Timeline for "Rocket elements" in bijections $f:\mathbb{N}\to \mathbb{N}$
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jul 14, 2019 at 22:19 | comment | added | Gabe Conant | It seems that the classification is: a subset of $\mathbb{N}$ is a rocket set if and only if it is either empty, or infinite and co-infinite. | |
| Jul 14, 2019 at 20:00 | comment | added | Greg Martin | @LSpice Aha, well spotted! And indeed (on further thought) no final segment $\{n,n+1,n+2,\dots\}$ with $n>1$ can be the rocket set of a bijection, because the preimage of $n$ would also be in the rocket set. | |
| Jul 14, 2019 at 19:48 | comment | added | LSpice | @GregMartin, I think that this doesn't work if $B$ is a final segment. | |
| Jul 14, 2019 at 19:23 | comment | added | Greg Martin | Just to tie off the loose end, here's a modification of the construction: instead of listing $A$ in increasing order, choose any way of writing $A=\{a_1,a_2,\dots\}$ as a sequence, subject only to the restriction $a_1>b_1$. Then the bijection given in the construction definitely has Roc$(f)=B$. | |
| Jul 14, 2019 at 18:36 | comment | added | Martin Sleziak | @GregMartin Thanks for the comment. I missed the fact that $a_1$ might be in $\operatorname{Roc}(f)$. | |
| Jul 14, 2019 at 18:35 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
correction pointed out by Greg Martin
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| Jul 14, 2019 at 18:27 | comment | added | Greg Martin | Indeed, this argument shows that any infinite subset of $\Bbb N$ is Roc$(f)$ for some bijection $f$ (and of course, it's impossible for a nonempty finite subset to be Roc$(f)$ for any function $f$). Although we need to be a little careful, because $a_1$ might be in Roc$(f)$ with this construction. | |
| Jul 14, 2019 at 17:00 | comment | added | Pietro Majer | The elevators in the Hilbert Hotel | |
| Jul 14, 2019 at 16:18 | comment | added | Dominic van der Zypen | Very nice construction, and nicely explained, thanks Martin! | |
| Jul 14, 2019 at 16:17 | vote | accept | Dominic van der Zypen | ||
| Jul 14, 2019 at 16:05 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
deleted 3 characters in body
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| Jul 14, 2019 at 15:45 | history | answered | Martin Sleziak | CC BY-SA 4.0 |