So, welcome to the Lecture Number 43 of Fuzzy
Sets, Logic and Systems and Applications.
In this lecture, we will discuss the Concentration
and Dilation of fuzzy sets and also we will
discuss a composite linguistic terms and the
related examples.
So, here let us first have the concentration
and then the dilation.
Concentration basically is nothing but its
a actually the concentration of a fuzzy set.
And when we say fuzzy set it means that we
have some linguistic value and which is characterized
by a fuzzy set or which is represented by
a fuzzy set.
Let us say A, then of course, we will have
its membership value or membership function
in case of continuous linguistic value. So,
we’ll have mu A of x. In case of x, we have
the generic variable. So if let us say we
have A which is a linguistic value, and when
we say linguistic value it means in fuzzy
system, we represents a linguistic value by
a suitable fuzzy set.
So we can say that it is nothing but a suitable
fuzzy set. So, if we have a linguistic value
which is represented by a suitable fuzzy set
A and if we are interested in having the concentration
of a fuzzy set A, so this concentration of
fuzzy set is nothing but A of k. So concentration
of a fuzzy set A is normally represented by
here, A raised to the power within bracket
k.
And if it is a continuous fuzzy set which
is being concentrated, then of course, this
will be represented by the integral sign and
then we have the universe of discourse capital
X and then we have mu A of x and since we
have the A of k so this k will be here as
well. So, this way the concentration of A
is represented by the expression A of k A
raised to the power within bracket k.
And similarly, if we have a discrete fuzzy
set which is being concentrated, we will use
sigma and then we use again the suitable universe
of discourse. So, in this case we have the
universe of discourse as capital X. So, for
continuous and discrete fuzzy sets, we have
the concentrated versions of these, the concentrated
fuzzy sets. So in summary, a concentration
of any fuzzy set basically is nothing, but
we get another set which has its membership
values or its membership functions raised
to the power k.
So, rest other things remains the same. Which
we can see here that we simply we raise the
power of membership function in case of continuous
fuzzy set and we raise the power of membership
values in case of discrete fuzzy sets. And
as I mentioned, rest other things will remain
the same.
So here as I mentioned, the power of membership
function is raised by k. So, k is here some
number. In general, we write k for concentration.
But for normal concentration are mentioned
here that when we do not mention any value
of k then normally, the for simpler concentration
the value of k can be 2 and the value of k
can be any value more than 1.
So if nothing has been mentioned, then we
can simply write the value we can take the
value of k as 2. So when we take a normal
concentration, we have here we have taken
k is equal to 2 and when we substitute the
k is equal to 2, what we are getting is here
for continuous membership concentration.
So, if we have a fuzzy set A which is a continuous
fuzzy set and if we are interested in finding
the concentration of fuzzy set A, then we
simply write here A raised to the power 2
within bracket. Why within bracket? Because
we are not exactly squaring the fuzzy set
A, we are only squaring here the membership
values or membership functions of the fuzzy
set A.
So here, this way this A raised to the power
2 within bracket is equal to the integration
sign and then we have the universe of discourse
and mu A of x raised to the power 2 oblique
x. So, this is going to be our concentrated
fuzzy set and when we have a discrete fuzzy
set. So for discrete fuzzy set, everything
remain the same except the summation in place
of the integration sign. So, this way we have
understood that what is the concentration
of a fuzzy set.
So once again, I would like to tell you that
if we are interested in finding the concentration
of any fuzzy set, we will simply write the
fuzzy set and we will try to find the value
of k if the value of k is given here, the
when we are interested in concentration, the
value of k will always be greater than 1.
This has to be noted, for concentration.
So, we will first look for the value of k
and if the value of k has been given, then
we will use the value of k which will be more
than 1. So we will use that, but if the value
of k is not mentioned, then we will go for
the normal concentration. Means, we will take
the value of k as 2. So, as we have done here.
So, concentration is very simple and we simply
use the value of k which is more than 1 and
this value of k basically increases the power
of the membership function for continuous
fuzzy set or the power of membership value
for discrete fuzzy sets.
And here, the notion of this concentration
is nothing but, when we concentrate any fuzzy
set whatever fuzzy set that we have here let
us say we have a fuzzy set A, let us say this
is a fuzzy set A, and if we are interested
in concentrating this fuzzy set so this the
concentrated fuzzy set will be something like
this, depending upon the value of k. So, I
can say this is my concentrated discrete fuzzy
set.
So, I can write here that this is a concentrated
fuzzy set A. So what do we see here? What
basically do we see here is that, when we
go for the resulting fuzzy set that is a concentrated
fuzzy set gets a squeezed. Means, they spread
gets reduced. So that needs to be understood
here, that whenever we concentrate any fuzzy
set, its spread is going to get reduced.
Let us take couple examples on the concentration
of fuzzy sets to understand the concept better.
So here we have a simple discrete fuzzy set
A, a discrete fuzzy set A with a universe
of discourse 1 to 5 and here we are interested
in the concentration of A and k has not been
given to us. So, obviously, we will have to
go for the value of k is equal to 2. So, I
am writing here since the value of k is not
given, so we will go for the normal concentration
and this means we would take the value of
k is equal to 2.
So, when we do this, here the concentration
of a discrete fuzzy set A, this can be written
as this equation. So, A raised to the power
within bracket 2 is equal to summation over
the universe of discourse capital X mu A of
x raised to the power 2. So the k comes here.
Here, k is equal to 2. So, the membership
values of this discrete fuzzy set will get
squared.
So when we go ahead, the we write the symbol
of the concentration as CON, so concentration
of A basically is equal to the summation over
X which is the universe of discourse, and
then mu A of x raised to the power 2 oblique
x and this is going to be equal to 0.1 raised
to the power 2 oblique 2 plus 0.7 raised to
the power 2 oblique 3 plus 0.8 raised to the
power 2 oblique 4 plus 1 raised to the power
2 oblique 5.
So we are doing nothing except we are squaring
the membership values of the corresponding
generic values of x. So, when we are squaring
this what we are getting we see here that,
we are getting 0.01 here. When we square
0.1 and we are getting 0.49 when we square
0.7, we are getting 0.64 when we square 0.8
and we are getting 1 when we are square 1.
So this way, we are getting a new expression
of the discrete fuzzy set which is here, corresponding
to the generic variable values, we are getting
the modified values of the membership.
So, we get the concentration of the fuzzy
set which we have taken as a discrete fuzzy
set. So, we can write here the concentration
of A is nothing but 0.01 oblique 2, then 0.49
oblique 3, then 0.64 oblique 4 then 1 oblique
5. So this is what is the new fuzzy set as
a result of the concentration of the discrete
fuzzy set that was given to us. All right,
so this was the example for the discrete fuzzy
set or I would say the concentration of discrete
fuzzy set.
Now, let us take an another example for the
concentration of a continuous fuzzy set. So,
if we have a linguistic term which is defined
by a suitable fuzzy set let us say it is named
as a Bright. So, fuzzy set for Bright. So,
fuzzy so, Bright is a fuzzy set, Bright is
a represented by a suitable continuous fuzzy
set and the universe of discourse here is
from 0 to 50. So, I can write here linguistic
term Bright is represented by a suitable fuzzy
set here.
Where, the membership of this Bright fuzzy
set is represented by mu Bright of x here
and this is nothing, but a gaussian function.
So, this is represented by gaussian x semi
colon 20 comma 5. And this is nothing but
the exponential of minus 1 by 2 into x minus
20 by 5 whole raised to the power 2. Where,
this 5 is nothing but the standard deviation.
And this 20 is nothing, but the mean.
So, this is the standard deviation and this
20 is the mean. Now, here we have been asked
to concentrate this fuzzy set Bright and Bright
is a continuous fuzzy set. So, let us do that.
So, we’ll write the concentration of Bright
fuzzy set by the CONs of Bright, the concentration
write concentration and then we will write
Bright like this.
So, when we do this, let us first take this
fuzzy set which has been given to us, let
us first understand the continuous fuzzy set
that is for the linguistic term Bright. So,
when we plot the fuzzy set here, the Bright
fuzzy set, this looks like this and this is
nothing, but the gaussian membership function.
So, we have the fuzzy set which is for Bright.
And this Bright has the membership function
as the gaussian with its mean 20 and its standard
deviation 5. We can see here, is 20. So, we
can represent the Bright fuzzy set like this.
Now, let us concentrate this fuzzy set Bright
and as I already mentioned that we can write
the CON of Bright means, the concentration
of Bright like this. And this can also be
written as Bright raised to the power within
bracket 2 and then this is again going to
be equal to since this is continuous fuzzy
set so, we will use the integral sign to represent
this fuzzy set.
So, integral sign over capital X which is
the universe of discourse, and then we will
write the mu Bright of x raised to the power
2. Here, this power 2 comes because we have
the normal concentration which is k is equal
to 2 and then oblique x. So initially, for
Bright we had simply the mu Bright of x. But
when we are concentrating the fuzzy set Bright,
then we will have to raise the power of the
membership function by k. And k since is not
given in this example so, we can take k is
equal to 2.
If k has been given, then we will use the
same value of k and this k since we are concentrating
the fuzzy set so the value of k is going to
be always more than 1. So here we are taking
k is equal to 2, we are squaring the membership
function. So, when we do that the whole fuzzy
set can be written as the integration sign
X and then within bracket the membership function
which is e raised to the power 1 by 2 into
x minus 20 over 5 raised to the power whole
raised to the power 2.
And then again whatever is here, as the membership
function is squared. Means raised to the power
2 and then we have the oblique x. So, this
is CON Bright, again.
Or in other words, we can write here the bright
and then within bracket 2. And here please
note that we have used k is equal to 2 only.
We have used the small k is equal to 2. So,
let us not get confused with the value of
k, as I have already mentioned if no value
of k has been given, then we will simply use
the value of k as 2 for concentration. So
when we plot the concentration of Bright,
means the fuzzy set which has come out of
the concentration of Bright, we get this fuzzy
set.
So, which is represented by the red color
here. So, this fuzzy set is the concentration
of Bright.
I can write it either this way or the CON
Bright. And this blue plot is for the Bright
fuzzy set. So, we can clearly see that when
we concentrate any fuzzy set, we get its spread
reduced or squeezed. So, here if we once again
go for further concentration of the concentrated
Bright fuzzy set, we will further get this
spread reduced. So, here the concentration
basically helps us in reducing the fuzziness
the uncertainties that is involved in the
fuzzy representation.
So, concentration of any fuzzy set basically
gives us reduces spread than that of the original
spread of the fuzzy set that was taken for
concentration. So, with this the discussion
on the concentration of a fuzzy set is over.
And in the next lecture we will discuss the
dilation and the composite linguistic terms
with some suitable examples.
Thank you.
