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In a practice test for my algorithms class I was asked to prove that the set $A = \{k^\frac{1}{k}| k \neq 0, k \in \mathbb{Z}\}$ is countably infinite. My professor provided me with the answer that we can create an image like the following

$$ 1 \rightarrow 1^\frac{1}{1}, 2 \rightarrow (-1)^{-\frac{1}{1}}, 3 \rightarrow 2^\frac{1}{2}, 4 \rightarrow (-2)^{-\frac{1}{2}} ..., k \rightarrow ((-1)^{k+1}\lfloor\frac{k-1}{2} + 1\rfloor)^{(-1)^{k+1}\cdot\frac{1}{\lfloor\frac{k-1}{2} + 1\rfloor}}$$

which shows that for every $k$ there is an unique value of $((-1)^{k+1}\lfloor\frac{k-1}{2} + 1\rfloor)^{(-1)^{k+1}\cdot \frac{1}{\lfloor\frac{k-1}{2} + 1\rfloor}}$, and thus the image is a bijection and the set is countable infinite.

I am familiar with the concept of how there has to be a bijection with the natural numbers for a set to be countably infinite, but I have no idea how my professor got to the expression after '$k \rightarrow$'. Can someone enlighten me on this? Or, perhaps, show a different way to proof that the set is countably infinite?

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  • $\begingroup$ Welcome to Math Stack Exchange. I'd find it easier to find a bijection for $k>0$ and $k<0$ separately and then use the fact that the union of two countably infinite sets ($k>0$ and $k<0$) is countably infinite $\endgroup$ – J. W. Tanner Jun 16 '19 at 19:27
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    $\begingroup$ If I were you I would ask my professor first. $\endgroup$ – uniquesolution Jun 16 '19 at 19:28
  • $\begingroup$ Do you know that the set of algebraic numbers is countably infinite? $\endgroup$ – J. W. Tanner Jun 16 '19 at 19:38
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    $\begingroup$ @J.W.Tanner I was not aware that the set of algebraic numbers is countably inifinite. The answer on that question seems like useful information. $\endgroup$ – Yousousen Jun 16 '19 at 20:04
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    $\begingroup$ On the other hand, if the values in $A$ are all supposed to be real, then $k^{1/k}$ doesn't mean anything when $k=-2$, so the professor's claimed bijection fails to even be a function in that case. $\endgroup$ – hmakholm left over Monica Jun 16 '19 at 20:06
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Your professor's expression is a (complicated way) of (attempting to) explicitly writing down a bijection. The strategy really has two pieces: firstly it shows that the positive integers is in bijection with all the non-zero integers, and then in turn that the non-zero integers are in bijection with the set $A$. The formula $$\left\lfloor \frac{k-1}{2}+1\right\rfloor = \left\lfloor \frac{k+1}{2}\right\rfloor = \left\lfloor \frac{k}{2}+\frac{1}{2}\right\rfloor$$ is a somewhat unhelpful way to say "Divide by two, and round up".

To see this, first suppose $k$ is even. Then $k/2$ is an integer, so adding a half to it and then taking the floor does not change its value. Hence the value of the expression for even $k$ is just $k/2$. On the other hand, if $k$ is odd, then $k/2$ is an integer plus $1/2$. Adding another half to this then taking the floor is the same as just adding the $1/2$, or, rounding up.

What this function does for us is that it matches each positive integers with two other positive integers, e.g. $1$ is matched with $1$ and $2$, $2$ is matched with $3$ and $4$, and so on.

Then the $(-1)^{k+1}$ part makes the sign of the output alternate, so now each positive integer is matched with one non-zero integer (as $2$ now goes to $-1$ instead of $1$, and $4$ goes to $-2$ instead of $2$, and so on).

Once we have a function that does this, you raise it to the power one over itself, which matched it up with elements of the set $A$.

Edit: It was pointed out to me that I have not actually shown that $A$ is infinite. What the above shows is that the map is surjective (or onto), but we would need to show that $A$ has infinitely many elements. This is slightly complicated by the fact that $A \subset \mathbb{C}$ but not $\mathbb{R}$. The first thing that occurs to me is to show that the function $f(x) = x^{1/x}$ for $x>0,x\in\mathbb{R}$ is decreasing for $x>e$ (as its derivative is negative there).

Edit: Note that in the definition of $A$, $k=2$ and $k=4$ give the same value: $\sqrt{2}$. Consequently, any map that tries to enumerate $A$ in the "obvious" way will not be injective.

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  • $\begingroup$ I think we're missing an argument here that the $k^{1/k}$s are all different -- or at least different enough that $A$ avoids being finite. $\endgroup$ – hmakholm left over Monica Jun 16 '19 at 19:52
  • $\begingroup$ For $k=2$ and $4$ they're equal $\endgroup$ – J. W. Tanner Jun 16 '19 at 19:56
  • $\begingroup$ @James Thank you very much! This cleared up a lot. $\endgroup$ – Yousousen Jun 16 '19 at 20:07
  • $\begingroup$ @HenningMakholm You are correct, let me fix that. $\endgroup$ – James Jun 16 '19 at 20:09

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