Hey everyone, welcome back to fuzzy logic lectures. In the previous video,
we learned about the different similarity methods.
Which is the cosine amplitude method and the max-min method. In this video,
we'll be focusing on features of membership functions and defuzzification to crisp sets
Let's start the video with the features of membership functions. Let's consider a fuzzy set A, which is depicted by a graph like this
So you can see you have μa(X) and X
Now the main features which are going to discuss is - core, support and boundary
Let's start with the first feature, which is the core. The core
comprises those elements X of the universe such that μa(X) is equal to 1
This means that the core comprises of the values where the membership function of the values is equal to 1
so you can see that in this particular region. ie.,
in this region, you can see that the membership value is equal to 1
So this particular region is called as the core of the membership function
The next feature is called a support. The support comprises those elements X of the universe
such that μa(X) is greater than or equal to 0
which means that the membership value of X should be greater than or equal to 0.
As you can see that this whole region, you can see here, μa(X) is equal to 0. Here
μa(X) is equal to 0 and in all this region μa(X) is greater than 0
Therefore this whole region, that is, from here
to here, this whole region
comprises of the support
Next we have boundaries. The boundaries comprise those elements X of the universe X
such that 0 is less than μa(X) which is less than 1
which means that the boundary it consists of those elements where the membership value lies between 0 & 1
And as you can see from the graph this particular region where 0 is less than X which is less than 1 and this region
0 less than X less than 1 these two regions, they constitute the bound. So this is a boundary and
this is also a boundary
Boundary can be given as the support minus the core
So these are the three main features of the membership function where you have the core, the support and the boundaries
next let's move on to normal fuzzy set. A
normal fuzzy set is one whose membership function has at least one element X in the universe whose membership value is unity
Which means that if you have a fuzzy set A,
atleast one of the values, one of the membership value should be equal to one. As you can see in this graph
Here you have the fuzzy set a and this particular value its membership value is equal to one
So you can say that this is a normal fuzzy set.
Because it has at least one value which is equal to one  In case of the fuzzy set A does not have any
Membership value which is equal to 1, then that fuzzy set is called as a sub normal fuzzy set
Okay? Subnormal
fuzzy set and
it can be depicted by this graph as this. So you can see that you have the fuzzy set a and
there are no values which is equal to one
So this is called as a sub normal fuzzy set. Next, let's move on to convex fuzzy set
Suppose by depicting the fuzzy set a with a graph like this where X Y Z are the elements, then we can say that, if
for any elements X Y and Z in a fuzzy set a the relation X is less than Y less than Z that implies that
μa(Y) should be greater than or equal to minimum of  μa(X) and μa(Z),
then A is said to be a convex fuzzy set
That is the minimum of the membership values of X and Z should be less than or equal to the membership value of Y
Or you can say that a convex fuzzy set is described by a membership function
whose values are
strictly increasing or
whose values are strictly decreasing
Or whose values are strictly increasing
then decreasing as the
elements are increasing in value. So this is how you define a convex fuzzy set
and this is depiction of a convex fuzzy set because here as we can see that our membership values are first of all strictly increasing
then it is decreasing and the elements are increasing in the same order. If this condition is not satisfied,
then the fuzzy set is called as a non-convex fuzzy set, that is, a non
convex fuzzy set and we can depict a non-convex fuzzy set by
this.
As you can see in this graph,
the membership value of y is not greater than or equal to minimum of  μa(X) and  μa(Z).
Moreover, if you observe you can see that the value of A or the membership value of A is strictly increasing,
then is strictly decreasing but then again, we notice that it's again increasing and decreasing
So this is not how it's supposed to be for a convex fuzzy set.
Either it can strictly increase or it can strictly decrease or it can strictly increase and decrease as the elements increase in value
So you can say that this fuzzy set is a non convex fuzzy set
Let's move on to crossover points
The crossover points of a membership function
are defined as the elements in the universe
for which a particular fuzzy set A has values μa(X) is equal to 0.5
That is the membership value of a will be equal to 0.5
So if you're representing the fuzzy set A as a graph, you can see that this particular point
It is equal to 0.5. That is it has a membership value equal to 0.5
So you can say that this point is called as the crossover point. This point is called as the crossover point.
And lastly we have the height of a fuzzy set
the height of a fuzzy set a is the maximum value of the membership function that this height of a is equal to
Maximum mu a of X that is when you take this particular graph
You can see that the maximum value of the membership value is equal to 1
so you can say that the height of this fuzzy set a is equal to 1 which means
This is the height of the fuzzy set
So these are the basic properties and features of the membership functions
the next topic which will discuss is the defuzzification the crisp sets now all of you know,
The fuzzification  is the process by which we convert a crisp set or a classical set into a fuzzy set
So defuzzification is when we convert a fuzzy set back to a crisp set
We're going to do this by a method called as the lambda cut method
We begin by considering a fuzzy set a and then we are going to define a lambda cut set a lambda
Okay where 0 is less than or equal to lambda less than or equal to 1
This is the criteria for the lambda cut set a lambda
then we can say that the set a lambda is a crisp set called as a lambda cut set of the fuzzy set a
Where a lambda is equal to X and the membership value of a of X should be greater than or equal to lambda
Let's take an example so you can understand it better
For example, we have a fuzzy set a and is given as 1 by a plus
0.9 by B plus 0.6 by C plus 0.3 by D plus 0.01 by E and 0 by F
Where 1 0.9. 0.6 0.3?
0.01 and 0 are the membership values of the elements a b c d e f
Respectively so now we are going to convert this fuzzy set into a crisp set by using the lambda cut method
so let's first take lambdas value as 1 and
According to this condition it is given that the membership value has to be greater than or equal to lambda
So when we look at a fuzzy set a only the element a has a membership value
which is greater than or equal to lambda, which is 1 so the new crisp set a
Will be equal to only a Y because only a has a membership value which is equal to 1 next
Let us take lambdas value as 0.9
So when lambdas value is taken as point 9 the membership value can be equal to 0.9 or can be greater than point 9
So we know that a and B both have a value which is greater than or equal to 0.9
So a new crisp set a will be having elements as a and B
Similarly when you take lambdas value is 0.6. Then the membership values have to be greater than or equal to 0.6
So here we know that a B and C have values greater than equal to 0.6
so a new crisp set has the values as a
B and C. So this is how you convert a fuzzy set into a crisp set by using the lambda cut method
similarly, you can put the value of lambda as point three zero point zero one or zero and then find out which all elements will
come in the new crisp set a
You can also express this relation in this form that is if lambda is equal to 0.9
Then you can write a is equal to 1 by a plus
1 by B plus 0 by C plus 0 by D
Plus 0 by E
Plus 0 by F
that is the membership values of a and B is given as 1 and all other elements is given as 0 which means that only
a and B belong to the crisp set a
Let's move on to lambda cut for fuzzy sets
So according to this let us define our lambda
Which is equal to X Y
where mu R of X Y should be greater than or equal to lambda as a lambda cut relation of the fuzzy relation are
since in this case
The fuzzy set R is a two dimensional array defined on the universes x and y any pair of X Y belonging to our lambda
Belongs the fuzzy set R with the strength of the relation greater than or equal to lambda
That is for a fuzzy set relation R, which is represented as a relation matrix
We can say that the ordered pair X Y should be greater than or equal to lambda
So to understand this better, let's take a simple example here
We have a fuzzy set R and it is a 3 into 3 relation matrix
So now we want to perform the lambda cut for this fuzzy set R
So let's take the value of lambda as point 8 now according to your condition
We have to check whether X Y value should be greater than or equal to lambda
So the first element 1 okay 1 is greater than or equal to 0.8
So a crisp set R is equal to 1 because 1 is greater than of equal to point it
Let us check four point eight since point eight is equal to 0.8
the value of point eight will be given as one because it fits the condition if
satisfy the condition
So the value is given as one as we all know crisp values or crisp sets can only take zero or one
One where it belongs to the set and zero if it does not belong to the set
So similarly when we check for point three, we know that point three is less than point eight
So the value of point three is given as zero similarly point eight is equal to one point three is equal to zero
Point nine since it's greater than point eight
It is given the value one point two is given as zero point is given as zero and zero
Is given as zero because zero is less than point eight. So this becomes our new crisp relation matrix
similarly, if you take the value of lambda as point three, then we'll get a new crisp set and that is given as
1 1 1 1 1 1 0
0 0
why because
Point 2 is less than point 3 and 0 is less than point 3
Less all the values are greater than or equal to point 3, so this becomes our new crisp relation matrix
So similarly you can take for any value of lambda and we can find out a crisp relation matrix
So this is how you do the lambda cut for fuzzy sets
I hope you all understood the concepts that were taught in this lecture
If you have any doubts, please feel free to ask in the comment section below in the next lecture
We are going to be discussing about the different methods for the defuzzification to scale us
Thank you for watching top early and have a nice day
