WEBVTT
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This is lecture 2 of module 2 in our course
on intelligent control. This is the module
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that covers the fuzzy logic concepts. Today,
we will be discussing fuzzy relations. In
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the last class, we discussed fuzzy sets and
some introductory ideas of fuzzy sets. It
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was different from crisp set- the concept
of membership function fuzzy operations. We
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gave some examples to illustrate fuzzy operation.
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Today, we will be talking about fuzzy relation.
Topics that we will be covering today are
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the extension principle of fuzzy sets, fuzzy
relations, projection of fuzzy relations and
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cylindrical extension of fuzzy relations,
fuzzy max min and max product composition
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operations.
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This is the extension principle of fuzzy set.
This is our fuzzy set A. As usual, in the
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last class, we said that fuzzy set is always
an ordered pair, where there are member and
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associated membership function. There are
n elements here; x1 x2 up to xn and similarly,
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y1 y2 up to yn. There is a one to one mapping
from set A to set B; that is, y1 is f of x1
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and on until yn is f of xn. This relationship
is also valid. That means there is a relationship
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between the members of the set B with members
in set A in the form of y equal to f of x.
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If that is true, that is the case now. We
know that A is a fuzzy set and B is a fuzzy
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set. The normal way they represent A is mu
A x1 upon x1, mu A x2 upon x2; that means,
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the members and associated fuzzy membership
function. Then similarly, B also; their associated
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membership function is mu B y1 mu B y2 mu
B yn. Then the question is that, do we have
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to again find what B is, if A is given? No,
because we know already y has a relationship
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with x in terms of y equal to f of x.
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If we know, what fuzzy set A is, fuzzy set
B is already known. How? y1 y2 yn are f of
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x1 until f of xn. We know what x is; x1 until
xn from x1, x2 until xn. We know y is y1 up
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to yn. All that we have to do is, once we
compute what is y1, y2 up to yn, because,
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we already know the information; that is,
between A and B there is one to one mapping.
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All that we have to do is that after computing
what is y1 y2 up to yn, instead of mu B y1,
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we just have to replace mu A x1; that is,
here mu B y1 is same as mu A x1, because this
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is one to one mapping. This is
how we compute the fuzzy set B from A given
the knowledge of fuzzy set A and the mapping
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from A to B, if you already know what should
be the fuzzy set B.
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Similarly, many to one mapping: In many to
one mapping, there is x1, x2 up to xn. They
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are all mapped to y1 and so on. That means
there are many members in x1 and they are
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mapped to the same member in B. Many members
in A are mapped; like here, three members
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in A are mapped to a single member in B. There
is many to one mapping through the mapping
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y equal to f of x. Simple example is y equal
to x square. In that case, minus 1 will map
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to same point in B minus 2 plus 2 will map
to same point in B. That is how many to one
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mapping is. Given A mu A x1 up on x1 mu A
x2 upon x2, until you know we have the last
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element xn in A. This is our fuzzy set A.
Similarly, fuzzy set B would be.
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If we have an element or let us put y; the
membership function associated with y1, y2
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until up to yn is computed by the relationship
y equal to f of x. When the associated member
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function is computed, what is the maximum
membership function associated with corresponding
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x here; that is, maximum mu A xj, where xj
is f inverse belongs to f inverse yi. For
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example, in this, if I want to find out what
mu B y1 is, in this case is maximum of mu
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A x1 mu A x2 and mu A x3. These are the three
members in A. They map to same element in
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B. Hence, the associated membership function
with y1 is the maximum membership function
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associated with its counterparts in A; that
is, in counterpart of y1 in A is not 1; It
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is many, because many to one mapping.
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This is the example; like here, as I said
f of x equal to x square is a good example
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of many to one mapping, where minus 1 maps
to 1 minus 2 maps to 4 and 3 maps to 9. In
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this case, if A fuzzy set is given like this,
where minus 1 has membership function associated
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0.2 minus 2 0.4 1 0.6 0.2. This is not 0.2,
this is 2.2 has 0.8 and 3 has 0.9; that is,
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the membership function B whose members are
1 4 9. Membership function would be like for
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1 the maximum membership minus 1. The maximum
is 0.64 minus 2 which is 0.4 and plus 2 is
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0.8. Maximum is 0.8, and then 9 of course
is single one to one mapping. Whatever is
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here, will come here. 0.9 will come here.
Then the answer is this; B equal to the fuzzy
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set B is 0.6 by 1.8 by 4.9 by 9. That means
it has 3 members 1 4 9 and associated membership
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functions are associated membership index
or indices are 0.6 0.8 and 0.9.
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Now, we will be talking about crisp relations.
When we talk about relation, that means it
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is more than one set; that is, two or more
than two sets are involved in this process.
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Let us take a simple case. You know, we can
have multiple sets, but we will first talk
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about crisp relation before we talk about
fuzzy relation. In crisp relation, the Cartesian
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product of two universal sets is determined;
X times Y is this. This Cartesian product
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is an ordered pair. An ordered pair x and
y, where x belongs to X, capital x, and y
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belongs to capital Y. The crisp relation mu
R x y is defined as this. Crisp relation is
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actually defined on the Cartesian product
space. mu R x y is either 1 or 0. This is
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1 if x y belongs to the Cartesian product
and 0 if x y does not belong to the Cartesian
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product space. What is the meaning of 1? 1
implies complete relation and 0 implies no
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relation. When the sets are finite, the relation
is represented by a matrix R, called relation
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matrix. I will give you a simple example.
Let us say, x is 1 and 2, and y is a and b.
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Relation means the Cartesian space is of course
x times y is a1 b2 and this is your Cartesian
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product space.
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Now, when I say relationship, that is, if
1 has no relationship with a, then the membership
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function mu R x y is the relation. They are
associated with this 0 because, that means
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1 is not related to a, but if 1 is related
to b and if the relationship is there, then
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we associate with this 1; that is, in the
relations space, the matrix R is the relation
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matrix which is x is equal to 1 2 and y is
equal to a and b.
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When I say the relation 1 and a are not related,
this means 0, 1 and b are related. 1, 2 and
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b are not related 0; then this is our relation
matrix and because we have finite number of
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elements in x and y, we can always represent
a relation by a matrix. But if it is a continuous
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set or a set having infinite points, then
we have difficulty in terms of representing
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a relation in terms of matrix. This is a crisp
relation; X is 1 2 3 and Y is a b c because,
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I said earlier which are 2 by 2 two elements
in X and 2 element Y and now it is 3 element,
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x3 element in Y. If I represent R, then if
R is 1, it implies all the elements in X are
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related with all the elements in Y.
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Each element in X is completely related to
each element in Y; that is 1. a1 b1 c1, c2
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a2 b2, c3 a3 b3 are all completely related;
whereas, in case of this particular relationship
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where we have 0 0 1 1 0 0 0 1 0. It implies
2a 1c and 3b. They are only related and others
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are not related. So, this is the meaning of
crisp relation. A crisp relation means either
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the relationship index is 0 or 1. There is
no in-between relationship. That is the classical
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relationship matrix.
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Now, the question is that if we have multiple
Cartesian product space and we know the relationship,
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then we can infer the relationship; for example,
here x y z are universal sets. Let R be relation
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that relates elements from x to y in the Cartesian
product space of x and y. R is defined. Similarly,
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in the Cartesian product space of y and z,
S is defined; the relation S is defined. Similarly,
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T is defined in the relationship. The Cartesian
product relationship x and R. T is the relation
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that relates same elements in x that R contains
to the same elements in z. S contains the
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Cartesian product space. Here, this is x times
z. This we can do, if R is given and S is
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given.
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Can I compute T? This is the normal question
you would ask. If I know the relation in the
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Cartesian product space, x y as well as Cartesian
product space y and z, can I infer the relation
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in the Cartesian space x and z? Because x
y is known and y z is known, can I know x
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and z? That is the inference; knowledge inference,
information inference. Yes. How do we do it?
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The given R and S, T are determined using
the principle of composition, where T equal
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to R composition S. This composition can be
found out either by method one which is max
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min product composition. In max min composition,
the associated membership function muT is
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computed as a maximum of minimum. mu R x y
mu S y z, I will give you a simple example.
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This is my R and this is my R S and then I
have to find out what is T. Let us say R is
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x x1 x2 and y is y1 y2. The relationship is
0 1 1 0. Similarly S is y and z y1 y2 z z1
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and z2. Let us say again the relationship
is 0 1 1 0. That is the case, now, I have
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to find out what the relation is between x
and z1 z2. How do I find out in this max min
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composition? I want to find out now, what
is x1 and z1, final x1 and z1.
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How do I find x1 and z1? 0 1 you bring here
0 1. It is like a matrix multiplication, we
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do this row. You put parallel to this column
over y1 and y2. Then 0 0 is the minimum and
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this is 0 mu R x y; mu R x y is 0 and mu S
y z is 0 and so minimum of that is 0. The
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first minimum is 0 and then second overall
y; similarly, x1 y2 x1 y2 and y2 z1. That
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is 1, x1 y2 is 1 and y2 z1 is 1. You find
that it is very simple. The basic principle
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is that this 0 1. You put parallel at 0 1
and find the minimum in each case. What I
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can do is that for 0 1, I have another 0 1
here in parallel, it comes from here and finds
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the minimum from 0, 0 is 0 and 1 one is 1
and find the maximum. From this, you put,
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x1 z1. Similarly, I have x1z1. Similarly,
x1 z2. How do I find? Again 0, 1. We put it
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here 0 1 0 1. Obviously, minimum is 0 minimum
is 0 maximum is 0. Similarly, x2 z11 0 1 0;
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this is 0, similarly 1 0 1 0 this is 1.
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The
max product composition will also lead to
the same thing, because, this is crisp relation.
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In crisp relationship, whether I follow max
product or max min, both will come and yield
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the same result. What is product? This is
multiply 0 1 0 1 multiply 0 0 is 0 1 1 is
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1 and max maximum of that is 1 here. Similarly,
0 1 1 0, here 0, if you multiply and then
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take the maximum, that is 0. Similarly, 1
0 0 1, you get 0 and 1 0 1 1. If you multiply,
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one element is 1, another element 0. The maximum
is 1. This is called a composition. The principle
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of composition - Given two relations R and
S, I can compute what T is, using these two
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rules. How to find out in this case? This
is max min composition, this is max product,
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this dot means product. I multiply mu R x
y into mu S y z.
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The example again I showed you, is little
more elaborated. We see that R is a Cartesian
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product, where x is 3 dimensional, y is 4
dimensional, S is 4 dimensional; y and z,
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where y is 4 dimensional and z is 2 dimensional.
Obviously, T has to be a relation from x to
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z which is 3 dimensional, x and 2 dimensional
z. Using the compositional rule, you can easily
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find out that if it is 1 0 1 0, you put here
1 0 1 0. Obviously, in every case minimum
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is 0 you get 0. Here, the maximum of that
is 1 1 0 1 0. You put here 1 0 1 0.
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Obviously, this will be 1 here and 0 0 0 1
0 0 0 1. Obviously, in this case, again all
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the 1 is there, minimum is 0, the total maximum
is 0. What we are trying to do is that, you
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take this row and take this column, you put
them like here. This column is 0 0 0 0 and
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this row is 1 0 1 0. To find out the relationship
between x1 and z1, what I do I take this row
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which is 1 0 1 0 and this column 0 0 0 and
find for each 1 what is minimum.
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The minimum in this is 0. Overall, what is
maximum? That is 0. This 0 entry comes. Now,
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I want to find out how this 1 entry came.
This 1 entry, to find out these 1 entry x1
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and z2 x1 means you take this row which is
1 0 1 0 and z2. z2 means the corresponding
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column that is 1 0 1 0. Now, you individually
find out what is the minimum of this. This
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is 1, this is 0, this is 1, this is 0 and
what is the maximum of all these elements?
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That is 1. This is called max min composition
and you can do all max product composition.
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You will get the same answer. The user should
note and verify that in case of crisp relation
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max min or max product will yield same result.
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Now, we will be talking about fuzzy relation.
If
x and y are two universal sets, the fuzzy
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sets, the fuzzy relation R x y is given. As
this is all ordered pair, mu R x y up on x
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y for all x y, belonging to the Cartesian
space x, you associate mu R x y with each
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ordered pair. What is the difference between
fuzzy and crisp relation? In fuzzy this is
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missing, where mu R x y is a number in 0 and
1. mu R x y is a number between 0 and 1. This
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is the difference between crisp relation and
fuzzy relation. In crisp relation, it was
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either 0 or 1. It is either completely connected
or not connected, but in case of fuzzy, connection
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is a degree; that is, it is from 0 to 1.
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The example is, let x equal to 1 2 3. Then
x has three members, y has two members 1 and
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2. If the membership function associated with
each ordered pair is given by this e to the
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power minus x minus y whole squared. You can
easily see, this is the kind of membership
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function that is used to know, how close is
the members of y are from members of x. Because,
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if I relate from 1 to 1 using this, then you
can see 1 minus 1 is 0 that is 1 and 1 very
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close to each other; whereas, 2 and 1 is little
far and 3 1 one is further far. This is a
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kind of relationship we are looking between
these two sets.
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Let us derive fuzzy relation. If this is the
membership function, fuzzy relation is of
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course all the ordered pairs. We have to find
out 111 2 2 1 2 2 3 1 and 3 2. These are all
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the sets of ordered pairs and associated membership
functions. You just compute e to the power
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minus x minus y whole square. Here, 1 1 1
minus 1 whole square, 1 2 1 minus 2 whole
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square, 2 1 2 minus 1 whole square, 2 two
2 minus 2 whole square, 3 1 3 minus 1 whole
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square, 3 2 3 minus 2 whole square and if
you compute them, you find 1 0.4 3 0.4 3 1
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0.1 6 0.4 3. This is your membership function.
This is one way to find relation.
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Normally, I know, it is easier to express
the relation in terms of a matrix instead
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of this continuum fashion, where each ordered
pair is associated with membership function.
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It is easier to appreciate the relation by
simply representing them in terms of matrix.
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How do we do that? This is my x 1 2 3 y is
1 21 the membership function associated was
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1 1 2 membership is 0.4 3 2 1 0.4 3 2 2 1
3 1 0.1 6 and 3 2 is 0.4 3 that you can easily
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verify here 1 3 0.4 3 0.1 6 and 1.
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The membership function describes the closeness
between set x and y. It is obvious that higher
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value implies stronger relations. What is
the stronger relation? It is between 1 and
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1, and they are very close to each other,
and 2 and 2; they are very close to each other.
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Closeness between 2 and 2, between 1 and 1
is actually 1 and 1. They are very close to
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each other; similarly, 2 and 2. If I simply
say numerical closeness, then 2 and 2 are
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the closest, and 1 and 1 are the closest.
That is how these are the closest. Higher
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value implies stronger relations.
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This is a formal definition of fuzzy relation;
it is a fuzzy set defined in the Cartesian
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product of crisp sets; crisp sets x1 x2 until
xn. A fuzzy relation R is defined as mu R
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upon x1 to xn, where x1 to xn belongs to the
Cartesian product space of x1 until xn; whereas,
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this mu R the fuzzy membership associated
is a number between 0 and 1.
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We will be talking about projection of fuzzy
relation. A fuzzy relation R is usually defined
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in the Cartesian space x and x and y. Often
a projection of this relation on any of the
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sets x or y, may become useful for further
information processing. The projection of
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R x y on x denoted by R 1 is given by mu R
1 x is maximum. So, y belongs to y mu R x
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y. The meaning is that if I have R, this is
x1 and x2 and this is y1 and y2, and this
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is 0.1 0.4 and this is 0.5 0.6. If these are
the membership functions associated with x1
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y1 x2 y2 is 0.4 x2 y1 is 0.5 x2 y2 is 0.6.projection,
which means for x projection, I find out what
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the maximum is. Overall, y in this case maximum
is 0.4 and for x2 the max maximum projection
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is if I took it here, 0.6. Similarly, if I
make projection of R, x, y over x, what is
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the maximum? This is 0.5 and this is 0.6.
This is called x projection and y projection
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of a relation matrix R.
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This is again for your benefit. We repeat
another example. We have x as 3 components
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1 2 3, y has 2 components 1 and 2. This is
the previous example that we had 1 0.4 3 0.4
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3 1 0.1 6 0.4 3. x projection would be 1 0.4
3 maximum 1 0.4 3 1 maximum 1 0.1 6 0.4 3
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maximum 0.4 3. This is x projection. Similarly,
y projection; look here, this maximum is 1
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and here maximum is 1. This is my y projection
of relation R and this is my x projection.
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Above figure illustrates x and y projection
of fuzzy relation. For x projection, the maximum
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value in each row is retained. What is the
maximum value in each row? Here, x projection
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maximum value in each row is retained, while
the maximum value in each column is retained
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for y projection.
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This is our formal definition of a fuzzy relation,
projection of a fuzzy relation R on to any
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of its set in the Cartesian product space;
that is in the Cartesian product space. This
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is our Cartesian product space and for that,
we can map this one to any of these i or j
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or k; whatever it is, for any value, then
is defined as a fuzzy relation Rp, where Rp
35:28.750 --> 35:35.750
is defined as maximum over Xi until Xk, where
this is our Xi Xj Xk and this is mu Rp.
35:43.010 --> 35:50.010
First, we talked about fuzzy relation projection
of fuzzy relation. Once we have projection
35:52.820 --> 35:59.820
of fuzzy relation, we can extend the projection
to again infer what should be the relation.
36:02.570 --> 36:08.369
This kind of technique may be useful in coding
the information, where we have a huge number
36:08.369 --> 36:15.369
of information and we want to transfer such
a kind of projection and from projection to
36:15.960 --> 36:22.960
extension would be beneficial for coding operation.
36:23.450 --> 36:28.940
Now, let us understand what is the extension
of fuzzy relation cylindrical extension, from
36:28.940 --> 36:34.460
an X projection? It means filling all the
columns of the relational matrix by the X
36:34.460 --> 36:39.750
projection. Similarly, cylindrical extension
from y projection means filling all the rows
36:39.750 --> 36:46.320
of relational matrix by the X projection.
This is my X projection in the previous example
36:46.320 --> 36:53.320
1 1 0.4 3. What I do is I create a relational
matrix between X and Y, then I fill all the
36:57.080 --> 37:04.080
rows with 1. Here, all the rows with this
1, then all the rows here 0.4 3; whereas in
37:05.109 --> 37:12.109
Y projection, if I extend to the relation,
then from Y, I fill all the columns here.
37:15.650 --> 37:22.650
This one is filled with all 1. That is how
we have actual relational matrix projection.
37:26.099 --> 37:31.270
Again from projection, obviously, this is
not a very good kind of extension, but this
37:31.270 --> 37:36.930
is known as cylindrical extension. Maybe other
methods of how to extend from projection to
37:36.930 --> 37:43.930
the actual relational matrix is a different
question altogether. What is the relation
37:45.599 --> 37:50.000
we learnt about projection and from projection
to extension?
37:50.000 --> 37:55.510
We will be talking about given two relations;
just like we learnt in crisp relation that
37:55.510 --> 38:02.510
given two relations, the third relation can
be inferred through composition rule max min.
38:10.080 --> 38:17.080
Let us define R1 over Cartesian space X and
Y, R2 over Cartesian space product space Y
38:17.430 --> 38:24.430
and Z, R3 over Cartesian product space X and
Z. If R1 and R2 are given, then R3 is inferred
38:27.040 --> 38:34.040
using either max min composition, as we have
already discussed our max product composition.
38:35.170 --> 38:41.910
The same principle - I will not explain this
again, because the only thing is earlier this
38:41.910 --> 38:48.550
mu R used to be in crisp relation. They used
to be either 0 1, whereas in fuzzy, it can
38:48.550 --> 38:52.470
be any number between 0 and 1.
38:52.470 --> 38:59.470
We will now explain, max min composition operation
using an example that makes things much more
39:01.530 --> 39:08.530
clear. This is my matrix, relational matrix
R1 relating x and y and R2 relating y and
39:11.690 --> 39:18.690
z. I have to find out the relational matrix
from x to z using fuzzy rule of composition.
39:23.640 --> 39:30.640
We normally write R3 is R1 composition R2.
Using max min composition, how do we compute
39:36.780 --> 39:41.220
R3? Very simple.
39:41.220 --> 39:47.130
I want to now build a relationship between
x and this. This is the membership associated
39:47.130 --> 39:54.130
with x1 and z1. Let me put it very precise,
x1 x2 x3 z1 and z2; if you look at what we
40:02.700 --> 40:09.700
will be doing here, This is my x1 row and
this is my z1 column. What I do, x1 row and
40:19.300 --> 40:26.300
z1 column; I put them parallel and find out
what is minimum. Here, minimum is 0.1 and
40:30.170 --> 40:37.170
here minimum is 0.2. After that, I find out
what is the maximum, which is 0.2. This is
40:38.770 --> 40:45.770
what maximum of minimum 0.1. 0.9 is minimum
0.2. 0.7 is 0.2. This is how we found out.
40:50.089 --> 40:57.089
The easiest way if I want to find out is this
one; this x1 x2 and z1. x2 means this row
41:01.520 --> 41:08.520
which is 0.4 and 0.5 and x2 and z1. z1 is
again 0.9 and 0.7. I will find out. Minimum
41:15.599 --> 41:22.599
here is 0.4, minimum here is 0.5 and maximum
here is 0.5. You get this 0.5 Similarly, we
41:32.099 --> 41:39.099
can compute all the elements in R3 using a
max min composition operation. As usual, any
41:40.599 --> 41:47.599
max min composition can follow certain properties,
associative and distributive over union. That
41:55.310 --> 42:02.310
is P fuzzy composition Q union R is P composition
Q union P composition R.
42:05.320 --> 42:12.320
Similarly, weekly distributed over union is
P composition, Q intersection, R is a subset
42:20.349 --> 42:27.349
of P composition. Q union P composition R
monotonic Q is a subset of R implies that,
42:40.720 --> 42:47.720
P composition Q is a subset of P composition
R.
42:48.030 --> 42:55.030
Fuzzy max product composition operation, earlier
we talked about max min. We will talk about
42:57.380 --> 43:04.380
max product as you know that, in case of crisp
set, both max product and max min yield the
43:05.890 --> 43:12.650
same result. In this case, it will not and
that we will see now. Max product means how
43:12.650 --> 43:19.650
we do? Now, again, the same example we have
taken R1, R2 and R3. Now, I want to find out
43:19.700 --> 43:26.700
from R1 and R2, what R3 using max product
composition is.
43:27.520 --> 43:34.520
Again I do the same thing. Let us say, this
is x1 x2 x3 z1 z2 z1 z2 and this is x1 x2
43:41.440 --> 43:48.440
x3 for x1. I take this row which is 0.1 0.2
and finding the relation the fuzzy membership
43:53.310 --> 44:00.310
associate x1 and z1. I take the column from
z1 which is 0.9 0.7 and I multiply them here
44:01.450 --> 44:08.450
0.1 0.9 is point 0 9 0.2 0.7 is 0.1 4 and
find out what is the maximum. This is the
44:11.589 --> 44:16.630
maximum 0.1 4.
44:16.630 --> 44:23.630
I take another example. Let us find out the
relationship between x2 and z2; for x2 the
44:24.740 --> 44:31.740
row is 0.4 0.5 and z2 the column is 0.8 0.6.
Corresponding to this, if I multiply I get
44:39.260 --> 44:46.260
0.4 0.8 is 0.3 2 0.5 0.6 is 0.3. Maximum is
0.3 2. This is 0.4 3 0.3 2. This is where
44:52.829 --> 44:59.829
it is 0.1. The answer is here, the R3 and
if I go back, if I look, R3 here is different.
45:05.300 --> 45:12.300
Now, we will consider another example of fuzzy
relation. A little more application oriented.
45:20.880 --> 45:27.880
Let us think of 2 different sets of objects.
1 set of objects are images collected from
45:30.300 --> 45:37.300
various objects like car, boat, house, bike,
tree and mountain. We have such images and
45:40.750 --> 45:47.750
now the geometry of these images can be compared
to various known geometries like square, octagon,
45:54.109 --> 46:01.109
triangle, circle, and ellipse like that.
Now, the question we may ask is that, if I
46:11.320 --> 46:18.320
have an image of any of these objects like
car or boat or house or bike or tree or mountain,
46:21.470 --> 46:28.470
how close these images are to an image of
a car? This is a question. How close an image
46:33.570 --> 46:40.570
of car is to those of either a car or a boat
or a house or bike or tree or mountain? I
46:43.270 --> 46:50.270
create a fuzzy set and I call that fuzzy set
to be a car. My fuzzy set A is a car fuzzy
46:51.790 --> 46:58.790
set. Obviously, when I compare the image,
we look at a car image. Obviously, the membership
47:01.420 --> 47:08.420
is 1, because car and boat may not look like
a car, but the fuzzy index is 0.4. Obviously,
47:17.470 --> 47:23.940
a house will not very less likely to look
like a car. The fuzzy index is 0.1.
47:23.940 --> 47:30.940
A bike may look like a car with fuzzy index
0.6 or more; whereas, the tree is similar
47:32.220 --> 47:39.220
like house very little resemblance with a
car with a fuzzy index 0.1, but a mountain
47:39.700 --> 47:46.700
may not have any similarity with the car,
so 0 fuzzy indexes. When I want to look at
47:48.760 --> 47:54.069
all these objects and infer whether they are
cars, the associated membership functions
47:54.069 --> 48:00.150
are 1 0.4 0.1 0.6 0.1 and 0.
48:00.150 --> 48:06.670
Similarly, I look at the various geometrical
features and I want to infer whether this
48:06.670 --> 48:12.829
geometry is a square. Obviously, if I am looking
at a square, actual square the fuzzy membership
48:12.829 --> 48:19.829
is 1, if it is octagon then it is 0.5, triangle
0.4, circle 0, ellipse 0.1.
48:23.619 --> 48:30.619
Now, the third is that this geometry is an
object; whether they have corners? The square
48:31.280 --> 48:38.280
will have 4 corners, octagon has 8 corners,
triangle has 3 corners, a circle has no corner
48:38.599 --> 48:45.530
and ellipse has no corner. All the 8 do not
have exactly corners, but it may appear like
48:45.530 --> 48:52.530
to have some corners. When I look at the fuzzy
set of a corner, then square will have fuzzy
48:55.270 --> 49:02.270
membership 0.6, octagon has maximum number
of corners 0.9, triangle has 30.4 circle has
49:03.970 --> 49:10.970
no corners 0 and ellipse 0.2. This is how
we specify the set car, square, and corner.
49:16.140 --> 49:23.140
Now, we would like to establish the relationship
between car and square. Square and corner
49:24.380 --> 49:31.380
and then infer the relationship between car
and corner using composition rule.
49:34.810 --> 49:40.079
Find a relation R between car and square?
Find a relation S between square and corner
49:40.079 --> 49:47.079
using max min composition rule? Find a relation
T between car and corner? This is the question.
49:49.540 --> 49:56.540
If I adjust the relationship matrix between
fuzzy set A and B, what was A? A was the set
49:58.750 --> 50:05.750
which is car and we defined this fuzzy set.
It will be this way. Car was 1 by car, 0.4
50:12.280 --> 50:19.280
by boat, 0.1 by house, 0.6 by bike, 0.1 by
tree, and 0 by mountain. This is my car subset
50:38.849 --> 50:45.849
fuzzy set and B which is my square subset
is 1 upon square. Octagon has 0.5 and triangle
51:02.500 --> 51:09.500
0.4 and circle has probably the membership
0 because, circle will never look like
a square. Ellipse is 0.1.
51:30.210 --> 51:37.210
How do I find the relationship R? You can
easily do so here. Here this is square; S
51:38.160 --> 51:45.160
stands for square, octagon, triangle, circle
and ellipse in this. This is my B, B set and
51:51.750 --> 51:58.750
this is my A set. In A set, the elements are
car, boat, house, bike, tree and mountain.
52:05.400 --> 52:08.069
How do I find these members here?
52:08.069 --> 52:13.569
All that I have to do is that each element,
because, I have to create an ordered pair
52:13.569 --> 52:19.990
and take the minimum membership function,
the car will be made ordered pair with the
52:19.990 --> 52:26.990
square octagon triangle, circle and ellipse,
car, circle, octagon, triangle, circle and
52:27.440 --> 52:34.440
ellipse and with car when I say square, the
minimum membership function is 1. Car with
52:35.280 --> 52:42.280
octagon minimum membership is 1, between 1
and 0.5 is 0.5. Similarly, car and triangle
52:44.450 --> 52:51.450
the minimum membership is between 1 and 0.4.
Similarly, car and circle the minimum between
52:53.319 --> 53:00.319
1. 0 is car and ellipse is between 1 and 0.1
minimum is 0.1.
53:00.920 --> 53:07.920
Similarly, I would find out the relationship
between B and all other elements in the boat
53:10.930 --> 53:17.930
in set A with all the elements in B. Boat
has a membership function 0.4 when I compare
53:21.310 --> 53:28.310
a square, obviously, the minimum is 0.40.4
and then with octagon 0.5, because this is
53:29.339 --> 53:36.339
0.4 0.5. I have 0.4 is 0.1. Like that I can
fill all these entries.
53:43.240 --> 53:49.160
What is the method? The method is that in
each ordered pair, I find out what is the
53:49.160 --> 53:56.119
minimum of the membership function. For example,
if I make an ordered pair between this and
53:56.119 --> 54:03.119
this. The minimum is 0.4 and 0.1 is 1 0.1
0.4 0.1 0.4 0.1 is 0.1. You can easily see,
54:07.960 --> 54:14.960
this is triangle and this is tree. This is
tree triangle and this is tree and the relationship
54:17.150 --> 54:24.150
is 0.1. This is how we find out the membership
function between A and B. Similarly, this
54:24.760 --> 54:31.760
is the relationship between B and C. This
is a set B and this is C set. C set B is a
54:42.000 --> 54:45.030
square and set C is the corner.
54:45.030 --> 54:52.030
Once I found what is R and S, the next is
to find out the relation T. The relation T
54:54.960 --> 55:01.960
between car and corner can be found out using
inference mechanism which is composition rule.
55:06.690 --> 55:13.690
We have used this as max min composition.
This has been found out using max min composition
55:17.599 --> 55:23.150
and as I explained earlier how to find out
max min composition, what I would like to
55:23.150 --> 55:28.839
show you here is the same matrix. That is
the relation matrix that I have got; I can
55:28.839 --> 55:35.839
also get directly from this. This is my A
and this CA stands for car.
55:40.319 --> 55:47.319
My car was 1 and 0.4 by boat, and 0.1 by house,
0.6 by bike, 0.1 by tree, 0 mountains. This
56:11.470 --> 56:18.470
is my car and C is corner which shows the
corner. The square is a 0.6. Octagon has maximum
56:35.569 --> 56:42.569
number of corners 9 0.9. Then triangle has
3 corners 0.4 circle has no corner and finally
57:00.440 --> 57:07.440
ellipse may look like have some corners. 0.2.
57:07.540 --> 57:14.540
Although we found by max min composition principle,
the same thing also, you can find out simply
57:14.950 --> 57:21.950
looking at these 2 sets; car and corner. You
can easily see that when I relate car with
57:22.230 --> 57:29.230
this these things, you should say because
car has membership function 1. All these membership
57:29.540 --> 57:36.540
function always will be minimum with respect
to 1. They will be there 0.6 0.9 0.4 0 and
57:38.930 --> 57:45.930
0.2. Next, when I go to boat 0.40.4 is minimum
here, 0.4 is minimum here, 0.4 is minimum
57:47.500 --> 57:54.500
here. Past 3 will be 0.4 and then 0 minimum,
here 0.2 minimum here. Exactly, whatever I
57:55.910 --> 58:02.910
got from inference mechanism, the same mechanism
and same thing I will get, using directly
58:03.770 --> 58:09.660
looking at car and corner.
58:09.660 --> 58:16.660
Finally, the summary: what we discussed today,
we extended the function mapping concept to
58:19.520 --> 58:26.520
fuzzy sets. The crisp relation and fuzzy relation;
the difference was discussed today, where
58:31.690 --> 58:38.690
we showed that crisp relation; the index is
either 0 or the membership grade is either
58:40.170 --> 58:47.170
0 or 1; that is, either complete relation
or no relation. Whereas, in fuzzy the relation
58:47.470 --> 58:54.470
has a grade from 0 to 1. Fuzzy composition
rule; max min composition max product composition
58:56.000 --> 59:02.920
unlike in crisp relation, where both max min
and max product gives you the same answer;
59:02.920 --> 59:09.920
whereas in fuzzy composition, max min and
max product will give two different answers
59:11.849 --> 59:18.849
and fuzzy projection and cylindrical extension.
We discussed and also we illustrated all these
59:21.480 --> 59:28.480
principles through various examples. Thank
you very much.
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