Re: Relational operators on intervals: bug?

• To: mathgroup at smc.vnet.net
• Subject: [mg128644] Re: Relational operators on intervals: bug?
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Tue, 13 Nov 2012 00:03:38 -0500 (EST)
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```On Nov 12, 2012, at 3:07 AM, Richard Fateman <fateman at cs.berkeley.edu> wrote:

> On 11/11/2012 12:59 PM, bertiiiiiiiiiiiiiiiiiiiiiiiiii at gmail.com wrote:
>
>> 1. Comparision of any Interval[___] to the empty interval Interval[]:
>>
>
> What makes you think that this is a notation for an empty Interval,
> whatever that is?

One way to define "interval" J mathematically in a (linearly) ordered set X such as the set of reals is to say J is a subset of X having the property that, for all a, b and c in X, if a and b are elements of J and if a < c < b, then also c is an element of J.

Using that definition, evidently the empty set _is_ an interval, as are rays in the real line. Which is fortunate, for otherwise the theorem that a subset of the reals is connected if and only if it is an interval would have to be restated with awkward restrictions.

Of course the OP's notation for the empty interval is spurious.

Here is the empty interval in Mathematica:

Interval[{1, 0}]

Indeed:

Resolve[Exists[x, IntervalMemberQ[Interval[{1, 0}], x]]]
False

---
Murray Eisenberg                          murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower            phone 413 549-1020 (H)
University of Massachusetts                      413 545-2838 (W)
710 North Pleasant Street                  fax   413 545-1801
Amherst, MA 01003-9305

```

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