# Thread: Involutions on the set of the natural numbers

1. ## 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?

AN involution is a function from N to N such that f(f(x))=x for every x.

2. ## 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?

3. ## 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.

4. ## 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.