Welcome to lecture 3 of the course on Fuzzy
Sets, Logic and Systems and Applications.
So, in today’s lecture we will discuss Fuzzy
Sets and logic Fuzzy Logic Toolbox in MATLAB.
So, before we discuss fuzzy sets let me introduce
classical sets just before going to fuzzy
sets, because we need to first understand
what is a set, I mean conventional set. And,
then we will from the classical sets we will
transition to the fuzzy sets and that this
is needed for better understanding. So, what
is a classical set? Classical set normally
is a collection of objects from the universe
of discourse.
So, if we take here classical set whose universe
of discourse is capital X. So, let X be the
universe of discourse and small x belongs
to the capital X. So, small x, x is an element
that belongs to the universe of discourse.
Then a classical set A can be defined as A
is equal to the collection of all the x’s
in the set and the condition here is that
the x should meet certain criteria or conditions.
And, as I mentioned that this x must be belonging
to the universe of discourse.
So, any element which is not belonging to
the universe of discourse should not be part
of this set. So, here we all know now that
conventional set A is the collection of all
the elements and these elements are from,
are drawn from the universe of discourse capital
X. So, this kind of collection is termed as
a set.
If we take an example here, like if we write
a set of positive integers less than 20 and
more than 15, so, and the universe of discourse
of course, that is X is set of all positive
integers that is I plus. So, if we are writing
a classical set A, so this classical set A
will be containing all the elements that satisfies
this criteria. So, if we write set A; set
A will be like this. So, A is equal to the
set the collection of all the elements that
is x which is satisfying this criteria.
What is this criteria? This criteria is the
all the elements must be greater than 15 and
less than 20 and as I mentioned earlier also
that this x must be drawn from the universe
of discourse. So, now, we can write this set
as the capital A which is the name of the
set, the classical set as collection of all
the elements in between 15 and 20. So, in
between 15 and 20 are coming out to be 16,
17, 18, 19 the part of this set and if we
look at the set here all the elements here,
then we find that these all the elements are
drawn from the positive integers, the set
of positive integers I plus.
So, this way we can say that A is a classical
set. Now, here a very important point that
we should note is that’s the element that
are present in the classical set are completely
present. What do I mean by completely means
16 is present with 100 percent, 17 is present
with 100 percent, 18 is present with 100 percent,
19 is present with 100 percent. So, 16, 17,
18 are completely present in the system. So,
the classical set, in classical set if we
have the element that present in the classical
set they all are present with 100 percent.
Now, if we look at the above classical set,
it is clear that any of the elements A are
either completely belongs to A, means they
are present with the with 100 percent or completely
does not belong to A. Means, the elements
that are not present in the set, means they
are not present in the set with 100 percent,
means their percentage is 0. So, any element
which is not present in the set, it indicates
here, it means here that the element is present
with 0 percent, means element is not present.
So, for the same classical set A let us let
me make this thing more clear, in other words
we make say that the membership value. So,
a new term is coming over here is membership
value. So, when I was mentioning 100 percent,
it means 100 percent is the membership value;
means here is that if we have classical set
or conventional set, so, this classical set
or conventional set, the element that are
present they are present with 100 percent
membership value.
So, the membership value here, the membership
value of 16 is 100 percent; although it is
not written, membership value of 17 is 100
percent, membership value of 18 is 100 percent,
membership value of 19 is 100 percent. But,
all these membership value are not written
in the classical set, because it is understood
that they are completely present. So now,
if we look at this set again, what we see
here the elements that do not belong to the
classical set A have their corresponding membership
value zero.
Of course, because the conventional set is
based on the Boolean logic. So, any element
which is present will have 100 percent presence
or element or any element which is not present
will have 100 percent absence. So, if we have
any element present in the classical set,
we will say that it is true and if the element
is not present with 100 percent means the
0 percent it is termed as false. So, there
is no possible case in between any of the
element.
So, this point is very important to be noted
because in classical set either an element
in the set is present or it’s not present.
But, if we talk of fuzzy logic, since fuzzy
logic is based on the multi-valued logic;
so, the element can be present in the set
in fuzzy set with varied membership value.
What does this, what does this mean here is,
that any element may be present with the membership
value more than 0 and less than 1; the elements
with this value may be present, but not completely
present. Any element which is present with
100 percent membership value is completely
present.
So, as I just mentioned that multivalued logic
here, with the multivalued logic in fuzzy
system; so, here not only the true or false,
the, so every element in a fuzzy logic A will
be assigned its membership value. So, let
me make it very clear that any element, any
element which has the membership value more
than 0 will be the part of a fuzzy set. And,
also if the element is not completely present
means if the element has the membership value
0, the element will not be part of the fuzzy
set.
So, let us now move to fuzzy set from the
classical set and in classical set we do not
write any membership value as I mentioned.
Because, it is assumed that any element which
is present in the classical set it is, it
is assumed, it is understood that the element
that are in the conventional set, or classical
set, or traditional set which is based on
the Boolean logic, the elements are with 100
percent membership value.
So, that is why it is not written, but if
we talk of fuzzy set here unlike the conventional,
traditional fuzzy logic, fuzzy logic, in fuzzy
logic the elements that are present in the
set they these will have the membership values
in between 0, in between 0 plus and 1. So,
the elements, these elements will be with
the respective membership values. This is
needed because, otherwise we may not be knowing
with what value, with what degree, with what
membership value these elements are present
in the sets.
So, here like in the classical set that we
have had just before we had 16, so, 16 was
100 percent present, 17 was 100 percent present,
18 was 100 percent present, 19 was 100 percent
present. And of course, all these elements
were drawn from the universe of discourse.
So, here also we will have a universe of discourse
and these elements must be from the universe
of discourse, some universe of discourse say
X. Now, in classical sets these all were present
with 100 percent means completely present,
but let us assume a case where this 16 is
not completely present, I would say partially
present. And, if we say partially present
means it is, it has some value, it has some
membership value less than 1.
So, if it has some membership value less than
1, say 0.6. So, this 0.6 is the membership
value of 16. So, this 16 and 0.6 these two
are very important for together to be included
in the in the set. And, since this is not
100 percent present; so, we need to know with
what degree it is present. So, that is why
the degree along with that element is needed
and that’s why if we see in a fuzzy set
every element is with some degree. So, every
element is paired with some degree and that
is how a fuzzy set is formed.
So, if we see here this fuzzy set A, now this
is not a crisp set, it is a fuzzy set because
the element 16, 17, 18, 19 they all are having
some degree associated with these element.
So, 16 is with 0.6, 17 is with 0.9, 18 is
with 0.2, 19 is with 1. So, if we see here
that fuzzy set is the ordered pair of all
the elements and this pair is nothing, but
the first element of the this pair is the
element and the second element of this pair
is the membership value. So, 16, 0.6 means
that this 16 is present with 0.6 membership
value, similarly 17 is present with 0.9 membership
value, 18 is present with 0.2 membership value,
19 is present with 1.0 membership value means
100 percent. So, this 19 is completely present,
18 is partially present, 17 is partially present
and 16 is also partially present. So, this
way fuzzy set is a set which has all the elements
with its membership values. So, we can also
say the same thing as it is written over here;
a fuzzy set A can be written as a set of ordered
pairs of the element and its belongingness,
and this belongingness is also called as membership.
So, this way fuzzy set can be written and
this is a transition from the classical set.
In classical set we saw that 16 was 100 percent,
17 was 100 percent, 18 was 100 percent, 19
was 100 percent. So, that is why there was
no need to include any membership value along
with these elements. So, that is why it was
not needed, but here since the since fuzzy
set fuzzy logic fuzzy system is based on fuzzy
logic which is of course, a multivalued logic
and because of that the no matter whether
a element an element is present with 100 percent
or not, all the element from the universe
of discourse must be included in the fuzzy
set. So, that is why any element which is
present, even if it is not even if it is not
present with 100 percent all the element has
been included here.
So, this is how we transition from the classical
set to a fuzzy set. So, we clearly see there
is a need of writing, there is a need of including
here the membership value with the elements
with element, without that it is difficult
for us to write a fuzzy set. So, that is why
we include a membership value and the membership
value here is represented as mu A, membership
value of any element is represented by mu
of A x. So, if we have any element x and the
corresponding membership value will be mu
A and this mu x and this A is nothing, but
the A signifies the particular the name of
the set. So, A is basically a particular set.
So, a fuzzy set in the fuzzy set representation
we can write a fuzzy set either in discrete
form or in continuous form. So, in discrete
form if we write a fuzzy set, this is the
way we write like, you see here that same
fuzzy set; that we already discussed could
be written as if we have let’s say are already
had fuzzy set A which was like this 16 with
ok. I have I am taking some other member membership
values like let’s say 0.2 and then 17 with
0.1 and 18, 0.5 say 19, 1 like this. So, if
we have this as a fuzzy set, this is a crude
representation of the fuzzy set.
So, the same fuzzy set can be represented
as you see first the membership value and
then corresponding membership corresponding
element and then with plus sign we add, but,
please understand that there is there will
not be any addition of thisthis values. So,
0.1, 17 and then 0.5, 18 plus 1, 19. So, the
same fuzzy set which you see here can be written
as A is equal to 0.2 oblique 16 plus 0.1 oblique
17 plus 0.5 oblique 18 plus 1 oblique 19.
So, we see that first we write the membership
value and then with oblique sign with line,
a slanted line we write here the corresponding
element.
So, if we have any discrete; so, let me first
make it clear that a fuzzy set can be the
discrete or a fuzzy set can be continuous.
So, if we are writing a discrete fuzzy set,
we write the discrete fuzzy set this way as
I just mentioned. And, then we see here the,
you know the representation wise here the
same thing can be written like this, like
form in which we write a fuzzy set. So, if
we have a fuzzy set A fuzzy set A should be
written as like A should be equal to mu A
x 1 mu A x 1 oblique x 1 plus mu A x 2, if
we have a x 1 x 2 x 3; all these the corresponding
generic variable value values drawn from the
universe of discourse.
So, this plus sign is very important here
because this plus sign this should be noted,
it should be noted that this plus sign this
plus sign is not here for addition. So, please
understand that although we have plus signs
here, all the elements are separated by plus
sign, but this plus sign is not here for addition.
So, please do not add these values with this
plus sign, if you add these values together
this will be, this will not be a fuzzy set.
So, this will become this will be wrong thing.
So, please do not add this, these plus signs
you leave as it is.
So, this is and please understand this is
another way of writing the same thing, you
can use summation also. Like if we do not
want to write it like this, we write we can
use a summation and then we can write universe
of discourse from which the all the elements
are drawn. So, this X will be written here
and then mu A x i over this x i can be written.
So, this way the discrete fuzzy set is written,
then if we want to write, if we want to represent
a continuous fuzzy set. So, a continuous fuzzy
set of course, this will be from the infinite
universe of discourse.
And, this fuzzy set will be written as you
see a fuzzy set A is written as A is equal
to we use here instead of this summation sign
we use the integration sign. So, please understand
here that this if we have continuous fuzzy
set, we use the integration sign. So, and
then after this integration sign we have again
the same logic that logic is first we write
the membership value. And, then with this
slanted line we separate and then we write
the corresponding generic variable value.
Please understand this mu A x i will be continuous
function, because if we are writing fuzzy
set in a continuous format of course x i will
be a continuous function. So, this is this
will be a continuous, this corresponding the
values here the generic variable values will
be x i. So, this way we write continuous fuzzy
set in a continuous, in a continuous format.
And, as it is written here that summation
and integration signs indicate the collection
of all the element in the universe of discourse.
So, along with their associated membership
values. So, please do not use this summation
and integration, so, summation for addition
and do not integrate the function that is
coming so, but you just leave it leave as
it is.
So, mu A x as I mentioned mu A x is nothing,
but a membership value. So, mu A x is termed
as the membership value for element x in fuzzy
set A and this gives a single value for every
element contained in fuzzy set. So, membership
value, values for every x can be found from
membership function. So, if we have a membership
function let us say mu A x i, it should be
x i here, it should x i. And, then if it is
a corresponding if this is the membership
value, membership function and if we want
to have a corresponding membership value.
If we have a generic variable let us say x
i, ith value i of the in the universe of discourse,
ith element some element x i. So, corresponding
that x i we will have mu A x i and then this
is a this will be a single value. So, and
one more thing that to be noted here is that
this membership values will be in between
0 and 1. So, the membership value which is
written here, that membership values lie between
the interval 0 and 1 for a normal fuzzy sets.
So, when we say a normal fuzzy set, it means
they it means there may be a fuzzy sets which
are not normal.
So, wenormally call those sets as subnormal
fuzzy sets. So, so fuzzy set which has at
least one membership value equal to 1 is a
normal fuzzy set. But, any fuzzy set which
does not have any of the values as equal to
any of the membership values equal to 1, it
is called subnormal fuzzy sets. So, in other
words you can say the subnormal fuzzy sets
are those sets whose which does not have any
membership value up to 1. So, with this I
would like to stop here.
And, in the next lecture we will be discussing
few examples on fuzzy sets and fuzzy logic
toolbox in MATLAB will also be discussed.
