So, in the last lecture we have learned about
different operations on fuzzy set. So, different
operation on fuzzy set that we have learnt
is basically given a two fuzzy set or a fuzzy
set how another fuzzy set can be obtained.
Now we are going to learn another concept
in fuzzy logic it is called the fuzzy relation.
So, by means of fuzzy relation we say that
if one element belongs to a fuzzy set then
how this element is related to another fuzzy
set. So, this basically the relation; between
the two elements, which belongs to the two
different fuzzy set.
So, fuzzy relations in many way related to
the crisp relation. Crisp relations means
the relation those are there on the crisp
set. So, will first learn about the crisp
relation and then whatever the operations
those are possible on crisp relation is basically
semi applicable to the fuzzy relations also,
but there are certain difference. So, it will
be better if will learn first operations on
crisp relation and then the operation that
can be applied to the fuzzy relation. Some
examples also will be considered in order
to understand the crisp relation and then
we will be in a position to know fuzzy relation.
Some examples operations those are possible
on fuzzy relation and finally, we clear our
idea with some examples.
So, these are the topics that we are going
to cover in this lecture.
So, let us first discuss about the crisp relation.
Now crisp relation basically is an collection
of order pair. So, if A and B are the two
sets then it’s order pair is denoted by
the Cartesian product A cross B and it basically
gives the collection of order pair a b such
that a belongs to the set A and b belongs
to the set B.
So, this order pair relation is important
to understand the crisp relation. Now a particular
mapping is basically belongs to a particular
relation, and we know so far the crisp relation
is concerned these are the different property
holds good. The first property is that A cross
B is not equals to B cross A; that means,
they are not commutative and here is basically
the number of elements which is belong to
the product is same as the number of elements
belongs to the constituent set A and the product
of the number of elements belongs to the set
B.
So, this also the equation hold goods and
as I told you A cross B essentially provides
a mapping the from a set from an element belongs
to set A to another element b belongs to set
B. So, it is basically a mapping and this
mapping is expressed by means of an order
pair and this particular mapping is called
a relation.
Now, we can understand this relation better
if we consider an example.
.
So, example suppose two crisp set A and B
and A is represented by this form and B is
3, 5, 7. So, these are the two sets A and
B and we can obtain their Cartesian product.
Cartesian product we can obtain for this two
phrase is all possible order pairs. So, it
is shown here. This is the Cartesian product
of that two fuzzy set A and B. So, here for
given 1, 2, 3, then 1 5 1 7 then 2 3 2 5 2
7 3 5 3 3 3 5 3 7 and so on.
So, these are the different what is called
the elements which belongs to the Cartesian
product of the two fuzzy sets A and A and
B and then the relation, I told you the relation
is basically a particular mapping. Now here
we express the relation and this is a this
is this is suppose the relation that we have
discussed here.
So, relation between the 2 order elements
in a order pair should satisfy the this equation,
if it is satisfy then it gives a particular
set or it is basically a relation which is
shown here. For example, if this is a relation
hold good for every element then the relation
that can be obtained is basically this one.
So, a relation is basically a collection of
order pairs which satisfy a particular mapping
or a particular definition.
Now, so this is the crisp relation and such
a relation can be expressed in a more compact
way and this compact way it is called the
matrix representation of a relation. Now the
matrix representation of the relation which
we have learned earlier, so R is 2 3 and 4
5 is denoted by this matrix. You can see 1
and 3 which is not belong to the set, so it
is 0 and the elements 2 and 3 belongs to the
set, so it is 1. So, here 0 and 1 are the
entries in the relation matrix. 0 indicates
that that order pair is not belongs to this
relation and 1 indicates that that order pair
belongs to the relation.
So, this way a relation can be represented
by means of a relation matrix.
Now, so operation on crisp relation so these
are the relations if it is available to us
we can apply different operations on it. Say
suppose R and S are the two operations defined
over x and y, where x is the some some elements
belongs to the universe of discourse x and
y is the element belong to the universe of
discourse y. So, R x y can be obtained as
a relation matrix similarly S x y can be obtained
by means of an another relation matrix.
So, if the two relation matrix, are available
to us then we can apply many operations on
them. So, the operation on them that can be
a applied can be applicable is union, intersection,
complement like this. Now you can find the
difference between the union of two fuzzy
sets and union of two fuzzy relations. So,
union operation of two fuzzy set is expressed
by this form. So, R is the relation and B
is the S is the another relation right and
this relation obtained over the two crisp
set say A and B then the union of the two
relations can be defined as a max of the two
entries in x and y in both relation.
So, one example can be given here. So, this
is the union operation likewise the intersection
operation is basically minimum values of the
entries and the complement is 1 minus the
entries in each elements.
Now, one example that can be consider here.
Suppose this is the one relation that can
be obtained over A and B and this is the another
relation obtained over the sets A and B. So,
it is basically A cross B with some relation
it is also A cross B with different relation
and then we want to obtain union of the two.
So, union of the two we can write R union
S. So, it is basically one matrix now union
operation as I told you it is a max. So 0
and 1, so you have to take the 1. 1 0 it is
a 1 then 0 0 for the first rows. Similarly
0 1 1 0, 0 0 1 1 and 0 0 0 1.
So, this is the another relation matrix that
can be obtained using the operation relation
operation of the two relations R and S. So,
this way we can obtain the relation.
Now, using the same concept you can find easily
the union of two, intersections of two and
then complement. Now, so far the complement
operation is concerned R complement this is
equals to it is basically complement value.
So, if the 0 then it will be 1 and it is 1
then 0. So you can complement of these is
this this this and then 1 1 0 1, 1 1 1 0 and
1 1 1 1. So, this is the complement of the
relation R.
Now, there is another important relation it
is called the composition and composition
relation is why they applicable in the context
of fuzzy relation. So, composition operation
it is denoted by this symbol R composition
S; that means, from two relation we can find
another relation, but. So, this R relation
suppose over two set, A and B and S relation
is over another say, A and C. So, here C is
the one common set then we can obtain R composition
S basically relation from A to B via C. So,
this is the concept that is called the composition.
Now, composition operation can be defined
mathematically using max-min calculation.
The max-min it is called the max-min composition
this is why and is denoted by this expression.
Max-min composition is basically this one.
Now it is little bit difficult to understand
at the moment. So, I can give an example.
So, that you can understand it basically follows
the similar concept of product of two matrix
actually. So, it basically take the first
minimum of corresponding entries and then
take for a particular entries the maximum
value.
So, let us have some example. So, that we
can understand the max-min composition or
simply a composition operation to relation.
Now this is a one example we have to consider
carefully. So, suppose x is the one universe
of discourse y is another and the relation
R xy defined over x and y and S is also another
relation defined over the same discourse x
and y. The relation that is there for a R
it is basically this is the relation and this
is the relation that is there in S.
Now, based on this thing we can easily obtained
the Cartesian product and then applying this
relation we can obtain this matrix and this
matrix. So, these are the two relations obtained
from the two fuzzy set through crisp set x
and y. Now having this relation we can find
the composition of the two. Now composition
basically we take first row wise and then
column wise just like a product to obtain
the first element here. Sorry, so this is
the row and this is the one to obtain the
first element here.
So, is basically take like, so 0 and 0 take
the minimum, so minimum is 0. So, 0 is a minimum.
Then 1 and 1 as 1 and 0 then it is a minimum
is 0. Then 0 and 0 so minimum is 0 and then
take the max. Max of this so this is 0, so
this is a 0. For the next element we can obtain
so this and then this one. So, 0 and 1 so
further next further next this one and then
this one, so you can get this element. So,
this and this we can go so; that means, 0
and 1 take the minimum it is 0.
And then 1 and 0 take the minimum it is 0
and then 0 and 0 take the minimum this one
and then maximum. So, it is 0.
Now, let us see how this element can be obtained.
So, again we have to take this one and then
this one. So, first 0 and 1 0 then 1 and 1
1 and 0 and 0 it is 0 and taking the maximum
this one. So, it get this one. So, these element,
again we can be obtained if we take these
one and the this last element can be obtained.
So, this way we can obtain the relation called
the max-min composition of the two relations
R and S.
So, it needs a little bit practice to understand
it. So, it will basically take the min corresponding
to this traversing and then taking the max
of all these will give a particular element.
So, this is the idea and now let us see, how
these operations those are applicable to the
crisp is also applicable to the fuzzy, but
in a different one.
So, difference between the fuzzy relation
and crisp relation lies in the terms of increase
in the relation matrix. In case of crisp relation
the entries in the matrix is either 0 or 1
whereas, in case of fuzzy relation the entries
in the matrix is any value in between 0 and
1 both inclusive.
Now, let us start with an example about the
fuzzy relation. Suppose two fuzzy sets which
is described over the 3 elements which is
here. One fuzzy set is X and another fuzzy
set is Y. The different elements in the fuzzy
sets are here. In X it is typhoid, viral,
cold. In fuzzy set Y, the elements are they
are running nose, high temperature and shivering.
Now you can understand what is the meaning
of these two fuzzy sets and the relation then.
So, basically here if the disease is typhoid
then what are the difference symptoms are
there. So, if the disease is typhoid the symptom
that running nose is, but with the strength
0.1, high temperature 0.9 and shivering 0.8.
So, every disease and the different symptoms
with the different membership values is represented
and by means of a relation matrix. So, if
it is a viral then what about the shivering.
The shivering is 0.7, if it is a viral then
running nose, but running nose with little
bit less uncertainty than the shivering that
is 0.2 and 0.7. So, the relation matrix basically
shows the different element which are belongs
to the different sets how they are related
to each other. So, this basically the physical
significant of the fuzzy relations and now
one thing it is clear that the element in
the fuzzy relations; that means, the entries
in the relation matrix is basically any value
in between 0 and 1 both inclusive.
So, this is the only difference between the
fuzzy relation and the crisp relation, otherwise
every operation those we have defined in case
of crisp also equally applicable to the fuzzy
sets. Now let us see what are the different
operations they are possible for the fuzzy
relations.
So, here so the fuzzy relation again defined
as in the crisp relation like the min operation.
So, it is the min actually. So, this is the
min operation.
Let us see one example. So, here say A and
B are the two fuzzy set. A and B are the two
fuzzy sets. Now I can find a relation. The
relation can be obtained as we have discussed
here, now relation operation that can be defined
over two fuzzy sets it is basically represented
by this expression that mu R xy where is a
membership values belong to relation and it
is denoted as A cross B as I told you Cartesian
product of x y and it basically takes the
minimum of two corresponding values in both
the set A and B for x and y respectively.
So, it is basically taking the minimum.
Let us have an example, so we can clear our
idea. This is an example can be followed to
explain the relation operation for the fuzzy
set in terms of Cartesian product A. Now here
A is the set which is defined like this. B
is another set which is defined like this
and then the relation are is basically Cartesian
product as I told you. So, it basically for
a1 and b1 a1 and b2 so a1 b1 a1 b2 now for
a1 b1 we have to take the minimum. So, 0.2
and 0.5 take the minimum, so it is a minimum
entries. Similarly a1 and b2 0.2 and 0.6 take
the minimum so 0.2.
Likewise a2 b1 so minimum is 0.5. a2 b2 the
minimum is 0.6 a2. So, this is a2 now a3 and
b1 so 0.4 and a3 b2 0.4. So, these way we
can obtain the relation matrix taking the
min operation that is there. So, this way
we can obtain the relation if the two fuzzy
sets are given to us.
Now, let us define different operations on
fuzzy relation like the different operation
in crisp relation. So, like union, intersection
and then complement these are the operation.
So, union operation on two fuzzy sets can
be defined using this expression. It is basically
taking the maximum value of the two entries
there. So, it will give the new matrix taking
the maximum of the corresponding entries.
Intersection basically taking the minimum
of the two entries and complement it is a
unary for one operation. So, you take the
if it is mu R b then the value or entries
in the relation matrix will be the complement;
that means, 1 minus mu R a b .
Now, some example can be followed to understand
this concept. Another composition I will discuss
the composition operation in details in regards
the fuzzy sets it is basically same as max-min
composition.
Taking the similar concept now better we if
we have one example. So, X is a crisp set
Y is another and Z is another. So, we can
consider they are the universe of discourse
of for the fuzzy sets may be and so R this
is the one relation defined over two sets
which is discussed over the universe of discourse
X and Y. So, this is the relation over the
two fuzzy sets and S is the another relation
which is defined over Y and Z.
So, these entries are given to you. Now if
it is given to you then we can calculate R
union S easily. Now again here we can say
R union S here basically elements those are
not same that is why we cannot apply R union
S, but the two I mean union operation is applicable
if the two relations are defined over the
same elements. So, if this relations is defined
over this one this one then another relation
should be defined this one then only we can.
For example, suppose I defined another relation
P and x1 x2 and x3 and y1 and y2 and say 0.1
0.2 0.2 and 0.5 0.3 and 0.4. So, this is a
relation.
Now, if we want to find the union of the two
relation. So, R P then relation basically
taking the maxima of the corresponding entry.
So 0.1 and 0.5 so in the first entry 0.5 and
then 0.2 and 0.1 so you should take 0.2, 0.2
and 0.2 so 0.2 and 0.5 and 0.9 0.9. Then 0.3
and 0.8 0.8 then 0.4 and 0.6 it is 0.6.
So, this is basically the relation obtained
over the union operation of the two relation
R and P. And you can note that R P this is
equals to same as P R, it means they holds
the commutative property. Now likewise the
intersection. Intersection we have to take
in, so it is basically take the minimum of
this on union where is the maximum and intersection
is the minimum.
And then complement also can be obtained.
For example, so complement operation this
R bar that can be obtained like. So, 0.5 then
0.9 then 0.8 then 0.1 0.2 and 0.4. I hope
you have understood how it is obtain it is
basically taking the complement 1 minus 0.5
1 minus 0.1 and this way.
So, this way the complement operation over
a relation R can be obtained. Now this is
another example that I am going to discuss
is a composition. So, R and S are the two
relation.
And we want to find another relation T which
is a composition of R and S. So, using the
max-min composition it is the composition
that we have discussed about the Cartesian
product finding. So, take the minimum first
and then minimum of all take the maximum again
we can follow it is these one and these traverse.
So, it will give this one; that means 0.5
and 0.6 we have to take the minimum so it
is 0.5 and 0.1 and 0.5 we have to take the
minimum 0.1 and taking the maximum of this
so 0.5. So this way 0.5 and likewise if we
traverse this one and this one the these element
can be obtained. these one and these one then
this can be obtained these then this element
can be obtained and so on.
So, this is the max min composition operation
that can be applied on two relation R and
B.
So, this is the relation operation and I can
give another last example in this direction.
Here these examples is very interesting to
note. So, P is the set with different element
P1, P2, P3, P4. D is the another set with
different element say D1, D2, D3, D4. Now
in the context of our real application, so
P basically consider a set of varieties of
paddy plant; that means, P1 is a one type
of paddy plant, P2 is another and so on.
Now, D the set D represent the different diseases
where D1 is a type of disease, D2 is another
and so on and say S is the another set with
basically set of symptoms, the symptoms are
S1, S2, S3, S4. Now how a particular plant
is related to the disease that can be given
by means of a relation. So, this is basically
a fuzzy relation showing that how a particular
plant is related to the different disease
that may have. For example, P2 is a plant
and is susceptible to disease D1 with the
0.1 certainty, D2 is 0.2, D3 0.9 and D4.
So, we can say that P2 is very much susceptible
to the disease D3 or D4 and less susceptible
disease D1 D2. It is the concept. So, these
are meaning of the fuzzy relation. Now having
this fuzzy relation are we can obtain another
relation by means of composition operation.
Say, S also another relation. It basically
showing the relation about disease and the
symptoms. So here different disease are there
and different symptoms and this is the matrix
showing how particular disease and related
to the different symptoms for that. Now having
this one then we can have the composition
operation. So, T is a set resultant which
is basically R composition S. So, R composition
S can be obtain previously we have discuss
about R and this is the S and taking the max
min composition then you can try and you can
check that this is the relation that can be
obtained.
So, this relation has the meaning. This meaning
is that if the R relation shows that which
paddy plant is susceptible to which disease
and if S denotes the particular disease and
what are the symptoms, then R S basically
shows a particular plant then what are the
symptoms that it basically corresponding to
some disease. So, this is the relation showing
a plant and then symptoms that they may be
affected. So, this is the one example and
we hope I hope you have understood the concept
of relation.
And this is the another example that is the
I can give it very quickly, so that you can
understand. If R is another relation showing
the relation from these sets to these sets.
Sorry. This is the R relation is this sets
to this set and another relation S showing
the relation from these element to this element.
If given this one then I can find a relation
from any to anyone, so via this one. So, alpha
beta gamma basically in between the two and
then we can find the relation to 2 a or relation
2 to b. So, that can be obtained by means
of Cartesian product and then relation composition
operation rather not Cartesian product.
So, it can be like this one. For example,
2 a, 2 is the element which is belongs to
set and these are the different and a. So,
you can find the relation from 2 to a by means
of max min composition or the max min composition
is can be calculated which is shown here.
So, it basically here. For example, 2 and
a there is a relation via other elements alpha,
beta, gamma. So, the relation that 2 is related
to a which strength 7 0.7. Similarly we can
calculate likewise the 2 and a relation between
2 and a. We can calculate relation between
1 and b or 1 and a with some what is called
the strength. So, this basically shows the
relation and it is the meaning it is there.
Now, so I think time is over. So, we can stop
it here. These portion can be discussed in
the next lectures.
Thank you.
