Extension principle of Fuzzy Set(English)

Hello friend in this chapter we are going to discuss about the extension principle of fuzzy set with example.In the previous lecture we have studied about the basic operation on the fuzzy sets you can read it from here -https://electricalvideolibrary.blogspot.com/p/fuzzy-logic.html
So lets start Extension principal:-
Lets take an Fuzzy set A:
A={(-1,0.5),(0,0.8),(1,1),(2,0.4)}

Now suppose somebody told you to apply the function F(x)=x^2 (^ means power) on fuzzy set A and form a new fuzzy set B.

So here you can easily square the element of Fuzzy set A but what will you do with membership value?,i.e
 (-1)^2=1,(0)^2=0,1^2=1 and 2^2=4

so fuzzy set B can be written as following as we do not know the membership value so we will leave it as it is-
B={(1,Ub(1)),(0,Ub(0)),(1,Ub(1)),(4,Ub(4))}
we can rewrite as the elements are repeating -
B={(0,Ub(0)),(1,Ub(1)),(4,Ub(4))}

This problem of membership can be solve by the extension principle:-

Extension Principle:-
With the extension principle you can extend the mathematical impression like function and algebra of the crisp domain(Normal set) to the Fuzzy Domain.
As in the above case you can see we are applying the function F(x)=x^2 of the crisp domain to a fuzzy domain A.
That is why it is known as extension principle  as it extend or pushes the normal mathematics in to the fuzzy domain.

How to apply extension principle?

So for the application of extension function you will be given a function like in our above example it is  F(x)=x^2 but it can be any function like inverse,logarithmic,trignometric,algebric,exponential,etc.
and Fuzzy Set as it is A in our above example but there can be more than one set as well depending upon the number of input your function take.

Now you can easily apply the F(x) on the elements of the given fuzzy set but not on the membership function as we did in the above case and we squared the element of fuzzy set A.

Now with the help of extension principle we have to find the degree of membership of the required Fuzzy set in above case-the degree of membership of B.

Ub(y)=Max{UA(x)} where x=F inverse(y)
Here First we have to find the inverse of the element in Fuzzy Set B so for the above example you can see (-1)^2=1 and (1)^2=1 so this is the function so inverse of 1 will be -1 and 1 Similarly the inverse of 0 will be 0 and inverse of 4 will be 2 in the above question.

After finding the inverse you have to write the degree of membership of the inverse.Now suppose if there is only one inverse then degree of membership will be one and that will be the desired degree of membership in B but if there is more than one inverse then you have to choose the degree of membership of that inverse which has the highest value or Maximum value as denoted in equation.

lets solve it for our above example to understand it more-
Ub(0)=Max{UA(0)}=Max{0.8}=0.8                 0=F inverse(0)
Ub(1)=Max{UA(-1),UA(1)}=Max{0.5,1}=1     {-1,1}=F inverse (1)
Ub(4)=Max{UA(2)}=Max{0.4}=0.4                  2=F inverse(4)

So the final Fuzzy set B will be-
B={(0,0.8),(1,1),(4,0.4)}

So in this way you can proceed for the other functions as well and this type of operation can also be perform on two or more than two set.