     A: X[0,1]
• 1. Fuzzy set theory permits gradual assessment of membership of elements in a set, described with the aid of a membership function-valued in the real unit interval [0,1].
• 1. The fuzzy set theory is an extension of classical set theory where elements have a degree of membership.
• 1. A fuzzy set theory is any set that allows its members to have a different degree of membership, called membership function in the interval [0,1]
• 1. Operations of a fuzzy set:
     Union: UAuB(X)=max[UA(X),UB(X)]
     =UA(X) ∨ UB(X)
     Intersection: UAuB(X)=min[UA(X),UB(X)]
     = UA(X) ^ UB(X)
     Complement: U ̅(X)=1-U(X)
• 1. The fuzzy set follows the same properties as classical set except for the law of excluded middle and law of contradiction.
     
     All the information are taken from the web.
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