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Concept of the pigeonhole principle

  1. Aug 7, 2005 #1
    This question applies with the so called "infinite" pigeonhole principle. Why is it possible to construct a one-one function out of two sets where the codomain has a length smaller than the length of the domain?
     
  2. jcsd
  3. Aug 7, 2005 #2

    Hurkyl

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    Could you go into a lot more detail? It's hard to answer appropriately if I don't know just where you're confused.

    The only measure of "size" that matters for one-to-one functions is cardinality.
     
  4. Aug 7, 2005 #3

    matt grime

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    "length" what does that mean in this context?
     
  5. Aug 7, 2005 #4
    Hello...

    This is what I understood so far...

    I was able to find that in the infinite case, "If n holes contains an infinite number of points, then at least one of the holes contain an infinite number. In particular, if the holes are labeled or ordered from 1 to n, then there must be a first hole with infinitely many points in it."

    What confuses me is regarding whether this concept I have researched applies to a one-one function...

    I was thinking of cramming Q+, the positive rationals into N the natural numbers with space left over to do it infinitely more times. Let p/q always be a reduced fraction in Q+ and define the map Q+--->N;p/q--->(2^p)(3^q). I can know that
    this is a 1-1 map by the fundamenntal theorem of arithmetic. (unique prime
    factorization) and no number that has any other prime in it's decomposition
    other than 2 or 3 is in the range.

    Also, I can construct a cartesian plane with all the values of y between -1 and 1, exclusively, ie., (-1,1) and all the values of x indefinitely. From here, I can assigned a one-one function in which every value in x corresponds to a unique value of y.
     
  6. Aug 7, 2005 #5
    Length here, is not that important since it doesnt determine size of the interval, since the interval will always have infinitely many points...
     
  7. Aug 9, 2005 #6
    You can also think it of as an infinite set broken into a finite number of subsets, then one of the subsets must be an infinite set.

    Pardon?

    ? Whats the problem ?

    Yes this is true, it follows from the schroeder-bernstein theorem. It goes to show how dense the set of (-1,1) can be.

    -- AI
     
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