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Proper Negation

  1. Apr 27, 2004 #1

    honestrosewater

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    I thought this would be easy...

    BEGIN QUOTE
    Definition. If there exists a 1-1 mapping of A onto B... we write A~B. ...This relation... is called an equivalence relation.

    Definition. For any positive integer n, Let J_n be the set whose elements are the integers 1, 2, ..., n; let J be the set consisting of all positive integers. For any set A we say:
    a) A is "finite" if A~J_n for some n (the empty set is also considered to be finite).
    b) A is "infinite" if A is not finite.
    END QUOTE

    I just want to define infinite by writing out the negation of a).
    I wasn't sure if I had to change "some n", so I tried to rewrite a) using what FOL I know, but couldn't figure out how to combine all the requirements of the equivalence relation. I end up with several sentences. Is there a simpler way?
    Ugh,
    Rachel
     
  2. jcsd
  3. Apr 27, 2004 #2

    matt grime

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    Rewriting a) it states

    A is finite if there exists an n such that A~J_n

    the negation is (ie A is infinite if)

    for all n A~/~J_n

    where use ~/~ to mean does not inject to J_n
     
  4. Apr 27, 2004 #3

    honestrosewater

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    Thanks :)
    So this excludes 'infinity' from being used as a positive integer... i.e. I can't start counting at infinity {n, n-1, n-2, ..., 1}. Just wanted to make sure.
    Happy thoughts
    Rachel
     
  5. Apr 27, 2004 #4
    Just use the rules for the quantifiers:

    [tex]{\neg}({\forall}x)({\neg}{f})x=({\exists}x)fx[/tex]

    and

    [tex]{\neg}({\exists}x)f(x)=({\forall}x)({\neg}f)x[/tex]

    So, let [tex]A[/tex] be an arbitrary set, [tex]f[/tex] be the predicate "is finite", [tex]J_n[/tex] be the set of all natural numbers less than or equal to n, and [tex]XBY[/tex] be a bijection between two sets [tex]X[/tex] and [tex]Y[/tex].

    So:
    [tex]{\exists}n:ABJ_n{\rightarrow}{fA}[/tex]

    Applying the negation:
    [tex]{\neg}{\exists}n:ABJ_n{\rightarrow}({\forall}n)({\neg}f)A}[/tex]
     
  6. Apr 28, 2004 #5

    matt grime

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    infinity isn't a positive integer, and you're best off avoiding all use of infinity if you can. just use the adjective infinite. if no one used infinity (which is entirely possible and desirable) we'd all be much better off.
     
  7. Apr 28, 2004 #6
    Why do you say that, Matt? I think infinity is a useful concept, despite its difficulties.
     
  8. Apr 28, 2004 #7

    matt grime

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    for the simple reason of the misunderstandings that arise, such as the one in this thread. Encouraging people to treat infinity as something physical (and i mean that in the same sense as 2, pi, e are physical numbers) only causes problems. I have no problems with people using it in compound names such as 'the point at infinity' or in phrases such as '1/x tends to infinity as x tends to zero', but many people misuse it to such an extent that I start to question its usefulness. even the otherwise admirable wolfram says infinity is 'something bigger than any natural number' which is a dangerous thing to say. In all those cases when we say infinity, we mean 'does not X finite Y' where X/Y could be "stop at some" /"number". It might be laborious to always say that but it would make it easier to deal with problems when they arise.
     
  9. Apr 28, 2004 #8
    I've never personally encountered someone seriously involved in maths who misunderstood infinity in the manner you describe. But that's not much of an argument...

    I think you're on to something when you point out that "Encouraging people to treat infinity as something physical (and i mean that in the same sense as 2, pi, e are physical numbers) only causes problems." My own thoughts are that it is dangerous to think of any number as such.

    One position I've always had in regard to mathematical pedagogy is that teaching mathematics should also involve teaching the philosophy of mathematics. If more people had insight into these issues, then I think a lot of the conceptual difficulties with things like infinity (is there anything really like it, though? ;) ) wouldn't be as notable.
     
  10. Apr 28, 2004 #9

    matt grime

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    do a quick search for the number of posts here and on sci.math for people who don't understand why you can't divide 0 by zero or infinity by infinity and you'll see what you've to deal with in the "lay" person. to any reasonable mathematician the answer is 'well you can't because the axioms do not allow you to, and you may as well ask what the quotient of paint by alfred the great is.'
     
  11. Apr 28, 2004 #10

    honestrosewater

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    The whole point of my looking closer at this definition of an infinte set was to avoid a misunderstanding about infinity :) I was hoping it exculded infinity from being a number. I think it's telling that, of the two ideas
    1) that you can start at 1 and, by repeated addition, end at infinity and
    2) that you can start at infinity and, by repeated subtraction, end at 1,
    though both must be equally incorrect, the error is much better hidden in the first than the second.
    BTW where would be the best place to discuss infinity?
    Happy thoughts
    Rachel
     
  12. Apr 28, 2004 #11

    matt grime

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    The error in the second is just as obvious: infinity isn't an integer, it isn't a member of the real numbers, so you can't take anything away from it as it is not part of an algebraic system that you know about.
     
  13. Apr 28, 2004 #12

    matt grime

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    SOmething I meant to mention before. The property of A~B is not an equivalence relation. It is not symmetric
     
  14. Apr 28, 2004 #13

    honestrosewater

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    Great, now I have to ask. I understand 0/0 but what does this have to do with infinity/infinity? I thought infinity wasn't a number :confused: or is that the point?
    Rachel
     
  15. Apr 28, 2004 #14

    honestrosewater

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    I meant the second was more obvious than the first. Have you never slipped and said "end at infinity" or seen other competent people slip?
     
  16. Apr 28, 2004 #15

    matt grime

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    a sloppy way of writig it is to say that 1/0=infinity, so that if we pretend this makes more sense is 0/0 = (0/1)*(1/0) = infinity/infinity.
     
  17. Apr 28, 2004 #16

    honestrosewater

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    The quote is from Walter Rudin's 'Principles of Mathematical Analysis', 3rd ed. I omitted his definitions of "1-1 mapping" and "onto", but, as I understand them, "1-1"=injective and "onto"=surjective, making "~" bijective, though he doesn't use those terms. Do you want me to type out the two definitions, they aren't that long?
    beginning of chapter 2, if you have it.
     
    Last edited: Apr 28, 2004
  18. Apr 28, 2004 #17

    honestrosewater

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    So you kill two birds with one stone- infinity is undefined and n/0 is undefined- nice trick. :biggrin:
     
  19. Apr 28, 2004 #18

    matt grime

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    no you just needed to say the quote in full, as it stands it is in correct in your first post because you omit the mention of surjection, hence you imply that A~B if there is an injection is an equivlance relation. now that you've explained you've omitted the other part of the defintion I'm happy. note that you only need injections to decide if a set is finite or not.
     
  20. Apr 28, 2004 #19

    honestrosewater

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    He uses ~ to define countable and uncountable sets, I figure this is why he adds the "onto".

    Only injections? I never thought of that. Is it because you cannot inject an infinite set into a finite set?
     
  21. Apr 28, 2004 #20

    matt grime

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    of course: if there is no injection there is certianly no bijection. moreover, if A injects to B and B injects to A then there is a bijeciton between them (schroeder bernstein)
     
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