Welcome to lecture number 42 of Fuzzy Sets,
Logic and Systems and Applications.
In this lecture we will discuss some examples
on Linguistic Hedges and then we will discuss
the Negation, Complement and Connectives.
So, let us take an example here to understand
this better.
So, if we have a linguistic variable bright.
So, we have a bright and as I already mentioned
that this bright is termed as a primary term,
this bright basically is a fuzzy value which
is represented by a fuzzy set.
So, if we have bright, simply bright here
and bright here is a discrete fuzzy set, this
is a discrete fuzzy set.
So, just to make you understand we have taken
this example.
So, let us now convert this fuzzy set here
which is characterized by a membership values
along with the corresponding generic variable
values 1, 2, 3, 4, 5.
So, let us now using the given bright a discrete
fuzzy set let us now find very bright and
then very very bright and then more or less
bright.
So, since we have been given here the primary
discrete set we can write primary discrete
fuzzy value or fuzzy set all these names can
be used interchangeably.
So, a bright fuzzy set has been given, now
let us convert this fuzzy set into very bright.
So, when let us say we are supposed to use
very bright word in order to communicate something.
So, how can we make use of this bright fuzzy
set and we can generate a new fuzzy set using
the fuzzy set bright for very bright.
So, very bright as I have already mentioned
very when very word comes here this means
we have to raise the power here by 2 on the
membership function and if it is a discrete
fuzzy set it is all the membership values
are squared.
So, very bright is very simple to get here.
So, since we already have 1 by 1 and then
plus 0.8 by 2 plus 0.6 by 3 plus 0.4 by 4
plus 0.2 by 5.
So, here what we have to do for very bright
is because we are only adding the adjective
very here on the bright, bright is already
given.
So, this very term has to simply increase
the power of increase the membership value,
that means the we have to square the values
of the membership.
And let us see what we are getting.
So, here we see that we have a squared the
values we do not have to touch we do not have
to change its corresponding membership, its
corresponding generic variable 1, 2, 3, 4
and 5.
We do not have to touch that we do not have
to change these values we only have to square
the values of the membership.
So, here we are squaring 1.
So, 1 will become 1 only and then 0.8 when
we square it we are getting 0.64 and here
is square of 0.6 will become 0.36, similarly
square of 0.4 will become 0.16 and then square
of 0.2 will become 0.04.
So, this way we see that we are getting a
new discrete fuzzy set.
So, this fuzzy set is a fuzzy set which is
for very bright I can write here the this
as the discrete fuzzy set for very bright.
So, what is interesting here is to note is
that we had bright only we were given the
discrete fuzzy set for bright and here we
are converting the bright into very bright.
And as I have already discussed when we add
very, before any linguistic value before any
primary fuzzy set.
So, then this bright becomes very bright here
by adding very and since we are adding very
and we have already seen that very comes with
squaring of the membership values or membership
function in case of continuous fuzzy set.
So, in case the fuzzy set is a discrete fuzzy
set then simply we have to square the respective
membership values, but if it is a continuous
fuzzy set, then we will have to simply square
the continuous membership function that has
been given for the primary set.
Let us now move to the second example which
is for very very bright.
So, bright has been given to us now we have
to convert the given bright discrete fuzzy
set into very very bright.
So obviously here we have two times very very,
so we have to square it two times, we have
to square the respective membership values
in twice.
So, when we square the membership values twice
this the respective membership values basically
becomes the original membership values raised
to the power 4 which you can see here, this
is for very very.
So, when we use the bright and we want to
have very very bright we this way by taking
the powers increased by 4, the powers of the
respective membership values by 4 we are getting
its membership values like this.
And please note that here no change will happen
to its corresponding generic variable values
which are 1, 2, 3, 4, and 5.
So, we do not have to change these values
only the change will happen to the corresponding
membership values in case of the discrete
fuzzy set simply we take the membership value
and we use the membership value raised to
the power 4 and then whatever value comes
we will write.
But if it is a continuous fuzzy set, then
the membership function will become the twice
square it means we will write the mu x raised
to the power 4.
Now, let us quickly go to the third part of
the example here and here the bright is given
to us and we have to find the more or less
bright.
So, more or less is a hedge as I have already
mentioned.
So, like we had very and then very, very,
very and then extremely like that.
Now, let us take more or less as hedge and
let us see what happens with more or less.
More or less basically we get when we change
its membership value we dilate its membership
value in other words I would say.
So, more or less means we rather than squaring
or raising the power we are decreasing the
power here.
So, decreasing the power means we are taking
the square root of the original membership
value or membership function.
So, in case of the continuous fuzzy set we
simply take the mu x raise to the power 1
by 2, whereas if it is a discrete fuzzy set
we simply take the square root of all the
respective membership values and here also
we will not touch any of the generic variable
values.
So, so all the generic variable values will
remain unchanged.
So, let us find more or less bright.
So, more or less bright is here more or less,
more or less bright fuzzy set is here.
So, what we are doing here is we had 1.
So, we are taking the square root of 1 and
then we are taking the square root of 0.8,
we are taking the square root of 0.6, we are
taking the square root of 0.4, we are taking
the square root of 0.2.
And this way we are getting when we are taking
square root of 1 we are getting 1, we are
getting here in this case when we are taking
0.8 and then we are taking square root of
0.8 we are getting 0.8944.
So, similarly we are getting all these values
when we are taking the square root and this
way we are forming a new set and here we have
formed a linguistic hedge.
So, hedges are coming out of the modifications
of the original or the primary fuzzy sets
by adding the adjectives or adverbs.
So, this way the bright fuzzy set is converted
into the more or less bright.
So, when we say more or less bright it means
we simply take the square roots of all the
respective membership values which are characterizing
the fuzzy set, this is discrete fuzzy set.
So, that way we have understood as to how
we managed to get the, a primary fuzzy set
converted into the linguistic hedges.
So, now let us understand here move to the
3rd class and 3rd class here is the negation,
complement and connectives.
So, when have been given a primary fuzzy set,
a primary term all these names can be interchangeably
used and when we try to find a negation of
it let’s say we have a primary set say middle
aged and then we say not middle ages, then
how to get the not middle ages fuzzy set out
of the given fuzzy set middle ages.
So, here if I have been given any fuzzy set
A, let us say A, the given fuzzy set, given
fuzzy set and this fuzzy set represents; this
represents basically a primary set which is
part of the term set.
So, if we are interested in finding NOT of
A, means as I mentioned if A is a middle aged,
then if you are interested in NOT middle aged
then we simply take the NOT of it and NOT
of it is represented by this sign here if
NOT is represented by this sign.
So, NOT is here and NOT of A can be symbolically
written like this and what is done here to
get NOT of A we have already learned this
when we have studied, when we have discussed
in the one of the previous lectures, the negation
the complement.
So, what we do here is we take we subtract
the corresponding membership values from 1
which you can see here.
So, simply when we have been given a fuzzy
set let us say if I write the given fuzzy
set A like this, if it is a continuous fuzzy
set we will be representing it like this the
integration sign and then the universe of
discourse just below it and then mu of x and
then x here.
And this is here what is done for NOT A is
simply we take the complement of it and when
we take compliment of it the membership function
is subtracted from 1 which you can see here
rest other things remain the same.
So, this way we can get NOT is as the negation
or the complement and here in other case we
can have the connectives like A and B when
we have let us say a fuzzy set A and fuzzy
set B both the fuzzy set have been given to
you, then how to connect both the fuzzy sets
together?
For example, I can say a middle aged and young.
So, we have two fuzzy sets and both the primary
terms are getting connected by AND.
So, AND is here a connective.
So, the connective can be either AND, OR,
so these two are the connectives.
So, here there could be A AND B or A OR B.
So, like middle aged and young or middle aged
or young.
So, whatever way any two fuzzy sets can be
connected.
So, this and can also be replaced by but,
so if we use, but or and both are same.
So, let us first take A AND B. So, when we
take A AND B, so this means what?
When we have already done this exercise in
one of the lectures previous lectures.
So, when we have two fuzzy sets let us say
and they are being connected by AND connective.
So, we simply use intersection.
So, A AND B will become A intersection B means
both the fuzzy sets are being intersected
means we have we will have to take the intersection
of the primary term set A and the primary
term set B.
So, this way when we do what is happening
with the corresponding membership function
is this we take the min of mu A x and mu B
x.
Similarly, when we connect A AND B by and
we take the union of A AND B and similarly
we use the max sign in place of the min sign.
So, here we connect both the membership function,
functions of A AND B like this like mu A of
x and then we take max mu B of x, means we
take max of mu A of x and mu B of x which
you can see here.
And please note again that the generic variable
values x will remain the same, will remain
unchanged.
Here these connectives are very interesting
and even negation also.
So, negation and connectives both are basic
connectives shown here, but since we have
already done in the previous lectures that
we have multiple kinds of negations, multiple
kinds of connectives.
So, like we could use for connectives various
kinds of t-norms and s-norms.
So, as an when it is required we can use that
also, but here if nothing is mentioned then
we simply use the basic s-norm and t-norms.
So, s-norm is used for OR and t-norm is used
for AND.
So, this way the negation complement and connectives
can be managed and this is quite interesting
to note here that any primary fuzzy set any
primary linguistic value, when we say linguistic
value linguistic value is nothing but a fuzzy
set, linguistic value is represented by a
primary fuzzy set.
So, any primary term set, any primary fuzzy
set, any linguistic value can be converted
into its hedges or in other fuzzy sets with
the negation or you know with connectives
you can manage to get a new fuzzy set.
So, let us take an example here to understand
the negation complement and connectives.
Here we have a two fuzzy sets both the fuzzy
sets are discrete fuzzy sets A and B and let
us using A and B let us find NOT A and then
let us find A AND B. Here this is a connective.
And in the third case also OR is connective,
so AND OR both are connectives.
So, let us now go one by one and try to find
NOT A first.
So, A has already been given.
So, we represent NOT of A by a negation A
and as I have already mentioned that simply
when we have mu A the membership function
given.
And then when it comes to negation of it we
subtract the membership values from 1 or if
it is a membership function we subtract this
also from 1.
So, when we do that we get NOT A like this
means here we are negating all these corresponding
membership values and this comes out to be
this.
So, NOT A is this, this is represented by
NOT A. So, a new fuzzy set NOT A given A is
this.
So, this way we find NOT A very quickly very
easily.
Now, let us find A AND B. So, A AND B as I
have already mentioned that AND is a connective;
AND is a connective.
We have been given fuzzy set A this is fuzzy
set A first fuzzy set and this is second fuzzy
set.
So, both the fuzzy sets have been given now
we have to connect these two fuzzy sets together
and as I have already mentioned that when
we have AND we have to take the intersection
of it.
So, when we take intersection, the basic intersection
uses the min.
So, when we use this we find the new fuzzy
set like this.
So, in the new fuzzy set is with min of both
the membership values.
So, when we use this the new fuzzy set is
coming out to be this.
So, we can say this fuzzy set is a new fuzzy
set; a new fuzzy set after connective AND
so this is for AND, now if we use OR.
So, OR can also be very quickly managed to
get we see that here when it comes to OR means
when we have two fuzzy sets and when we have
to make a new fuzzy set by ORing both the
sets.
So, A union B gives us A OR B and here we
use the max sign as a basic union.
So, you see here the max sign the inverted
open triangle and this way we have A OR B
and this A OR B is given here as after taking
max of the corresponding two membership values
we are getting these values.
So, the new fuzzy set, the new fuzzy set which
is coming after connecting A AND B as OR by
OR we are getting a new fuzzy set here the
by OR.
So, OR is the connective.
So, this way we see that we have been able
to manage to get new fuzzy sets.
So, either taking the negation of the primary
set primary fuzzy set or by connecting the
two primary sets or maybe even further we
can connect two or more fuzzy sets by AND
or OR or any other connectives and we can
get the expression for the fuzzy set.
So, this way we are able to manage to get
the, a new fuzzy set.
So, with this I would like to stop here in
this lecture.
And in the next lecture we will discuss the
concentration and dilation of linguistic values.
Thank you.
