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Hello there and welcome to the second lecture
of our second module, random events. In
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today’s lecture, we will cover set theory
and set operations, which is very important
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in
understanding the concept of probability theory.
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So in this class, we will know the
concept of set and different set operations,
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which will be useful to describe the different
probability of different set and subset of
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the outcome of a particular experiment.
.
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First, we will know the definition of the
set and set theory, basically a set is a collection
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of physical or mathematical objects called
as members or elements. So the elements of
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a
set do not require to have any similarity
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among them except that they belongs to the
same set. So, basically a set, if we want
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to define that, it is a collection of a particular
group of elements and those elements can be
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physical or mathematical in terms of
different outcomes of a particular experiment,
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if we think in the context of our
probability theory.
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.Now the set theory says the theory concerned
with those properties of a set, which are
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independent of the particular element belong
to the set, so when we say set theory, those
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theories are generally applicable to the whole
set, it is not related to a particular element
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of a set. So, the overall the set, what are
the different operations that we can do, what
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are
the different theories that is applicable
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to the full set, and that consist that is,
we call as
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set theory.
.
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So, with this, if we see that different examples,
so if we start with our classical example
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one for example, if we take a coin and if
it is tossed twice, then the outcome of this
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experiment can be: either the first and second
both are head, or head-tail, or the first
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one
is tail and second one is head or both are
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tail. So the outcome of this events, that
is the
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full, all the collections of this experiment
is one set. If an experiment is carried out
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to
determine the strength of concrete, in particularly
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in the civil engineering, then all the
experimental values, generally they form a
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set. In the other example of the stage height,
generally, we measure the stage height at
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a river gauging site, there if we take that
the
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river gauging site, if we take the stage height
at a particular gauging site, then the all
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possible values of the stage height form the
set.
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So, here you can see that these things can
have any value depending on the feasible range
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of the strength of the concrete depending
on various factors its water cement ratio
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or the
time and type of curing and all these things,
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it can take any values. For this type of set,
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.we can say, whenever we are saying that the
stage height, this is generally called, one
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side it is bounded by 0, so physical or in
the mathematically, it can take any value
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starting from 0 to the plus infinity.
.
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So, all this outcome if we take a total, this
is generally this form, generally a set. There
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are different properties that we will see
now one after another, the first thing is
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that when
we say that both the sets are equal. So two
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sets are, we generally say, these are equal
if
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all the elements of one set belong to the
other set and vice-versa. Here in this pictorial
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representation, there are two sets are there;
one is drawn by the red circle and another
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one is drawn by the black circle, if we see
that these are exactly same, so whatever the
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entries that belongs to the red set, there
are some red dots and the black dots and also
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there are some… So, all these dots belong
to both the set S as well as set T. Now, so
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we
can say that this set S and set T are equal.
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In the example of the throwing a dice, it
is
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having the six different outputs, now if I
say the one set is even and the outcome is
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even
number and if in other words, if I say the
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outcome consists of 2, 4 and 6, then we can
say
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that these two sets are equal.
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..
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This is very important which is known as the
subset; a set S is a proper subset of another
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set if we can designate that T when we say
that all the elements of this set T belongs
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to
the former one that is S, but the opposite
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is not true, that is the vice versa is not
correct.
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So if you see this pictorial representation,
this the bigger ellipse is the T set and inside
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this there is another set, which is shown
in a red circle, so this S belongs to this
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T, so this
S is known as the subset of this the larger
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set T.
So, in the example of the river gauging site
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if I say that the extreme values of the stage
height, then I can say that, if I say that
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all the stage height that is above say 10
meter or I
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can say that another set, which takes all
possible values. So, the all possible values
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means that can take any value from 0 to whatever
the feasible maximum available
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height in the historical record or it can
be even higher than that. And another set
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is that
the just extreme value which I put a threshold
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of the 10 meter, then I can say that the
threshold, which is more than 10 meter is
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a subset of the full set, which can take any
value from 0 to mathematically plus infinity.
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..
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The second thing that is known as a null set,
when we say that there is a particular set,
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which does not have any element in it, then
we say that this is a null set and it is defined
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as that when the set contains no element and
this is denoted by this kind of symbol you
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can see here that there is a set, where there
is no element then we can say that this is
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a
null set. In the probability concept, we can
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say that this null set is generally a set
that is
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the impossible set, that is impossible outcome;
for example, if I say that the strength of
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concrete, then if I say some value which is
negative, then we can say this is an
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impossible event and this set is a null set,
which does not have any entry, which belongs
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to the negative value, less than 0.
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..
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Generally the sets are classified by its size
according to the number of elements that they
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are having; a set a may be finite or infinite,
a finite set consists of a whole number of
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elements, which is known as the cardinal number.
Now if any set that is having the all
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the with the cardinal numbers, is called the
countable set; and infinite set, generally
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infinite set, when we cannot count or the
number of elements that belongs to it is infinite
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or sometimes another term is used that is
called the countably infinite. It can be
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countable, but it can be infinite in number,
those set, those size of set is known as
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countably infinite set. So, these are all
the basic concept that will be useful to understand
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the other properties and the operations of
set.
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..
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The second one, another important thing is
ordered, a set is generally said to be ordered,
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if a relation like the greater than or less
than can be defined for any two elements of
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the
set. For example, if I say, if I take two
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elements a and b, I can say that a is greater
than b
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or a is less than b is possible, then this
kind of set is known as the ordered set. We
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can
even conclude that if we say that a is less
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than b, and b is less than c, then I can say
that
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the element a is less than element c, so this
kind of set is known as the ordered set.
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.
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.These two are two operations, which is very
important and frequently used; These are
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known as – one is union and another one
is the intersection, these concepts are very
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important to understand the later part of
this class. First, we take this union, a union
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of
two events that is A and B, here you can see
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that this is your, full set; this is your
full
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sample space and in this space, there are
two events, one is this the left hand side
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circle
and another one is the right hand side circle,
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which is denoted as B and the earlier one
denoted as A. So, this two are events in the
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sample space S; now the union of A and B is
generally denoted as A this symbol B, this
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is generally read as the A union B; this A
union B consist of all the outcomes that is
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either in A alone or B alone or both in A
and
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B. So you can see this three regions, where
I am pointing now, this area belongs to A
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only, this area belongs to the B only, and
this in between belongs to both A and B. So
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if I
say that A union B, then this area which is
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highlighted in red, this area in total is
known
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as the A union B.
.
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On the other hand, if I say the intersection,
again this two events are defined here, one
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is
A and another one is B, and this is its intersection
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denoted as this symbol , and it is read
as the A intersection B. So, we call that
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intersection is defined as all the outcomes,
those
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are in both A and B, so it must be both in
A and B. So obviously the highlighted area,
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the
red highlighted area is known as the intersection
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between A and B. So once again the
union is the total area that consist of either
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B or A or both A and B, that is the union;
and
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intersection means, it is both on A and B.
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..
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There are other few concepts that are also
useful; here one is the mutually exclusive
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set,
this mutually exclusive set between two events
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A and B, we can call that this A and B
are mutually exclusive, when no outcome, no
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element in A are in B and of course vice
versa. So there is no overlap between these
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two sets. So if we say that there is no overlap
between these two sets that means. if I take
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the intersection between A and B, now you
know this symbol is an intersection, that
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means in this intersection there is a null
event,
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where there is no element existing in this
part.
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So you can say that if this two are completely
separated, there is no overlap area, these
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two events known as the mutually exclusive
event, in the example of throwing a dice.,
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If
I say that one set consist of all the even
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numbers and another set consist of all the
odd
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numbers, then you can easily say that there
is no intersection between these two events,
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and so all the set consist of the even number
and the set consist of the odd number are
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mutually exclusive sets.
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..
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There is another very important and very frequently
used concept, which is known as
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the collectively exhaustive sets. Now the
collectively exhaustive sets means, when I
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take
some subsets or some events or some sets;
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and if I take the union of all this events
then I
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can say, if the union of this sets comprise
the whole set, whole set means that whole
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sample space if it consist of, now you can
see that there are few subsets defined here
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say
for example, this area is your A1 , this area
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is your A2 , in this way it is going on, and
there are some subsets like this.
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Ultimately, if I take the union then obviously,
the full union consists of this whole
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shaded, red shaded area, so this whole red
shaded area is nothing but you have the full
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sample space. Here one thing is important
to note that the intersection of any two sets
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need not be null sets, so you can see that
this A1 and A2 the intersection is null but
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this
A1 and if I say that this is another set which
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is a A3 , so there are some intersection point.
So we are not considering the intersection
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point here we are just considering the union.
So, if I can somehow say that all this unions
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consist of the full set, then I can say that
this
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is a collectively exhaustive set. In the example
of throwing a dice if I say that one event
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is less than 4 and another event is greater
than two, and if I take the union of this
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two
obviously, the union will consist of all possible
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outcomes starting from 1 to 6. So the two
sets, which is that less than 4 and greater
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than two, there may be some intersection but
if
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I take the union of this two, it consists
of 4 full possible sets. So, these two events
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are
mutually exclusive sets.
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..
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Then the partition of a set. Now this partition
of a set, suppose that we have one set in
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hand, and I want to make it partition. Here
I just make this is say the another subset
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A1 ,
this is A2 and in this way it is going on
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up to n, so n number of subsets are there
in such
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a way that two conditions must be satisfied.
The first condition is that if I take the
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union of all the sets, it should consist of
the full
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set, which is here shown as S, the full set.
And if I take the intersection of any two
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sets as
long as i is not equal to j, then this intersection
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A i intersection A j should be a null set.
You see here no events are intersecting each
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other, so the intersection between any two
event, any two subset is the null set, then
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we can say that this S is partitioned by A
1 A 2
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A 3 up to A n. So if partition U of the set
S is the collection of …, that will be one
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word
of the collection of mutually exclusive and
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collectively exhaustive. So, this is a new
word again we will just explain it in a moment.
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So the mutually exclusive, this mutually exclusive
means, all this unions consist of this
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full set, again mutually exclusive and collectively
exhaustive subset are A i, of the whole
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set such that as I told there are two conditions
one is that A 1 to A n, if I take the union
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of
all the set it consist of the full one and
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any two set in the section is 0. So in this
way if
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satisfying this two condition, if we can make,
if we sub divide the sets, then this is
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known as the partition of a set.
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..
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Now, the set operations; there are different
set operations; the first thing is the
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complement. A complement of a set A are the
outcome in S, those are not contained in
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A. So, suppose that this is your full sample
space A, and there is one event A. So, this
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is
your event the complement of A is the area,
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is the set, which does not belong to A. So,
if
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I say that the even numbers of throwing a
dice experiment if the even number consist
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of
one event, then the complement of this set
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is the odd number of the outcome.
So, one thing is very important to note here
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is that if I take the union of a set and its
complement then we get the full set, full
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feasible set, which is S. This complement
is
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generally denoted by either A
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C
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or A prime or A bar, so all this notations
are just a
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notational difference, but these are all the
meaning the same, the complements of a
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particular set. In this class, we will use
this concept, which is A bar to denote that
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which
is the complement of a particular set.
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..
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Another set operation, which falls on this
De Morgan’s law; okay, before I go to this,
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I
should tell that this unions and intersections
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00:20:40,250 --> 00:20:44,049
are also the different set operation, which
is
211
00:20:44,049 --> 00:20:49,490
also very important, just it is covered and
the complements also we covered, and using
212
00:20:49,490 --> 00:20:55,159
this two things we will now explain a very
important law which is known as the De
213
00:20:55,159 --> 00:21:03,200
Morgan’s law and that is very helpful for
different kinds of operations on different
214
00:21:03,200 --> 00:21:09,340
sets.
So, this is De Morgan’s law; when in a set
215
00:21:09,340 --> 00:21:13,490
identity, all the subsets are replaced by
their
216
00:21:13,490 --> 00:21:23,059
complements, all unions by intersection and
all intersections by unions, the identity
217
00:21:23,059 --> 00:21:28,280
is
preserved. May be when I am just telling this
218
00:21:28,280 --> 00:21:35,090
one it may not be very communicable, but
we will just explain it, what all these things
219
00:21:35,090 --> 00:21:42,240
means. Before that, just look at this diagram,
where there are two sets one is A and another
220
00:21:42,240 --> 00:21:50,710
one is B, and if we just tell once again,
recapitulate what just now we have seen. So
221
00:21:50,710 --> 00:21:58,279
this is your A, this area is your B, so we
know that this is in between the intersection
222
00:21:58,279 --> 00:22:02,050
between this area, which is is little bit
gray
223
00:22:02,050 --> 00:22:07,640
looking, this is the A intersection B. Here
when we write that A B side by side it
224
00:22:07,640 --> 00:22:13,650
indicates that this is the intersection; and
when we write this symbol, this is generally
225
00:22:13,650 --> 00:22:25,150
the union. So, this area is your A intersection
B; now so if I say this pink area around this
226
00:22:25,150 --> 00:22:31,039
one, then it is A complement intersection
B complement.
227
00:22:31,039 --> 00:22:37,580
So what is A complement here? So, whatever
the set that we can see here outside of this
228
00:22:37,580 --> 00:22:44,360
area is your A complement, and here it is
B, and outside of B is your B complement.
229
00:22:44,360 --> 00:22:50,500
Now all this outside area, outside B that
is the complement B and complement A, if we
230
00:22:50,500 --> 00:22:57,810
.take the intersection between these two,
then only this area where my curser is moving
231
00:22:57,810 --> 00:23:07,260
now, only this area is your A intersection
as A complement intersection B complement.
232
00:23:07,260 --> 00:23:14,590
Now if I say that A B complement A intersection
B that complement, then it consist of
233
00:23:14,590 --> 00:23:21,960
this set, this area as well as this area,
which is not in this intersection but the
234
00:23:21,960 --> 00:23:25,520
rest of this
A and rest of this B, which is not in this
235
00:23:25,520 --> 00:23:28,450
one. So this understanding of these two that
is A
236
00:23:28,450 --> 00:23:33,890
complement intersection B complement and A
intersection B that complement is
237
00:23:33,890 --> 00:23:40,880
important. So, in this A intersection B complement
consist of this pink area as well as
238
00:23:40,880 --> 00:23:47,409
this A area apart from this intersection and
this B area not belonging to this intersection;
239
00:23:47,409 --> 00:23:53,970
these three area consist of A intersection
B complement and only this outside area which
240
00:23:53,970 --> 00:24:01,670
is not in A, not in B, not in A and B, that
is that A complement intersection B
241
00:24:01,670 --> 00:24:06,990
complement.
Now if we know these three area, then we can
242
00:24:06,990 --> 00:24:14,090
just appreciate these two relationship just
you can see that if A and B are the subsets
243
00:24:14,090 --> 00:24:25,470
of a set S, then A union B complement means,
this A first of all we will see, where is
244
00:24:25,470 --> 00:24:31,220
the A union B, this A union B is nothing but
this
245
00:24:31,220 --> 00:24:41,059
area and its complement means, this area what
just now I explained, which is nothing but
246
00:24:41,059 --> 00:24:49,390
here that A complement intersection B complement,
so this relationship. we can show it
247
00:24:49,390 --> 00:24:57,680
pictorially. Similarly, if I say that A intersection
B, then the A intersection B is nothing
248
00:24:57,680 --> 00:25:03,880
but only this area now if I take the complement
of A intersection B, which is shown by
249
00:25:03,880 --> 00:25:09,820
this bar here which is nothing but the summation
of three areas; one is this area plus this
250
00:25:09,820 --> 00:25:19,730
area plus this area and this three area is
nothing but your A, so this three area is
251
00:25:19,730 --> 00:25:22,720
shown
here now again if I see that this one, the
252
00:25:22,720 --> 00:25:32,410
A complement union B complement that is that
A outside B outside, if you take the union
253
00:25:32,410 --> 00:25:36,950
of this two area, then this three area will
be
254
00:25:36,950 --> 00:25:41,120
added, so this area plus this area plus this
area, so this two relationships also holds
255
00:25:41,120 --> 00:25:44,980
good.
Now again I am just reading this De Morgan’s
256
00:25:44,980 --> 00:25:53,059
law, when in a set identity all the subsets
are replaced by their complements, all the
257
00:25:53,059 --> 00:25:59,840
unions by their intersections, all the
intersections by unions, the identity is preserved.
258
00:25:59,840 --> 00:26:05,529
So, I am just making all the symbols
just opposite in the sense that if I say that
259
00:26:05,529 --> 00:26:11,590
intersection is opposite to the union or the
subsets are opposite to their complements,
260
00:26:11,590 --> 00:26:15,110
then if I just make this conversion, then
this
261
00:26:15,110 --> 00:26:20,289
identity is preserved, this will be more clear
in the next slide, when I just see this
262
00:26:20,289 --> 00:26:21,739
relationship.
263
00:26:21,739 --> 00:26:22,739
..
264
00:26:22,739 --> 00:26:32,110
This is a demonstration of that what just
now we told is that this is your A intersection
265
00:26:32,110 --> 00:26:38,231
B
union C, and A intersection B union A C; you
266
00:26:38,231 --> 00:26:45,490
can even draw another figure just to see
that whether these two quantity are same or
267
00:26:45,490 --> 00:26:52,880
not; you can see by a simple Venn diagram
that these two quantity are same. Now you
268
00:26:52,880 --> 00:26:57,279
see this relationship, and you just see the
last
269
00:26:57,279 --> 00:27:03,600
line first before you go through all these
steps, just see the last line here. Now this
270
00:27:03,600 --> 00:27:07,850
A
here, this A is replaced by its complement,
271
00:27:07,850 --> 00:27:11,880
this intersection is replaced by the union,
this
272
00:27:11,880 --> 00:27:20,309
B is replaced by its complement, this union
is replaced by intersection, this C is replaced
273
00:27:20,309 --> 00:27:25,409
by this C complement.
Similarly on the right hand side, this A is
274
00:27:25,409 --> 00:27:33,270
replaced by its complement, intersection is
replaced by union, B replaced by B complement
275
00:27:33,270 --> 00:27:40,230
similarly, this union replaced by this
intersection, A by A complement intersection
276
00:27:40,230 --> 00:27:47,260
by union and C by C complement. So
according to the De Morgan’s law if this
277
00:27:47,260 --> 00:27:51,390
two are equal, then again this two also will
be
278
00:27:51,390 --> 00:27:56,081
equal.
Now, let us see that in between, how we can
279
00:27:56,081 --> 00:28:04,289
show this thing. Just in the previous slide
we showed that this A B, A intersection B
280
00:28:04,289 --> 00:28:10,450
complement is A complement union B
complement. So using this one and we have
281
00:28:10,450 --> 00:28:18,980
described this one with this simple Venn
diagram, using this relationship here that
282
00:28:18,980 --> 00:28:24,880
A intersection B complement equals to A
complement union B complement. If we use this
283
00:28:24,880 --> 00:28:34,380
identity and we want to put this one is
that A intersection B union C full complement,
284
00:28:34,380 --> 00:28:38,590
so this is one set A, and this is B union
C
285
00:28:38,590 --> 00:28:44,780
.is another set, so this is A corresponds
to A here and B union C corresponds to another
286
00:28:44,780 --> 00:28:52,210
set B here, then using this relationship we
can write that this A complement union B
287
00:28:52,210 --> 00:29:02,400
union C complement. Again this in the previous
slide another identity that A union B
288
00:29:02,400 --> 00:29:08,559
complement is equals to A complement intersection
B complement; using this identity,
289
00:29:08,559 --> 00:29:16,470
we can even this B union C from this right
hand side, we can just write that A
290
00:29:16,470 --> 00:29:23,200
complement union B union C using this one
now, this B corresponds to A and C
291
00:29:23,200 --> 00:29:30,659
corresponds to B, we can write that A complement
union B complement intersection C
292
00:29:30,659 --> 00:29:36,139
complement.
Similarly, so from this one I just prove that
293
00:29:36,139 --> 00:29:40,299
this is equals to this quantity; similarly,
if I
294
00:29:40,299 --> 00:29:48,090
use that A intersection B union A intersection
C and its complement, this will be equal
295
00:29:48,090 --> 00:29:55,311
to, using this identity again; they are the
A intersection B corresponds to A and A
296
00:29:55,311 --> 00:30:01,159
intersection C corresponds to B, then using
this identity, I can explain that this is
297
00:30:01,159 --> 00:30:06,230
A
intersection B its complement intersection
298
00:30:06,230 --> 00:30:14,730
A intersection C its complement, which is
again equal to by using this back, sorry,
299
00:30:14,730 --> 00:30:24,190
using this identity; A B equals to A complement
union B complement; we just put this one here,
300
00:30:24,190 --> 00:30:33,820
so that we will get A complement union
B complement and A complement union C complement.
301
00:30:33,820 --> 00:30:40,559
Now you see this relationship that is this
one is equals to this, and this entity this
302
00:30:40,559 --> 00:30:45,790
set is
equals to this entity. Now, I have started
303
00:30:45,790 --> 00:30:49,130
with the relation that this set, this full
set is
304
00:30:49,130 --> 00:30:54,289
equals to this full set. Now, so if these
two are equal, then their complements are
305
00:30:54,289 --> 00:30:58,520
also
equal; now this is nothing but the complement
306
00:30:58,520 --> 00:31:03,650
of the left hand side, and this one is
nothing but the complement of this one. So
307
00:31:03,650 --> 00:31:10,460
if this two are equal then this two are also
equal, so I can write that this entity is
308
00:31:10,460 --> 00:31:15,630
equals to this entity. So that way what I
can write
309
00:31:15,630 --> 00:31:22,610
is that I am just writing this one here and
this entity here. So I am finally, getting
310
00:31:22,610 --> 00:31:26,669
this
statement, which if I compare with the first
311
00:31:26,669 --> 00:31:29,920
statement, then all the sets are replaced
by
312
00:31:29,920 --> 00:31:35,630
their complements, all the intersections replaced
by their union, all the unions replaced
313
00:31:35,630 --> 00:31:41,909
by their intersection and still we see that
if we do all this replacement, then still
314
00:31:41,909 --> 00:31:45,970
the
identity is preserved; this is what is the
315
00:31:45,970 --> 00:31:52,200
De Morgan’s law.
There is, using this De Morgan’s law, we
316
00:31:52,200 --> 00:31:56,600
can we can utilize this De Morgan’s law
for
317
00:31:56,600 --> 00:32:03,450
another concept, take this concept further,
to what is known as the duality concept; and
318
00:32:03,450 --> 00:32:07,179
that duality concept is written here.
319
00:32:07,179 --> 00:32:08,179
..
320
00:32:08,179 --> 00:32:13,980
It is a, this is a simple extension of the
De Morgan’s law; here, this is the S is
321
00:32:13,980 --> 00:32:19,299
the full set,
so if I say that S is the full set, then if
322
00:32:19,299 --> 00:32:21,690
I take its complement, which is nothing but
a null
323
00:32:21,690 --> 00:32:28,419
set. So S consists of everything, so if I
take that its complement, obviously its
324
00:32:28,419 --> 00:32:36,181
complement will be a null set and vice versa
if I take the complement of a null set, then
325
00:32:36,181 --> 00:32:39,059
I
will get back the full set, which is known
326
00:32:39,059 --> 00:32:46,909
as S. Now if in a set identity, all the unions
are
327
00:32:46,909 --> 00:32:55,429
replaced by intersection, all the intersections
by their unions, the set S by the null set
328
00:32:55,429 --> 00:33:00,840
and
the set and the null set by its set, then
329
00:33:00,840 --> 00:33:10,240
the full set S, the identity is preserved.
So, using this A intersection B, which is
330
00:33:10,240 --> 00:33:13,730
equals to this A B just in the previous slide
we
331
00:33:13,730 --> 00:33:19,191
have just shown the this one, then this will
lead to, I will just simply change this
332
00:33:19,191 --> 00:33:29,120
intersection as union and union as intersection
on both side, left hand side and right hand
333
00:33:29,120 --> 00:33:34,000
side, then we get the another side, another
identity which is also preserved.
334
00:33:34,000 --> 00:33:43,930
Then S union A is equals to S obviously the
full set if I take the union of any sub set
335
00:33:43,930 --> 00:33:47,149
of
this S, then that also leads to this S, which
336
00:33:47,149 --> 00:33:50,059
leads to, if I just replace this whatever
I told
337
00:33:50,059 --> 00:33:57,010
here, the S by the null set, union by the
intersection, and A by this one, then it will
338
00:33:57,010 --> 00:34:04,240
written as this S is null set; so you see
that the null set intersection A is the null
339
00:34:04,240 --> 00:34:08,040
set. So
this is known as the duality principle, which
340
00:34:08,040 --> 00:34:14,070
is an extension of the De Morgan’s law.
341
00:34:14,070 --> 00:34:15,070
..
342
00:34:15,070 --> 00:34:22,010
There are few other concepts also which are
also useful in the set operation, say for
343
00:34:22,010 --> 00:34:29,000
example, that first one is the symmetric difference;
the symmetric difference for any two
344
00:34:29,000 --> 00:34:38,770
sets A and B, the set of elements which exist
only one of the set not in both called the
345
00:34:38,770 --> 00:34:45,810
symmetric set. Here you can see there are
two sets; one is the black rectangle which
346
00:34:45,810 --> 00:34:49,220
is
denoted as A and another set is denoted by
347
00:34:49,220 --> 00:34:55,000
B which is a red rectangle; here there are
few
348
00:34:55,000 --> 00:35:01,430
elements, which are common both in A and B
in this area that you can see and which are
349
00:35:01,430 --> 00:35:10,540
not common, which are located in this area
or in this area. So, now if I prepare another
350
00:35:10,540 --> 00:35:22,851
set, which consists of all the elements, which
falls either in only A and or only in B, then
351
00:35:22,851 --> 00:35:28,700
the set that is the symmetric difference between
this A and B.
352
00:35:28,700 --> 00:35:35,760
..
353
00:35:35,760 --> 00:35:44,920
The second one is the cartesian product; for
any two sets of events A and B, the set
354
00:35:44,920 --> 00:35:51,600
comprise of all possible ordered pair, this
is a new term will just explain it in a moment;
355
00:35:51,600 --> 00:35:59,069
the all possible ordered pair A B, where a
is a member of A and the small b is a member
356
00:35:59,069 --> 00:36:07,180
of B, B set, then this all the possible ordered
pair is called the cartesian product of A
357
00:36:07,180 --> 00:36:14,829
and
B, and which is denoted as A cross B. Now,
358
00:36:14,829 --> 00:36:17,660
what is this ordered pair? An ordered pair
is
359
00:36:17,660 --> 00:36:24,859
a collection of objects having two coordinates,
such that one can always uniquely
360
00:36:24,859 --> 00:36:33,770
determine the object; thus the first coordinate
is A and the second coordinate is B, then
361
00:36:33,770 --> 00:36:40,400
the ordered pair is A B. So always the first
entity comes from the first set and the second
362
00:36:40,400 --> 00:36:47,030
entity comes from the second set, which is
obviously different from the B A. So if we
363
00:36:47,030 --> 00:36:54,359
maintain the first entity of all the sets,
all the elements, the first entity always
364
00:36:54,359 --> 00:36:56,700
comes
from the first set and second entity always
365
00:36:56,700 --> 00:37:02,560
comes from the second set, then this is known
as an ordered pair.
366
00:37:02,560 --> 00:37:11,720
So once again the cartesian product if we
see, it states that for any two sets of events
367
00:37:11,720 --> 00:37:16,200
A
and B, the set comprise all possible ordered
368
00:37:16,200 --> 00:37:27,030
pair A comma B, where a is a member of A
and b is a member of B is called the cartesian
369
00:37:27,030 --> 00:37:31,029
product of A and B, which is denoted as A
cross B.
370
00:37:31,029 --> 00:37:36,700
..
371
00:37:36,700 --> 00:37:42,210
The third one is the power set; the power
set, sometimes this kind of terminologies
372
00:37:42,210 --> 00:37:47,690
are
used frequently to have the concept – the
373
00:37:47,690 --> 00:37:55,250
power set of any particular set A is the set
whose elements are all possible subsets of
374
00:37:55,250 --> 00:38:00,810
A. I repeat the power set of any particular
set
375
00:38:00,810 --> 00:38:09,800
A is a set whose elements are all possible
subsets of A. So if I just take one set that
376
00:38:09,800 --> 00:38:14,839
is 5
6, then the power set of this particular set
377
00:38:14,839 --> 00:38:23,000
can consist either the null set or the one
element, the first element or the second element
378
00:38:23,000 --> 00:38:30,650
or another subset can consist of the full
set, the 5 6, so there are 4 possible subset
379
00:38:30,650 --> 00:38:40,570
that can happen. So this is known as the power
set of this set 5 6. Generally, we say that
380
00:38:40,570 --> 00:38:46,510
this any particular set is one subset of its
own
381
00:38:46,510 --> 00:38:48,170
power set, which is obviously true.
382
00:38:48,170 --> 00:38:49,170
..
383
00:38:49,170 --> 00:39:04,391
There are two more things that we say, that
open set and the closed set, the open set,
384
00:39:04,391 --> 00:39:10,200
a set
S is called open set, if any point A in S
385
00:39:10,200 --> 00:39:12,760
always belongs to the set even though it moves
to
386
00:39:12,760 --> 00:39:18,660
some particular direction; and the closed
set, a set is called the closed set if its
387
00:39:18,660 --> 00:39:24,300
complement is an open set. So the concept
of open set is more important and the closed
388
00:39:24,300 --> 00:39:32,610
set we generally say that, if the complement
of a set is open set, then that particular
389
00:39:32,610 --> 00:39:38,500
set is
closed set. Before I go to this example, if
390
00:39:38,500 --> 00:39:47,650
simply we say that a particular set that consists
of all the integer numbers between 5 to 10,
391
00:39:47,650 --> 00:39:56,329
then this is a closed set, and if I say its
complement then it consists of the countably
392
00:39:56,329 --> 00:40:02,160
infinite number and that is obviously an
open set.
393
00:40:02,160 --> 00:40:09,780
Coming back to this particular example, which
is taken from the Wikipedia, the point a
394
00:40:09,780 --> 00:40:15,510
b, if they satisfy this relationship that
is a square plus b square equals to r square
395
00:40:15,510 --> 00:40:19,720
which
nothing but an equation of one circle and
396
00:40:19,720 --> 00:40:27,359
which is shown by this blue line here, this
circle. Then a square plus b square is exactly
397
00:40:27,359 --> 00:40:35,570
equal to r square is colored blue here the
points a b satisfying the a square plus b
398
00:40:35,570 --> 00:40:38,369
square less than equal to r square, these
are
399
00:40:38,369 --> 00:40:45,339
colored red. So, there are this red area consists
of many points, which satisfies the
400
00:40:45,339 --> 00:40:56,119
relationship a square plus b square less than
r square, and this circle, this blue circle
401
00:40:56,119 --> 00:40:59,170
also
consists of many points, which satisfies this
402
00:40:59,170 --> 00:41:10,320
relationship a square plus b square is equal
to r square. This red points forms an open
403
00:41:10,320 --> 00:41:13,240
set, this red points, what we can see that
in
404
00:41:13,240 --> 00:41:21,181
between the inside this blue circle that forms
an open set. The union of this red and
405
00:41:21,181 --> 00:41:29,400
the…, sorry this will be blue, the red and
the blue points forms a closed set. The union
406
00:41:29,400 --> 00:41:30,400
of
407
00:41:30,400 --> 00:41:36,910
.red and blue points forms the closed set,
because if I take the complement of this union
408
00:41:36,910 --> 00:41:42,050
so this union is nothing but this full area,
red area including this blue area if I take
409
00:41:42,050 --> 00:41:45,540
the
complement of this set, then it is completely
410
00:41:45,540 --> 00:41:51,570
outside so that is is an that is an open set;
so
411
00:41:51,570 --> 00:42:00,320
that is why this union is a closed set.
.
412
00:42:00,320 --> 00:42:09,040
There is another important concept that we
generally say that this is a borel set, the
413
00:42:09,040 --> 00:42:17,200
concept is like this, suppose there are some
events n number of sets A1, A2 is an infinite
414
00:42:17,200 --> 00:42:28,130
set, in some class F. Now, if you say that
the union of any number of sets here from
415
00:42:28,130 --> 00:42:32,670
this
A1 to An or intersection between any two or
416
00:42:32,670 --> 00:42:38,930
more than two sets, all this possible cases,
all the unions all the intersection of this
417
00:42:38,930 --> 00:42:42,710
sets also belongs to same class F, then the
F is
418
00:42:42,710 --> 00:42:51,020
known as the borel field or borel set. So
this is important, when we say that we in
419
00:42:51,020 --> 00:42:56,910
particularly in the probability theory, when
we say that there are some events, and which
420
00:42:56,910 --> 00:43:03,080
belongs to class F whether that probability
can be applied to their unions, to their
421
00:43:03,080 --> 00:43:11,079
intersection, so to apply that one this condition
must be satisfied that is that particular
422
00:43:11,079 --> 00:43:20,140
set
should be a borel set or a borel field.
423
00:43:20,140 --> 00:43:21,140
..
424
00:43:21,140 --> 00:43:30,480
Another important thing is a sigma algebra,
a sigma algebra is a set S, is a non empty
425
00:43:30,480 --> 00:43:37,900
collection capital sigma here of sub sets
of S, including S itself obviously, S is also
426
00:43:37,900 --> 00:43:46,900
included that is closed under complementation
and countable unions of its members. So
427
00:43:46,900 --> 00:43:53,339
here importantly there are three conditions;
one is that that set must be non empty this
428
00:43:53,339 --> 00:43:57,010
is
the first condition, second condition it should
429
00:43:57,010 --> 00:44:02,470
be closed set under its complementation. If
I take its complements, then the complements
430
00:44:02,470 --> 00:44:13,700
should also be closed; and the countable
union of the set also is it is closed under
431
00:44:13,700 --> 00:44:18,470
this countable unions also. So there are three
conditions, for a set if these three conditions
432
00:44:18,470 --> 00:44:25,540
are satisfied, then that set is known as the
sigma algebra. So, suppose here that this
433
00:44:25,540 --> 00:44:32,609
S consist of a, b, c, d is one possible sigma
algebra on S is that this particular set,
434
00:44:32,609 --> 00:44:36,990
the null set, first two, second two and these
full
435
00:44:36,990 --> 00:44:45,109
set. So this is a possible sigma algebra on
the S where this your that collection of the
436
00:44:45,109 --> 00:44:48,330
particular set, this is a sigma algebra.
437
00:44:48,330 --> 00:44:49,330
..
438
00:44:49,330 --> 00:44:59,990
The probability space, this probability space
here is important, because this is the way
439
00:44:59,990 --> 00:45:05,740
how we link between the set theory and the
probability concept. So in this probability
440
00:45:05,740 --> 00:45:15,400
concept, the sample space S or sigma is called
the certain event; it is elements
441
00:45:15,400 --> 00:45:21,309
experimental outcome and it is subsets are
events. The empty set, null set is called
442
00:45:21,309 --> 00:45:25,520
the
impossible event, and the event ζ consist
443
00:45:25,520 --> 00:45:28,890
of a single element is an elementary event.
So,
444
00:45:28,890 --> 00:45:38,130
the full samples space or its sigma algebra
is a certain event, in from the set theory
445
00:45:38,130 --> 00:45:41,940
the
full sample space consist of the certain event,
446
00:45:41,940 --> 00:45:46,720
its elements are the experimental
outcomes; so whatever the experimental outcome
447
00:45:46,720 --> 00:45:51,850
that we have taught in the previous
classes; so its elements, the elements of
448
00:45:51,850 --> 00:46:01,260
the set is the experimental out come and its
subsets correspond to some events. The empty
449
00:46:01,260 --> 00:46:07,131
set, again here if I say, some set is empty
that means that is an impossible event, so
450
00:46:07,131 --> 00:46:16,900
for that experimental outcome that particular
outcome is not feasible, is not possible;
451
00:46:16,900 --> 00:46:26,680
and obviously if it consist of a single element,
single outcome, that is an elementary event.
452
00:46:26,680 --> 00:46:27,680
..
453
00:46:27,680 --> 00:46:30,210
In the application of the probability theory
to the physical problems, the identification
454
00:46:30,210 --> 00:46:34,740
of
the experimental outcome is not always unique.
455
00:46:34,740 --> 00:46:40,170
This is important in the sense that is how
you are interpreting the particular outcome
456
00:46:40,170 --> 00:46:44,900
of an event that that generally consist of
how
457
00:46:44,900 --> 00:46:56,030
you are representing that particular outcome,
for example, if I say that I am just
458
00:46:56,030 --> 00:47:03,609
interested to know whether the outcome of
a throwing a dice is even number or odd
459
00:47:03,609 --> 00:47:08,369
number, then my possible sets or possible
outcomes are two, either it is even number
460
00:47:08,369 --> 00:47:13,270
or
it is odd number, but if you say that, if
461
00:47:13,270 --> 00:47:19,470
I say that the whether the outcome is 2, 4
or 6, I
462
00:47:19,470 --> 00:47:25,080
am also looking for the even number but my
possible outcome here that 2, 4 and 6, so
463
00:47:25,080 --> 00:47:33,099
there are three possible outcome in this representation.
Again if I say that if I throw a die and its
464
00:47:33,099 --> 00:47:40,271
coordinate, if I just fix up some coordinate
system, and say that the outcome is two and
465
00:47:40,271 --> 00:47:47,190
along with its location, where the two
comes. So, that location of die itself, its
466
00:47:47,190 --> 00:47:54,050
coordinate that is x coordinate and y coordinate
on a plane obviously and the outcome is two.
467
00:47:54,050 --> 00:48:03,059
Then the number of possible outcome is
infinite, so this is the way how a particular
468
00:48:03,059 --> 00:48:05,569
representation of a particular application
of
469
00:48:05,569 --> 00:48:10,670
this probability theory to a physical problem,
generally this identification of the outcome
470
00:48:10,670 --> 00:48:16,330
is not unique, the way we want to represent
it, it becomes in that particular way.
471
00:48:16,330 --> 00:48:22,200
A single performance of an experiment which
we generally call is as a trial. So, a
472
00:48:22,200 --> 00:48:26,250
particular, in a particular trial whether
a particular outcome will come that depends
473
00:48:26,250 --> 00:48:29,710
on
this, how the experiment is conducted, and
474
00:48:29,710 --> 00:48:35,680
that is the way we generally assign the
475
00:48:35,680 --> 00:48:41,720
.particular probability. So a single performance
of an experiment is known as a particular
476
00:48:41,720 --> 00:48:45,430
trial.
.
477
00:48:45,430 --> 00:48:51,850
In the concluding remarks, in this class,
we have seen that different set theory, the
478
00:48:51,850 --> 00:48:58,220
definition of set and the set theory that
defines the various inter relationship among
479
00:48:58,220 --> 00:49:05,069
the
elements or the subsets of a set. The set
480
00:49:05,069 --> 00:49:09,990
operations also classify the different types
of
481
00:49:09,990 --> 00:49:19,550
sets, set properties of the elements. So,
with this, in following lecture, in the next
482
00:49:19,550 --> 00:49:24,940
lecture,
we will explain about the axioms of probability
483
00:49:24,940 --> 00:49:30,160
theory which will be discussed in the
next class in details; that will very important
484
00:49:30,160 --> 00:49:32,880
for the probabilistic properties of events
and
485
00:49:32,880 --> 00:49:34,590
set. Thank you.
486
00:49:34,590 --> 00:49:34,590
.