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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Fri, 18 Oct 2024 14:22:37 -0700
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On 10/18/2024 2:15 PM, FromTheRafters wrote:
> Chris M. Thomasson presented the following explanation :
>> On 10/18/2024 1:55 PM, FromTheRafters wrote:
>>> Jim Burns brought next idea :
>>>> On 10/18/2024 2:13 PM, Chris M. Thomasson wrote:
>>>>> On 9/16/2024 2:38 PM, Jim Burns wrote:
>>>>>> On 9/16/2024 4:15 PM, Chris M. Thomasson wrote:
>>>>
>>>>>>> WM is a strange one.
>>>>>>> If no point is between them,
>>>>>>> then they are the same.
>>>>>>
>>>>>> If no point is between different points,
>>>>>> then the points aren't in the complete line.
>>>>>
>>>>> If p0 and p1 are the same point then p1 - p0 is zero.
>>>>>
>>>>> Say:
>>>>>
>>>>> p0 = (1, 2, 3)
>>>>> p1 = (1, 2, 3)
>>>>>
>>>>> pdif = p1 - p0
>>>>>
>>>>> pdif would be (0, 0, 0)
>>>>>
>>>>> See?
>>>>
>>>> I wouldn't choose to say that 0 is between 0 and 0
>>>> If necessary,
>>>> I would correct what I'd said so that
>>>> it could not be taken to mean that.
>>>
>>> Yeah, you said different as opposed to unique. I simply took it to 
>>> mean not the same point.
>> [...]
>>
>> if p0 and p1 are different in my example then pdif will not be all 
>> components zero.
> 
> For some value of 'different'.
> 
> This is a dot . this is a different dot . but neither dot is unique. Two 
> different elements from a multiset,each might not be unique (both dots) 
> but may be distinct (rightmost dot, leftmost dot).

Well, I was thinking of p0 and p1 being 3-ary vectors in my example. 
Three components, x, y and z.

So, if p0 and p1 are different, then pdif will not have its x, y and z 
components all zero.