Re: Infinity......


zuhair wrote:

Yes, the have the same size as determined by f when f is bijective.
but when f is not bijective then they don't have the same size_f.

If they do not have the same size then A changes size when
you rename its elements.

Can you be more clear about that.


We have a valid name change for A if for every element a
of A
-we have a new name for a

-if a_1 and a_2 are elements of A
then the new name for a_1 is not the same
as the new name for a_2

Example

A = {1,2,3}

A valid name change is:
change the name of 1 to 4, the name of 2 to 5 and the name of 3 to 6
to get the set

A'={4,5,6}

One invalid name change is
change the name of 1 to 4, and the name of 2 to 5 to get the set

A'= ?

This name change is invalid because we did not specify a new
name for every one of the elements of A (specifically we did
not specifiy a name for the element named 3)

Another invalid name change is:
change the name of 1 to 4, the name of 2 to five and the
name of 3 to 4 to get the set
A' = {4,5,4}

This name change is invalid because we used the same new name
(4) for more two different elements (the element named 1 and the
element
named 3).

Recall, we know nothing about size except for the
fact that if we get A' by applying a valid name change
to A, then A and A' have the same size.


lets see this example.

A= {1,2,3,........}
B={ 2,4,6,........}

Now let f:B->A, f(x)=x
Now according to this f, I cannot rename all of A elements by replacing
them without repetition by elements of B.


It is true that there is a way of using the elements of
B that does not give a valid name change.
However, the fact that there is a way
to use the elements of B so you do not have
a valid name change for A, does not change the fact that
there is a way to use the elements of B that gives a
valid name change for A.

All we need is *one* way of using the elements of B
to give a valid name change for A. Since this
does not change the size of A, we can conclude
that A and B have the same size.


and this is the meaning of
(size E < size N)_f.

so what do you really mean by your statement : if they do not have the
same size then A changes size when you rename its elements. but here we
cannot rename all the elements of A by elements from B when B is
smaller in size than A.

No. There is more than one way to use
the elments of B to rename the elments of A.
The fact that there is one way in which we cannot use
the elements of B to rename all the elements of A does not change
the fact that there is a way to use the elements of B to
rename the elements of A. So B is not smaller in size than A.



see what I wrote. you are not differentiating between reltive and
absolute size.
that's your problem

If the two sets A and B do not have
the same absolute size, then the absolute size of A
changes when you rename its elements.

This is weard, I can't understand that. how I understand it is as
fellows;

if A and B do not have the same absolute size, then we cannot change
all the elements in A by elements in B if A has an absolute size bigger
than the absolute size of B.
we can change all the elements in B by elements from a proper subset of
A but this will not change the the absolute size of B. it will remain
the same.

It seem that you think that for any two sets, if we can change the
elements of one of them by elements from the other without affecting
size, then they are equal in size.

No. We start with the statement (here I quote you)

we cannot change the size of any set
by merelly renaming each of it's elements.

So rather than

if we can change the elements of one of them by elements from
the other without affecting size

we have

if we can change the elements of one of them by elements from
the other *then this does not change the* size

So

if we can change the elements of one of them by elements from
the other then this does not change the size and therefore
they are equal in size.


But if we change elements of one of them
them by elements from the other and this results in size change of the
set who's elements are replaced, then these two sets are inequal in
size.

We cannot do this. If we have a valid name change for A, then
the size of A does not change. Your example above does
not work. The function f:B->A, f(x)=x does not define a valid name
change on the
set A (some elements of A are not included).

- William Hughes

.

Relevant Pages

  • Re: Infinity......
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  • Re: Infinity......
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