Re: infinity
- From: "zuhair" <zaljohar@xxxxxxxxx>
- Date: 1 Nov 2005 12:59:12 -0800
Just becasue this post was indiscriminative, my remarks are not
discriminated from those
of Randy's I will rewritte it:
Randy poe wrote:What can "in what they are bijected" mean, if not the
elements? A bijection is a map between elements.
----------------------------
My reply:
yes it is! but regarding what ? cardinality or ordinality?
------------------------------------------------------
> so for example if i use a
> cardinal bijection
You are creating new undefined terms in every post. I have
given you the definition of a bijection. It is a map
which is injective and surjective. If you want to
call that a "cardinal bijection" and distinguish other
types of bijections, you are going to have to provide
a definition.
---------------------------------
My reply:
I almost did below.but I will do it now.
The set A = { k,b,c} is cardinally bijected to set B = { e,f,g}
if their exist a fucntion f:A-->B defined as below:
f( k xor b xor c ) = e xor f xor g
f( Remainder ( k xor b xor c ))= Remainder ( e xor f xor g)
f( Remainder (( Remainder ( k xor b xor c ))) = Remainder ((
Remainder ( e xor f xor g ))
..
..
..
..
If we reach to f(0)=0 at the same step then their exist a cardinal
bijection between
sets A and B.
Example: set A = {1,2,3} , set B={ 9,8,6)
f works in the squence below:
f( 1 xor 2 xor 3 ) = 9 xor 8 xor 6 suppose this was true for 1
and 8
Remainder denoted by R
Now: R( 1 xor 2 xor 3 )= 2 xor 3
R( 9 xor 8 xor 6 ) = 9 xor 6
Now f ( 2 xor 3 ) = 9 xor 6 , for example suppose it was between 2
and 6
R ( 2 xor 3 ) = 3
R ( 9 xor 6) = 9
so f (3) = 9
Now R (3) = 0
R(9) = 0
so their is bijectoin becasue we reached to f(0) =0
suppose if set A are compaired in that way to set B and at the end we
reached
and f:A-->B
to f(0)= x were x<>0
then this means that set B is bigger than set A in cardinality
However if it happesn that f(x)=0 were x<>0 then this means that set A
is bigger than set
B in cardinality.
This F defined in the stepwised logical manner is what I mean by
Cardinal bijective function.
While the functions you and your colleges widely use in that group are
in reality ordinal bijective functions since they don't fulfil the
cardianl bijective function above.
For example sets A= {1,2,3} and B={2,3,4}
Now f:A -->B
f(x)= x+1
it is thought that f(x) is a bijective function which is true, but it
is ordinal bijective function
and not cardinal bijective fuction.
because f ( 1) = 2 , it cannot be f(1)=3 or f(1) =4
while the cardinal bijective funciton f( 1 xor 2 xor 3 ) = 2 xor 3 xor
4
That's what I mean that f(x)= x+1 is an ordinal function.
it can biject sets of coarse but sure the sets it is bijecting are put
in an ordered arrangment
(series )and in reality what such and the alike functions are bijecting
is the ordinal number
of the series they are bijecting, and not the cardinality of them.
Cardinal bijective functions are logical algorithmic fucntions , they
are not like the functions
used in numerical math, all these later functions are ordinal functions
and they only assure
ordinal bijection and not cardinal bijection.
---------------------------------------------------------------------------------------------------
> then if it is fullfilled betwee setA and setB it only means
> that these two sets are similar (or equal) in cardinality
Yes, since that is how equal cardinality is defined.
> while if you biject setA put in a specific order to set B also put in
> a specific order then we are bijecting ordinality of these two sets
> not cardinality.
We do not "biject in a specific order". Here is the definition
of a bijection:
-------------------------------------------
My reply:
like when I put set S ( the pairs set I defined in an earlier post)
according to original order remember.!!!
The field of a series is the set of terms in the series. but if I
display that set in the original
order of its series and then try to biject this is what I mean by
biject in a specific order.
doing that for my example on S , revealed impossiblity to biject it
with N. remember.
--------------------------------------------------------------------------------------------
Map f:A->B is a bijection if and only if it is an
surjection (for every y in B, there exists x in A such
that f(x) = y) and an injection (f(x) = f(x') implies
x = x').
There is no mention of order there. "Biject in a specific order"
is yet another term you want to use but are unable to define.
----------------------------------------------------
My repy:
yes their is no mention in order there because this is a definiiton of
bijection
and not of Ordinal bijection,if you use the same definition above and
say that f is an ordinal function ,then this is what I call a
definition for ordianl bijection.
----------------------------------------------------------------------------------------------
> That's what i meant,read preciselly again.
I can not "read precisely" sentences that contain undefined
terms. There is no precision possible where terms are
not defined.
> > It only means elements of one set can
> > be used as labels for elements of the other. It is used
> > as a definition of "cardinality".
> yes true if it is cardinal bijection.
> > > when we say the cardinality of { Lion , elephant,dog} = x then this
> > > means also
> > > that it is the same for any arrangment of that set
> > No, we don't mean that. That is something which can be proven
> > for finite sets, but there is nothing in the definition
> > of cardinality which implies that to be true in general.
> What are you garbling about it seems you don't know the real definition
I'm garbling that the definitions of cardinality and bijection
do not talk about order. So nowhere in the definition of
equal cardinality is there a statement that "equal cardinality
means the cardinality is the same for all arrangements of
the set".
Sets do not have arrangements. When we talk about cardinality of
a thing that does not have "arrangements", we most CERTAINLY
are not making any sort of statement about "arrangements", since
there are no "arrangements" inherent in the definition of
any set.
> Cardinality is a property the set irrespective of the individual
> properties of its members and their order.
That is correct. Therefore it does not mean anything in
particular relative to order.
> What Cantor means simply that if you rearrange the set in any manner
> then cardinality is still the same. Also if you change the members of
> the set in such a way that every new member has one-one relation with the
> old replaced member then the new formed set has the same cardinality.
You have just restated the bijection definition of cardinality. It
follows directly from the statement you just made that your set
S of ordered pairs has the same cardinality as N.
> > We define cardinality in terms of bijections. One-to-one
> > correspondence (injection) is necessary but not sufficient
> > for a map to be a bijection.
> Randy it seem you don't know that the term one to one correspondance
> means bijection and not injection
That's right. I don't "know" that.
http://mathworld.wolfram.com/One-to-One.html
Hmmm. I see a statement agreeing with both of us. This author
says "one-to-one" means "injection", but "one to one correspondence"
means "bijection".
He also notes the terminology is confusing. So I prefer to avoid
confusion and use precise terms on which we both agree. I will
continue to use the terms bijection, surjection, and injection.
> one to one which is injection and one to one correspondance which is
> bijection
Yes, you are apparently correct.
> According to them ( not me) a set A is similar(this is Bertrand
> Rusell's term not me but i know you in your infinite wisdom
You are in a math forum. Even if I take your word to be correct
about what a philosopher says, in my "infinite wisdom" I will
do my best to use mathematical rather than philosophical idioms
in a mathematical discussion.
> prefare the term
> equal) to set B in cardinality. if every element of set A has one-one
> relation with elements in setB
You see? That's exactly where the confusion can sneak in.
A "one-to-one map" or relation is an injection. See the
Mathworld link again, where he says that a map which is one-to-
one is an injection. Confusingly, it seems only the word
"correspondence" distinguishes the two terms.
> > > if we have X={a,b,c}, Y={e,f,g} and their is one-one relation between
> > > all their members then this is one-one correspondance ( bijection)
> > > Now the above is cardinal bijection and it implies similarity in
> > > cardinality.
> > It is the DEFINITION of EQUAL cardinality.
> As you like EQUAL.
> > > However their is another kind of bijection
> > No, there isn't.
> yes their is
No there isn't.
- Hide quoted text -
- Show quoted text -
> only wait to see what I mean. don't make what you usually
> make in all posts what I call it premature responses.
> > > For example:A= 4,5,6 bijected to B=1,2,3 , this kind of bijection is
> > > not cardinal bijection.
> > No, a bijective map between these two sets is precisely the
> > same as one between the above two sets: it is an injection
> > and a surjection. What possible difference do you see?
> > > For example if a,b,c is bijected to 1,2,3 this means that a,b,c are of
> > > the same sequence as 1,2,3
> > What the heck does "of the same sequence" mean?
> let me tell you what that means for example if i biject the SET {
> lion,elephant,dog} to the SERIES 1,2,3 this doesn't mean as you think
> that the set { lion,elephant, dog} is a set of three animales. not it
> mean
It means that it is a set of three elements. That is in fact
how we define the cardinality of a finite set: that there
exists a bijection to {1,2,...,n} for some natural n.
> dear Randy that : lion is the first animal, elephant is the second and
> dog is the third.
No it doesn't. There exist 6 bijections from {lion, elephant, dog}
to {1,2,3}, and only in one of them is the above association
made. Do I need to remind you again that a set has no order?
When you make this statement:
"i biject the SET {lion,elephant,dog} to the SERIES 1,2,3 "
there is nothing in this statement that implies that lion
is mapped to 1, elephant to 2, and dog to 3.
> In that way you didn't fullfil cantor's definition of Cardinality a
> property of a set devoid of order and member individual properties.
Um, yes you do. Cantor's definition of finite cardinality is
that there exists a bijection to {1,2,...,n}. No order is
required of the elements, no property of the elements is
used.
> The real bijection
There are six bijections. Any of them will serve. They are
equivalent.
-------------------------
My reply:
All of them should serve.
-----------------------------------
> which should prove that the set of the animals above
> has the same CARDINALITY as the cardinality of 1,2,3 is the bijection
> between the set { lion, elephant,dog} and the FIELD of the SERIES
You haven't defined that either.
- Randy
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- From: zuhair
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