Standard Operation on Fuzzy Sets,Alpha Cut Set,Equivalent Fuzzy Set(English)

Hello friends! In this blog we are going to discuss following things about the Fuzzy Set:
1.Alpha Cut Set or Level Set
2.Equivalent Fuzzy Set
3.Subset of Fuzzy Set
4.Standard Operation on Fuzzy Sets-Compliment,Union,Intersection

इसी  ब्लॉग को हिंदी में पढ़ने के लिए यहां:- क्लिक करे 
Lets start by taking one by one each.
1.Alpha Cut Set or Level Set:
Alpha cut set or level is the set of those fuzzy element whose degree of membership is greater than or equal to Alpha where Alpha is the number between zero and one.Alpha cut set or level set is not a fuzzy set but it is a normal set because here we only write the element not there degree of membership.Lets understand its notation and its example.

Fig 1.Alpha Cut Set or Alpha level set

Here we see that in figure 1 that the alpha set is written for fuzzy set A where the value of Alpha is 0.6 and only those element are kept in to the set whose degree of membership is greater than or equal to 0.6.

2.Equivalent Fuzzy Sets:
Two fuzzy sets A and B are said to be equivalent when their degree of membership are same for all the corresponding element.Its notation can be given in this way:-
It can be read it out in this way-Two fuzzy Set A and B are equivalent when Mu A is equal to Mu B for all x belong to X.Here Mu is Degree of membership,x is the fuzzy set element and X is the universal set.

3,Subset of Fuzzy Set:
Fuzzy set A is set to be the Subset of fuzzy set B If the degree of membership for any element in the universal set is less in fuzzy set A than the fuzzy set B.Lets see its notation.

Fig 2.Subset of Fuzzy Set

Standard Operation On Fuzzy Sets

1.Compliment of Fuzzy Set:

Complement of a fuzzy set A can be found by subtracting the degree of membership of each element from 1 and replacing the result with their existing degree of membership.Lets take an example to understand it well..  

Fig 3.Compliment of Fuzzy Set

2.Union of Fuzzy Sets:

To find the union set of two fuzzy set we first write the element and then secondly we write the maximum value of degree of membership from both the set for that particular element,we repeat this process for all the elements.Lets understand it by an example..

Fig 4.Union and Intersection of Fuzzy Set

lets look at the element 10 in fuzzy set A,B and the Union Set.Here you can see its degree of membership in A and B is 0.3 and 0.6 respectively so in the union set we have written its degree of membership as 0.6 as this is the maximum value between the two set.Similarly one can extend the same logic for all the element. 

3.Intersection of Fuzzy Sets:

 To find the union set of two fuzzy set we first write the element and then secondly we write the minimum value of degree of membership from both the set for that particular element,we repeat this process for all the elements.You can understand it  with the same example as in fig 4.Its just like Union but the only difference is maximum is replaced by the minimum.