Re: Cardinality and injection


William Hughes wrote:
> zuhair wrote:
> > William Hughes wrote:
> > > zuhair wrote:
> > > > quasi wrote:
> > > > > On 13 Nov 2005 09:36:36 -0800, "zuhair" <zaljohar@xxxxxxxxx> wrote:
> > > > >
> > > > > >
> > > > > >zuhair wrote:
> > > > > >> I have a question:
> > > > > >>
> > > > > >> Injection seems to serve as a determiner of equality or inequality of
> > > > > >> cardinality. If set A have injection to set B then card A < card B. if
> > > > > >> set A have injection and surjection (bijection)with set B then card A
> > > > > >> = card B.
> > > > > >>
> > > > > >> Now if A and B are finite sets, their is always one mapping either
> > > > > >> injective or bijective
> > > > > >> so one can easily determine the comparison of cardinality of two sets
> > > > > >> by injecting them
> > > > > >> and seeing what result will happen? is it injection or injection with
> > > > > >> surjection? and accordingly
> > > > > >> determines the equality or inequality of cardinalities of the two
> > > > > >> finite sets.
> > > > > >>
> > > > > >> But if A and B are infinite sets, their is always infinite mappings
> > > > > >> some are injective from A to B , some are injective from B to A and
> > > > > >> some are bijective between A and B.
> > > > > >>
> > > > > >> Accordingly: card A < card B AND card A > card B AND card A
> > > > > >> = card B
> > > > > >>
> > > > > >> Now which one of these infinite mappings is the determiner of
> > > > > >> cardinality of infinite sets
> > > > > >> and why? and if non of them then how we determine the cardinality of
> > > > > >> infinite sets.
> > > > > >>
> > > > > >>
> > > > > >> Zuhair
> > > > > >
> > > > > >
> > > > > >Now I reached to the bottom of the difference between my understanding
> > > > > >of cardinality and the standard one.
> > > > > >
> > > > > >1) I define Cardinality in a different way , so in reality I am talking
> > > > > >about another version of cardinality.
> > > > > >
> > > > > >2) My version of Cardinality was not present in infinite sets because
> > > > > >it uses the
> > > > > >meaning of proper subset as the bases for inequality, and since at
> > > > > >infinite
> > > > > >level there are injections between a set and its proper subset then my
> > > > > >type of cardinality
> > > > > >disapeare or become inconsistent at infinite level. That's why I always
> > > > > >called for another
> > > > > >axiom like the Axiom of Congruence in order to limit the inconsistency
> > > > > >my definition of cardinality arrives at infinite level.
> > > > > >
> > > > >
> > > > > It's exactly because any infinite set can be injected into a proper
> > > > > subset of itself that we can't use the existence of an injection from
> > > > > A to B to define the concept "A has smaller cardinality than B".
> > > > >
> > > > > Thus, the problem with your definition (call it card', not card) is
> > > > > that you can get the contradiction card'(A)<card'(A) when A is
> > > > > infinite.
> > > > >
> > > > > Intuitively, if A is a subset of B, A can't have larger cardinality
> > > > > than B. Thus, any definition of cardinality which allows comparison
> > > > > should at least require that if there is an injection from A to B,
> > > > > then card(A)<=card(B).
> > > > >
> > > > > Also, it's reasonable to expect if there is a bijection between 2
> > > > > sets, then the sets should have the same cardinality.
> > > > >
> > > > > Finally, the general definition of cardinality should give the
> > > > > expected answers for the comparison when the 2 sets are both finite.
> > > > >
> > > > > This motivates the general definition of cardinality.
> > > > >
> > > > > quasi
> > > >
> > > >
> > > > According to the standard definition of cardinality , if I want to
> > > > define "proper subset" by extention, then I should say A is a proper
> > > > subset of B if A is a subset of B and
> > > > card A <= card B.
> > > >
> > > > while at finite level A is a proper subset of B if A is a subset of B
> > > > and card A < card B.
> > > >
> > > > Well I will try to define subcardinality( I don't know weather this
> > > > term is used in standard fields of mathematics) as below:
> > > >
> > > > 1)For any f(x)=y, f" is defined as f"(rang f) = dom f .
> > > >
> > > > 2) For any two sets A and B , if there exist f:A->B and f":B->A were
> > > > both are injective
> > > > then f is called subbijective.
> > > >
> > > > 3) For any two sets A and B , if there exist f:A->B were f is injective
> > > > and f":B->A
> > > > were f" is not injective ,then f is called subinjective.
> > > >
> > > > 4) For any two sets A and B , if there exist f:A->B were f is
> > > > subbijective and f = f"
> > > > then f is called id-subbijective
> > > >
> > > > 5) For any two sets A and B , if there exist f:A->B were f is
> > > > id-subbijective
> > > >
> > > > Then subcard A = subcard B
> > > >
> > > > 6) For any two sets A and B, if there exist f:A->B were f is
> > > > subinjective then
> > > > subcard A < subcard B, provided that 5) is exluded
> > > >
> > > > 7)For any two sets A and B, if there exist f:B->A were f is
> > > > subinjective then
> > > > subcard A > subcard B, provided that 5) is excluded
> > > >
> > > > 8) For any two sets A and B , if there exist f:A->B were f is
> > > > subbijective
> > > > then subcard A = subcard B, provided that 6) and 7) are exluded.
> > > >
> > > > Results:
> > > >
> > > > Now if A and B are finite sets , the definitions above gives the same
> > > > results
> > > > as cardinality.
> > > >
> > > > So:-
> > > >
> > > > 0) if A is finite then card A = subcard A always.
> > > >
> > > > If A,B,C are infinite sets then :
> > > >
> > > > 1) subcard A = subcard A
> > > > 2) subcard B = subcard A => subcard A = subcard B
> > > > 3) subcard A = subcard B , subcard B=subcard C => subcard A = subcard
> > > > C
> > > >
> > > > 4)subcard A = subcard B => [{ A . B} C {A \/ B} ] /\ [{A . B} is a
> > > > finite set]
> > > >
> > > > ( "." intersection , "C" belong to as a proper subset , "\/" or , "/\"
> > > > and )
> > > >
> > > > 5)if A and B are two infinite sets and subcard A <> subcard B then the
> > > > intersectional
> > > > set of A and B is infinite and is a proper subset of at least one of A
> > > > or B.
> > > >
> > > > subcard A <> subcard B => [{ A . B} C {A \/ B} ] /\ [{A . B} is
> > > > an infinite set]
> > > >
> > > >
> > > > I am not sure of these definitions really, I only wrote them in hast ,
> > > > may be they are gravely erronous? I don't know? however I will be more
> > > > than happy if any one points errors in them and especially in their
> > > > results( In reality I expect these definitions to be contradictive and
> > > > the results I wrote to be inconsistentely derived by them).
> > > >
> > > > Enjoy brain train !
> > > >
> > >
> > > Your definition of f'' leaves more than a little to be desired. (even
> > > after
> > > your correction). f'' is pretty much the inverse of f, but it is not
> > > clear if this is exactly what you mean.
> > >
> > > Anyway, let A be the natural numbers and B be the even natural numbes.
> > >
> > > i) Let f:A->B be f(x)=4x. Then f''(x)=x/4. f is an injection from A
> > > to B, f'' is
> > > not an injection from B to A. Thus f is subinjective and
> > > subcard(A) < subcard(B)
> > >
> > > ii) Let f:B->A be f(x) = x. Then f''(x) =x. f is an injection from B
> > > to A
> > > and f'' is not an injection from A to B. Thus f is subinjective and
> > > subcard(B) < subcard(A)
> >
> >
> > That's the basic point, I am with that conclusion
> > subcard A <> subcard B , here what I mean by <> is bigger or smaller or
> > both
> > bigger and smaller ( you see < means not >= , > means not <= ,
> > accordinglty
> >
> > the truth values of < > are
> >
> > < >
> >
> > T F T
> > F T T
> > T T T
> > F F F
> >
> > the last condition occures with [ not(not >=)] and [ not(not =<) ]
> > which means
> >
> > >= and <= wich means = , so only = is always false and not
> >
> > implied by <> .
> >
> > If you compair between two infinite series A and B of the same
> > cardinalities then of coarse
> > the subcardinality will be subcard A <> subcard B.
> >
> > But if you compair between two infinite series of different
> > cardinalities like
> > N and R then subcard N < subcard R.
> >
>
> > What I am trying to say is when A and B are infinite sets and subcard A
> > <> subcard B
> > then the intersectional set of A and B is infinite.
>
> No. Let A be the even naturals and B be the odd naturals.
> (Do you ever try your definitions on obvious simple cases?)

OOpps :

Correction: the intersectional set of A and B CAN be infinite.





>
>
> > Accordingly by the concept of subcardinality one can define proper
> > subset by extention as
> >
> > "A is a proper subset of B , if A is a subset of B and subcard A <>
> > subcard B".
> >
>
>
>
> In fact thiis correct, but for the utterly boring reason that
> if A is a subset of B and A and B have the same cardinality,
> then subcard A <> subcard B, unless A=B (you create a special
> exception for this case). So the only way to show subcard A = subcard
> B
> is to show A=B.

No A do not necessarily equal B.

see set A = 1,2,3,4,5,...............
set B = 1,1/2,1/3,1/4,1/5,................

These are also set which has subcard A = subcard B

Also set A = 1,2,3,4,5,............
and set B = -1,-2,-3,-4,-5,..............

subcard A = subcard B.



So if we can show that A=B, we can conclude that
> A=B. I am underwhelmed.
>
> > This is a general definition of Proper subset by extention , equally
> > applicable
> > for finite and infinite sets.
> >
> > I think the importance ( if valid ) of this definition comes when we
> > cannot define proper subset by intention ( the standard way ), and I
> > expect that to happen with future infinite calculus.
>
> You still have not said what you find problematic about the
> standard definition (A is a proper subset of B if A is
> a subset of B and B\A is not empty). Nor have you
> said why you think some cardinality type measure
> should have anything to do with this.
>
>
> >
> > Zuhair
> >
> >
> >
> >
> >
> > >
> > > (Note you can replace A and B with any two sets of the same
> > > cardinality.
> > > You will always have an injection from A to B that is not a surjection
> > > and an injection from B to A that is not a surjection)
> > >
> > > You remain convinced that the reason that mathematicians have
> > > not replaced cardinality with a better generalization of number
> > > of elements to infinite sets is because they are crooked, stupid
> > > and/or lazy. You are wrong. I would like to wish you luck in
> > > your quixotic search for a cardinality replacement but I cannot.
> >
> > Here I am not replacing cardinality. I am defining something else which
> > can actually be related to cardinality. The definition of
> > subcardinality
> > is not intended to replace cardinality, nor it will.
> >
>
> What do you want subcardinality to do?
>
> >
> > > You are looking for something that does not exist.
> >
> > So cardinality is the only concept present in math of the infinite ???
> > huh , no new defintions of something else are accepted at all??? what
> > is that Dictatorship of Cardinality.
>
>
> If I said to you, "you will never find an even prime greater than 2"
> would you ask "What is that, Dictatorship of 2"?
>
> -William Hughes

.

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