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| Date | 2023-06-15 17:06 -0700 |
| References | (23 earlier) <19ad04ee-38d5-47c6-b9d8-1a453c4f6b0fn@googlegroups.com> <adb974b4-f192-4a39-a9e0-a5314df6f99fn@googlegroups.com> <d3d96577-746b-4b6c-8505-c5d44734bcd6n@googlegroups.com> <fed0e46e-8421-4007-9381-9469f04d959cn@googlegroups.com> <86b2b992-195a-4350-bc93-4d83861cf59cn@googlegroups.com> |
| Message-ID | <46943df9-46c2-423c-b15d-f13d98fe3bc3n@googlegroups.com> (permalink) |
| Subject | Re: Some results about unit fractions |
| From | Fritz Feldhase <franz.fritschee.ff@gmail.com> |
On Thursday, June 15, 2023 at 6:26:26 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Donnerstag, 15. Juni 2023 um 15:01:04 UTC+2:
> > On Thursday, June 15, 2023 at 1:53:14 PM UTC+2, WM wrote:
> > >
> > > Can you imagine two terms of an inclusion-monotonic sequence (M_n): M(n) ⊇ M(n+1) which have no element in common?
> > >
> > Ja, Wenn für zwei Mengen A, B, A ⊇ B gilt, dann ist A n B = B. (*)
> >
> > Fordern wir also A n B = { }, so folgt B = { }. (Where A may be any set.)
> >
> [...] A last question: Why does this not hold for all terms of the sequence of endsegments?
Because it does not hold for all terms of the sequence of endsegments.
______________________________
An the other hand, the result from above can be generalized to any (finite) number of terms.
Not a straight proof, but a reasonable explanation:
Assume (for any n e IN) that we have A_1 ⊇ A_2 ⊇ ... ⊇ A_n. Then by (*) (...(((A_1 n A_2) n A_3) n A_4) ... ) n A_n = (...((A_2 n A_3) n A_4) ... ) n A_n = (...(A_3 n A_4) ... ) n A_n = ... = A_n-1 n A_n = A_n. In short: A_1 n A_2 n ... n A_n = A_n
Hence if we require A_1 n A_2 n ... n A_n = { }, this implies A_n = { }. Where A_1, ..., A_n-1 may be any sets.
Now this generalization does not "work" for _infinitely_ many terms A_1, A_2, A_3, ..., since there is no "last" term (A_n, in the "proof" above) in this case.
You see, there's no last/biggest natural number, and an infinite sequene (A_n)_(n e IN) does not have a last term.
______________________________
Actually, A_1 n A_2 n A_3 ... = { } does NOT imply that there is a k e IN such that A_k = { }, even though A_n ⊇ A_(n+1) may hold (for all n e IN).
The sequence (E_n)_(n e IN) with E_n = {m e IN : m >= n} (for all n e IN) is a "counterexample".
1. We have E_n ⊇ E_(n+1) for all n e IN.
2. E_n =/= { } for all n e IN.
3. INTERSECTION_(n e IN) E_n = { }.
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Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-04 20:39 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-05 07:12 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-05 07:41 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-06 07:46 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-06 07:55 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-06 11:06 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-06 13:18 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-06 13:33 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-06 13:52 -0400
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-05 13:25 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-06 11:04 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-05 15:49 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-06 11:11 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-07 12:25 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-08 03:46 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-08 04:13 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-08 05:51 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-08 07:15 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-08 09:34 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-08 16:48 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-08 16:50 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-09 03:43 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-08 14:48 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-08 13:50 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-08 18:39 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-09 03:48 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-09 12:26 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-09 13:36 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-09 18:25 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-10 05:05 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-10 15:10 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-11 06:47 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-11 10:24 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-11 11:37 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-11 11:53 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-11 13:24 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-11 15:21 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-11 15:38 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-12 11:52 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-12 11:52 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-12 13:16 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-12 14:03 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-12 14:45 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-13 07:20 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-13 07:35 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-13 14:02 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-13 14:50 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-13 20:12 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-12 14:48 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-11 15:11 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-11 13:28 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-11 17:12 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-12 11:52 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-13 13:25 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-13 14:07 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-13 20:11 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-14 04:55 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-14 05:58 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-14 11:30 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-14 12:55 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-15 04:53 -0700
Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-06-15 13:22 +0100
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-15 05:40 -0700
Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-06-16 01:17 +0100
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-16 06:32 -0700
Re: Some results about unit fractions Python <python@invalid.org> - 2023-06-16 16:27 +0200
Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-06-18 02:09 +0100
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-18 07:15 -0700
Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-06-20 01:34 +0100
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-20 08:34 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-20 10:09 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-20 10:18 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-21 07:17 -0700
Re: Some results about unit fractions Python <python@invalid.org> - 2023-06-21 16:52 +0200
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-22 05:48 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-22 13:20 -0700
Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-06-22 00:51 +0100
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-22 05:58 -0700
Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-06-23 01:05 +0100
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-22 17:17 -0700
Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-06-23 02:26 +0100
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-23 09:01 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-23 08:53 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-23 09:52 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-15 06:01 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-15 09:26 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-15 09:52 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-15 10:59 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-15 12:40 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-16 06:07 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-16 06:22 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-16 06:41 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-16 08:54 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-16 12:37 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-17 05:54 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-17 08:46 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-18 06:57 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-17 11:56 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-18 06:59 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-18 14:40 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-19 07:01 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-19 07:34 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-20 08:30 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-20 10:30 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-19 13:55 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-20 08:36 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-20 10:24 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-20 14:03 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-21 07:22 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-21 12:26 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-22 05:54 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-22 09:38 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-23 08:44 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-23 12:37 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-23 09:53 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-23 15:39 -0400
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-16 08:53 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-15 17:06 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-16 06:18 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-16 06:53 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-17 08:10 -0700
Re: Some results about unit fractions Python <python@invalid.org> - 2023-06-17 17:17 +0200
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-17 08:30 -0700
Re: Some results about unit fractions Python <python@invalid.org> - 2023-06-17 17:33 +0200
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-16 06:25 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-14 08:03 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-14 10:58 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-14 18:30 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-15 04:51 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-09 14:18 -0700
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