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I need to construct bijections of few sets to show that they have the same cardinality and prove their correctness. I have already done $f: \mathbb{N}\rightarrow\mathbb{Z}$, which was not very hard, but I am struggling with a bit more complex examples.

In the following examples $a \bot b$ means, that $a$ and $b$ are coprime numbers.

Ex. 1: $\mathbb{N}$ and $\{\langle n,m\rangle \in \mathbb{N}^+\times\mathbb{N}^+ \:|\: m\bot n\}$
It should be a function of the form $g: \mathbb{N}\rightarrow \mathbb{N}^+\times \mathbb{N}^+$, but I do not know how should a function of one number create a pair of coprime numbers.

Ex. 2: $\{\langle n,m\rangle \in \mathbb{N}^+\times\mathbb{N}^+ \:|\: m\bot n\}$ and $\mathbb{Q}^+$
There the function should have the form $h:\mathbb{N}^+\times\mathbb{N}^+\rightarrow \mathbb{Q}^+$ and I was thinking about a function which would divide $n$ by $m$, which in my opinion would be pretty understandable, but I am not sure whether it is a correct idea.

I would like to get some tips how should I get a grasp in solving such problems, as well as some hints how to solve and prove these two examples.

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  • $\begingroup$ No, sadly my lectures did not involve them. $\endgroup$ – whiskeyo Jan 6 at 13:10
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    $\begingroup$ For the general problem, it's useful to know constructive proofs of the Cantor-Schröder-Berstein theorem, but I suspect that in these exercises you're supposed to come out with the appropriate bijection out of thin air. $\endgroup$ – Git Gud Jan 6 at 13:25
  • $\begingroup$ Hint: For Ex.1 do you know of that diagonal argument that proves that (for example) $\mathbb{Q}$ is countable, or that $\mathbb{N}^{2}$ has the same cardinality as $\mathbb{N}$. If so can you adapt that proof? $\endgroup$ – Adam Higgins Jan 6 at 13:51
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For $2$ we have the obvious map sending $(m,n)\in C$ to $\frac{m}{n} \in \mathbb{Q}^+$, where $C$ are the co-prime pairs of positive integers. It's onto and 1-1 because every positive rational has a unique representation as a quotient of co-prime positive integers (after cancelling common factors in numerator and denominator).

As to $1$, did you do a bijection from $\mathbb{N}$ to $\mathbb{N}^2$? You could use that for the subset $C$ as well: If $f$ is that earlier bijection, define $g: \mathbb{N} \to \mathbb{N}$ recursively by $g(0) = \min \{n: f(n) \in C\}$ and $g(n+1) = \min\{n : n > g(n) \text{ and } f(n) \in C\}$ and then $h(n) := f(g(n))$ is a bijection from $\mathbb{N}$ to $C$.

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  • $\begingroup$ When it comes to the 2nd example, I understand it well, as it is pretty close answer to what I thought at first. I have had a bijection from $\mathbb{N}^2$ to $\mathbb{N}$, which is Cantor's pairing function on 2D array, but I did not have an inverse function to that, also when I read about inverting it on Wikipidia, it makes me confused and I do not understand how it is done. $\endgroup$ – whiskeyo Jan 6 at 15:34
  • $\begingroup$ @whiskeyo if $f$ is a bijection from $\mathbb{N}^2$ to $\mathbb{N}$ then $f$ restricted to $C$ is a bijection to an infinite subset of $\mathbb{N}$ which you then enumerate to get a bijection with $\mathbb{N}$ (map each $n$ in $f[C]$ to its “rank”. $\endgroup$ – Henno Brandsma Jan 6 at 16:11

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