I am a new student in Fuzzy sets & Fuzzy relations. I can found that the Level of a fuzzy set as
ΛA = {α/μA(x) = α for some x belongs to X} , Please provide an example of this with a set.
a-level sets of a fuzzy set shows objects which have the membership function more than a. This set is a crisp set.
If A={(x,0.1) , (y,0.5) , (z,0.88)}, then 0.4-lelel set A={y,z}
IN FACT there are 2 definitions of a-level sets of a fuzzy set:
1- {y in X; membership degree (y) > or = a}
2-{y in X; membership degree (y) > a}
the second one is often called strong a-level set.
so it depends on you.
Level cuts are very useful tool to study the theory of fuzzy sets and their applications. It convert the whole fuzzy system to crisp system ( i.e. classical set theory system) which was given by George Cantor.
Dear Anes ; Plz see the following link
http://mazsola.iit.uni-miskolc.hu/DATA/diploma/brutoczki_kornelia/fuzzy13.html
Please click the link to have more information regarding fuzzy sets
https://www.academia.edu/1899926/Fuzzy_Mathematics_and_Its_Importance_in_Technology
Plz go through the following my ppt.
https://www.academia.edu/1899926/Fuzzy_Mathematics_and_Its_Importance_in_Technology
The level set is a horizontal representation of alpha cuts but sometimes technically incorrect as with alpha cuts we go with the notion of >= and with strong alpha cuts we use > (strictly greater).
Level set also known as α-cut is a crisp set for a fuzzy set consisting of all members of universe of discoure have membership value greater than or equal to α
If X={a,b,c,d} and FS A={(a, 0.2),(b, 0.8),(c, 0.9)}, then alpa-cut are
A0.2={a,b,c}, A0.8={b,c}, A0.9={c}. Note that A=UaAa where a belong to [0,1]
The strong alpha-cut is A0.2+={b,c} etc.
The value of $\alpha$ which explicitly shows the value of the membership function is in the range of [0,1]. Thus the “level set” is obtained as the collection of all levels in a set.
For detail the book of (First course on fuzzy theory and applications) by KH Lee
Level set for a fuzzy set is the set of all alpha cut of that fuzzy set.
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