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Infinite sets statements equivalence

  1. Aug 1, 2013 #1
    The problem statement, all variables and given/known data

    Let A be a set, prove that the following statements are equivalent:

    1) A is infinite
    2) For every x in A, there exists a bijective function f from A to A\{x}.
    3) For every {x1,...,xn} in A, there exists a bijective function from A to A\{x1,...xn}

    Relevant equations

    The first and only thing that comes to my mind is that (I've read this in my textbook, but I have to prove it) if A is infinite, then it admits a bijection with a proper subset; but I'm not sure if proving that would help me to automatically say that 1) implies 2) and 3) because A\{x} and A\{x1,...,xn} are proper subsets of A, but how do I know that these are the indicated proper subsets, I mean, the statement says that if a set is infinite, it admits a bijection with A proper subset, not every proper subset.
    Now, when it comes to prove that 2) or 3) imply 1) I am totally stuck. So, that's all that I have, if I think of something, I'll post it here.
  2. jcsd
  3. Aug 1, 2013 #2


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    Homework Helper

    Do the 1 -> 2 proof so long, you seem to have solid ideas about that. When you've done that, you'll likely know more and will find a proof for 2 -> 1.

    PS. I say this but the Dedekind-infinite -> infinite direction will be tough, probably you'll need to look it up.
    Last edited: Aug 1, 2013
  4. Aug 1, 2013 #3
    These are not necessarily equivalent. For example, let ##A=\emptyset##. If ##A## is empty, the bijection in 2 and 3 is just the empty function given by ##f:\emptyset\to\emptyset##.
    Last edited: Aug 1, 2013
  5. Aug 25, 2013 #4
    Sorry, I forgot I've posted this. Now I am trying to solve the problem again. I think that in the case A is the empty set, 2 and 3 don't make any sense, because what is A-{x} or A-{x1,...,xn} if the empty set doesn't have any elements which belong to it?
  6. Aug 25, 2013 #5
    It's still the empty set. It's what we call a "vacuous truth." The exact statements say "for every ##x## in ##A##..."

    If there are no elements in ##A##, then the statement must be true for "all" the elements in ##A## because there aren't any elements to satisfy it.

    Even if we just have the statements without the "for every ##x## in ##A##" requirement, the empty function is trivially a bijection.
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