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Re: Why does Reduce work this way ...?

• To: mathgroup at smc.vnet.net
• Subject: [mg67234] Re: [mg67225] Why does Reduce work this way ...?
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 14 Jun 2006 06:28:36 -0400 (EDT)
• References: <200606130507.BAA23801@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

On 13 Jun 2006, at 14:07, jackgoldberg at comcast.net wrote:

> Hi folks,
>
> This post is related to a previous series of posts about ReplaceAll  
> in a set of inequalities.
> I have "reduced" the problem to this unexpected situation:
>
> In[1]  FullForm[-3<=x<=1]
> Out[1]  LessEqual[-3,x,1]
>
> In[2] Reduce[-3<=x<=1]
> Out[2]  -3<=x<=1
>
> In[3]  FullForm[%2]
> Out[3]  Inequality[-3, LessEqual, y, LessEqual, 1]
>
> In[4] Reduce[ LessEqual[-3,x,1]
> Out[4] -3 &#8804; x &#8804; 1
>
> In[5] FullForm[%4]
> Out[5]  Inequality[-3, LessEqual, y, LessEqual, 1]
>
> This defies my understanding.  What is qoing on with Reduce?
>
> Jack
>

Mathematica has a (not very well documented) function Inequality  
which can be used to express complicated inequalities in a compact  
form; for example:

In[8]:=
Inequality[a, Less, b, Less, c]

Out[8]=
a < b < c

In[9]:=
Inequality[a, Greater, b, Less, c]

Out[9]=
a > b && b < c

In[10]:=
Inequality[a, Greater, b, Less, c, Equal, d]

Out[10]=
a > b && Inequality[b, Less, c, Equal, d]

In[11]:=
Inequality[a, Less, b, LessEqual, c]

Out[11]=
Inequality[a, Less, b, LessEqual, c]


As you can see, sometimes these functions reduce to a simpler form,  
like a < b < c, which has FullForm: Less[a,b,c] and sometimes not.  
The forms
LessEqual[a,b,c] and Inequality[a,LessEqual,b,LessEqual,c] are  
semantically equivalent (and have the same but syntactically distinct  
but normally they evaluate to themselves. The relation between them  
is a little unusual; note than using the Cell:Convert To menu (in  
other words the FrontEnd) and converting Inequality 
[a,LessEqual,b,LessEqual,c] to StandardForm or to TraditionalForm  
will result in Less[a,b,c] but doing this:

In[12]:=
Inequality[a,Less,b,Less,c]//StandardForm

Out[12]//StandardForm=
a<b<c
In[13]:=
FullForm[%]

Out[13]//FullForm=
Inequality[a,Less,b,Less,c]

does not.

Note than the Inequality form is much more general than forms with  
Heads Less, Greater, LessEqual etc, which I think is the reason why  
Reduce often returns expressions with head Inequality . If you find  
this a problem you can always use LogicalExpand@Reduce instead of  
Reduce, which will take longer and give you a more complex answer but  
which, I think, will not contain any expressions with Head Inequality.

Andrzej Kozlowski

• References:
  • Why does Reduce work this way ...?
    • From: jackgoldberg@comcast.net