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}

So Azadeh,

if a fuzzy set A = {(x,0.1),(y,0.5),(z,0.88),(a,0.4) the 0.4 level set (Crisp set) contains

A = {y,z,a} . Am i right ?

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.

Thanks Azadeh ,

Your answer is perfect and clear my doubts

thanks

Other name of level set is Alpha-cut.

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 Poonam,Pushpinder , Azadeh

Thanks for your valuable advise

Please explain The difference between Fuzzy Relations and Crisp relations?

I only notify the Member ship value thing . please explain in detail

Thanks,

Anes

Dear Anes ; Plz see the following link

http://mazsola.iit.uni-miskolc.hu/DATA/diploma/brutoczki_kornelia/fuzzy13.html

it is the  ordinary set corresponding to the fuzzy set

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.