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Thread: Involutions on the set of the natural numbers

  1. #1
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    Involutions on the set of the natural numbers

    Hi,
    I want to find the cardinality of the set of all involutions on the set of all natural numbers N.
    I don't even see a single involution on N.
    Can i get a clue for the solution?
    Thank's in advance.

    AN involution is a function from N to N such that f(f(x))=x for every x.
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  2. #2
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    Re: Involutions on the set of the natural numbers

    This is an involution on N with the property that f(x) and x are different for all x in N:
    f(x)=x-1 for even x,and f(x)=x+1 for odd x.
    How can this be used to show that for every infinite subset A of N there is an involution g on N such that g(x) is different from x only on the elements of A?
    Thank's in advance.
    Last edited by hedi; Dec 26th 2016 at 03:54 PM.
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  3. #3
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    Re: Involutions on the set of the natural numbers

    So your first statement, "I don't even see a single involution on N" is no longer true? Good!
    (A really obvious one is f(x)= x.)

    "show that for every infinite subset A of N there is an involution g on N such that g(x) is different from x only on the elements of A".

    First define f(x)= x for any x in the complement of A. Now for A, use the fact that the N, and every infinite subset of N, is "denumerable". That is, we can order A as {a_1, a_2, a_3, \cdot\cdot\cdot\}, and define [tex]f(a_i)= a_{i+1}[tex] if i is odd, f(a_i)= a_{i-1} if I is even.
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  4. #4
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    Re: Involutions on the set of the natural numbers

    I ment non-identity function.the indeces you defined in your answer is an example for such function.
    Thank's a lot,you were very helpful.
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