Need help with Countable Set

Discussion in 'Undergraduate Math' started by Herb, Feb 6, 2005.

  1. Herb

    Herb Guest

    Show that :
    every infinite set has countable subset which is equipotent to N(atural numbers).

    Could someone give me a hint or sketch of the proof ?

    Thanks in advance.

    Give a example :
    Infinite set that is union of countable number of finite set.

    My answer : \/_{n=1 to infinity} {n}

    Is it correct ?
    Herb, Feb 6, 2005
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  2. As the set S is infinite, there is a injection f:S -> S
    with f(S) proper subset S. Pick x in S\f(S).

    Show { f^k(x) | k in N } is countable.
    Yes, it'll do.
    What's incorrect is to write hard to read equations like
    I rarely bother with posts like that for there are so many more
    that are easier to read, they usually occupy my evening's time.
    William Elliot, Feb 6, 2005
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  3. Herb

    Herb Guest

    Thank you. Sir
    Yes, I believe this, too.
    But, I cannot think method for proving this.
    How can I prove above fact ?
    Herb, Feb 8, 2005
  4. It's Cantor's definition of infinite.
    What definition of infinite are you using?
    William Elliot, Feb 8, 2005
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