Hello everybody on the previous lecture there was a little mess when I tried to explain what infimum is.
So all this.
Notations, supremum, infimum
are very important for this module. So let us consider another example about supremum.
Suppose X is the open interval around 2:00.
And supremum of Works is obviously.
Equal to two. So capital a = 2. Usually you have to guess what supremum is, but after you guess.
You must check the definition. I mean, you must justify your answer.
So firstly, for any X the value is smaller than.
2.
For any eggs.
X is for any X from the set. The value is smaller than two. This is clear.
The second step is less clear.
For annual number, why there is such eggs? Or which is bigger than well then why?
Take an arbitrary away smaller than two.
Simple case, if a Y is smaller than one.
But then.
You can.
Show many values of X.
From this initial set, which are bigger, for example 3 / 2 is a point belongs to Capital X and is bigger than Y.
Second case, if Y is bigger than one.
Like here.
Then one can take eggs.
In the middle.
2 + y / 2.
But remember why is bigger than one smaller than two?
So.
In this example, I take X being one plus half or why?
And one has to show that this X belongs to the original set and is bigger than Y.
So X belongs to initial set becausw.
X.
Which is 1 plus half a. Why is smaller than one plus?
2 / 2 becausw. Why smaller than two?
And Secondly, X is bigger than 1 + 1/2 because of. Why is bigger than one?
So are these two.
Inequalities.
Show that X.
Little X belongs to the initial set.
And Secondly, you must show that X is bigger than a Y.
To do this?
Will you consider the difference?
X.
This is X.
And minus y = 1 minus half a Y.
And since Y is smaller than two, the value is bigger than one minus.
2 / 2 which is 0, so the difference is.
Pawsitive
and the second inequality is proved.
So please understand other definition and.
You must be able to calculate simple.
In films and Supremums.
Next section is about countable and uncountable sets.
So I hope you understand the word.
121 correspondence mean.
So one to one correspondence.
If you have two sets, for example finite sets.
A.
105
and be.
Hey BE.
One to one correspondence means that every one element from one set.
Is put in correspondence with element in another set. You can put five to be.
You can put 02.
E does not matter, so but every one point from a must be involved and every point from B.
So two sets have the same cardinality.
Or the same number of elements, if they exists one to one correspondence. Here A&B have the same cardinality.
A.
And be.
Here.
But the same.
Cardinality.
So the idea is.
Obvious.
There is no need to count the number of elements of the elements.
Assets may be infinite.
But if there is a one to one correspondence, then we say that.
So their sets have the same cardinality, the same number of elements.
For example, if there are people in the room.
And there are cheers in the room.
Then there is no need to.
Count.
The cheers and people you can.
See, people sit down.
And if.
Everyone cheer is occupied.
And all people are seated and there are no free cheers.
So then you can be sure that the number of cheers and number of people coincide on the.
Sit.
Of people
and they said of cheers.
Have the same cardinality.
So if such a.
121 correspondence exists or then it is fine.
For infinite sets.
This definition can.
Result in.
A bit strange consequences or the set of even numbers, which is a proper subset of natural numbers, is still.
Has still the same cardinality, the one to one correspondence is obvious.
If you try to construct a one to one correspondences and you are unsuccessful, it does not mean much becausw another trial can be successful.
If at least one trial is successful, like here then.
You can be sure you that is the proof that one set has the same cardinality as another set.
If I said has the same cardinality is.
Natural numbers, so we say that we said A is Denumerable.
So what is interesting for this?
Theory about ordinal Cardinal numbers was developed by George counter.
Russian German mathematician. He lived at the end of 19th century.
And one of his famous pros is that.
Interval 01 is not denumerable. There are much more real numbers or then.
Natureal numbers.
The proof is.
Simple.
It is not necessary for your module.
But it would be good if you understand it.
The proof is by contradiction. Suppose there is a table.
1st.
Real number is in correspondence with Naturile number one. Another real number is in correspondence with.
No nature or #2 and so on.
So suppose.
There is such one to one correspondence.
In this case, well, it is example one real number, his number one, another real number, his number 2 and so on.
But in this case you can construct so-called diagonal.
Video number
so a bill one in this example is different from a one.
B2 is different from 5.
And so on.
So B in is different from the diagonal digit ANM.
So this.
Real number represented as an infinite.
Sequence of 1234.
This decimal my number.
Is.
And number from the interval 01.
And it is not in the table because it is of course different from Maxwell.
Becausw bill one is different from one. It is different from X2 Becausw B2 is different from 5 and so on.
So this contradiction shows that.
In your one such table does not exhaust all real numbers.
So another definition, if there is a one to one, there is no one to one correspondence between A&B, but a has the same cardinality as a proper subset of B. We say that cardinality of B is bigger than that of A.
Well, this can be illustrated.
On
fine it says.
For example a.
Is.
Who won two?
And be.
Chris, four elements ABCD.
Clearly.
There is no chance to construct one to one correspondence.
But
there is 121 correspondence between.
Hey B.
Subset of B.
This is a.
There is always one to one correspondence between A and subset of B. So in this case B.
Cardinality would be is bigger than a.
In the finite case, it means that B has more elements than A.
So cardinality of real numbers 01 is bigger then.
Natureal numbers becausw as it was proved.
As it was proved.
Of the interval 01 is not denumerable. There does not exist one to one correspondence.
Between real numbers and.
Nature or numbers, but of course one can find easily a subset of real numbers, for example of 1/2, one third, one quarter.
This subset.
Or for real numbers from the interval.
Is in a one to one correspondence with natural numbers, so it's his number one. This is number 2. This is number 3, and so on.
So.
On the real numbers from the open interval 01.
From zero to 1.
Got the same cardinality.
Is all real numbers.
Minus Infinity Plus Infinity.
It looks a little strange because this interval is very small and the real straight line is very big.
But according to the definition.
The cardinality's coincide becausw they exists for one to one correspondence.
One to one correspondence of the following.
Shape.
So that which is given by function term.
So done.
Boiled eggs.
This cross by.
There is 01.
And.
This function.
It looks like that.
So for any real number.
You can find.
The corresponding.
Point on the interval 001 and for any point on the interval 01 you can find.
The corresponding.
Real number on the straight line.
So this is the picture illustrating that.
This function 10 provides a one to one correspondence.
What do you have next?
Next counter proved that.
If you have an arbitrary set or then.
There is no one to one correspondence between X and they said.
Why?
Two 2X collection of all subsets of X.
So here is an example.
Effects 'cause three elements are then the number of all subsets is 8.
Including empty, including the total set.
Singletons pairs, triples, and so on. So that's why the notation is 2, two X if X is has three elements.
Why?
The set of all subsets. CAS 8 elements 223.
So the proof is for your independent reading.
It is equipped with asterisk means it is not necessary for this module, but it would be good if you understand it.
Again, it is theorem due to counter.
And it says now that if you.
They can use it.
Infinite, very big Infinite Sir. Then you can easily construct a set which has bigger cardinality.
So you can take collection of all subsets.
And the cardinality or will be bigger, so there is endless.
Sequence of different.
Infinite sets and every next one is bigger than the previous one.
So the proof is not difficult.
But
you should read all the.
Controller is at least.
If we have.
Collection of all subsets.
Then cardinalities bigger becausw there is no.
121 correspondence.
With Y, but there is 1 to one correspondence.
Movies subset of Hawaii.
So in this example.
In this example.
But that is 1 to one correspondence.
There is a one to one correspondence between X.
Or which is ABC?
And subset of Why?
Which is collection of all singletons.
Collection of all singletons is subset of over.
So there is 1.
21
Go despondence
this is illustration to the Corollary.
On page 7.
X has the same cardinality as the subset of Y containing all singletons.
A look at this.
Picture.
Illustrating on this corollary.
So the last remark today.
Is about interval interval 01 has the same cardinality as the set of all subsets of integers, the subsets of integers have the bigger cardinality and that is exactly cardinality of.
Real numbers 01.
That is all about introduction to the set theory.
Next time or we will start really measure theory.
Um?
Man, that man basic object here or will be Sigma Field.
That is, for the next election.
Thank you for your questions, good luck.