1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    verty

    User Avatar
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Infinite sets statements equivalence
  1. Equivalent sets (Replies: 3)

  2. Infinite Sets (Replies: 2)

  3. Infinite set (Replies: 11)

Loading...