       Re: LessEqual vs Inequality, was ..Re: Replacement Rule with Sqrt in denominator

• To: mathgroup at smc.vnet.net
• Subject: [mg114724] Re: LessEqual vs Inequality, was ..Re: Replacement Rule with Sqrt in denominator
• From: Richard Fateman <fateman at cs.berkeley.edu>
• Date: Tue, 14 Dec 2010 06:55:42 -0500 (EST)

```On 12/13/2010 11:56 AM, Andrzej Kozlowski wrote:
> On 13 Dec 2010, at 17:53, Andrzej Kozlowski wrote:
>
>>> FullSimplify [Less[x,y,z]-Inequality[x,Less,y,Less,z] ]
>>> does not reduce to zero, as it should.  So Mathematica, dare
>>> I say, exhibits a bug.
> In fact, I am (almost) amazed to see you claim that. There are certainly advantages to having a mathematics education for I am sure that nobody with that would ever expect this to work. What sort of algebraic structure is this subtraction taking place in, according to you? What sort of algebraic transformations is FullSimplify supposed to use here?
The same structure that allows  (3<4) - (5<6)  to come out as 0.  That
is, in Mathematica, True - True is zero.
You may argue that truth values do not support subtraction as an
operation, but then you are imposing either
an algebraic structure (mathematically speaking)  or a type system
(programming-language speaking -- or
perhaps mathematically, a category theory), on Mathematica.  There is
very little evidence of such sensitivity
in Mathematica, and I, for one, am not a keen fan of imposing such a
structure in a computer algebra system
that is kind of open ended.

The algebraic transformation that FullSimplify can, and I think should,
apply,  is <object X>  - <object X> --> 0.

This is ordinarily, but not universally true
.  e.g. Indeterminate - Indeterminate is not zero.
Nor is it zero for X= Interval[{-1,1}].
But  True - True seems to me kind of OK to simplify to 0.
After all, if someone has asked you to do subtraction on these things, he
might know what he is doing. Maybe he is just a physicist, and not
particularly educated in modern algebra or programming language types.

(I could also suggest circumstances in which True-True is not a great idea,
and what should be used is  TautologyQ[p==q]  rather than p-q.   But in this
case, especially because it means that I disagree with you, I will defend
the usage FullSimplify[p-q],  which could be interpreted as
Tautology[p==q].)

> Good grief!
>
> As I already pointed out:
>
>
> Refine[Less[x, y, z], Inequality[x, Less, y, Less, z]]&&
>   Refine[Inequality[x, Less, y, Less, z], Less[x, y, z]]
>
> True
>
> This simply means that truth of each expression implies the truth of the other - which is all that you could expect Mathematica to tell you.
I don't understand why you think that.  I can expect Mathematica, when
asked, to convert each of these expressions to
a canonical form.  I would prefer, at the moment, to have the
simplification be Less[x,y,z] --> Inequality[x,Less,y,Less,z]  or
perhaps Comparison[x,Less,y,Less,z]  or  even
And[Comparison[x,Less,y],Comparison[y,Less,z]].

Comparisons, especially inequality, of more than 2 items is tricky.    1
!= 2 != 1 is, by convention, false, and is not the
same as  not[1==2==1].

> And please do not point out that
>
> (2<  3) - (5<  7)
>
>   0
>
> for you surely know that what is going on here has is an entirely different thing.
Not at all.  It means exactly what I intend it to mean, which is True -
True, which is a simple way of comparing two objects
to see if they are isomorphic.

Mathematica also can subtract strings,  "abc" -"abc" is 0.

Mathematica also can subtract Graphics objects.  Plot[Sin[x],{x,-1,1}] -
Plot[Sin[x],{x,-1,1}]  is 0.

If you don't like this behavior, I suggest you report it as a bug.

RJF

```

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