       Re: Truth in inequalities

• To: mathgroup at smc.vnet.net
• Subject: [mg29390] Re: [mg29347] Truth in inequalities
• From: Mianlai Zhou <lailai at nikhef.nl>
• Date: Sat, 16 Jun 2001 02:48:01 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Hi Jack

I can explain to you a little bit. In the internal notation of
Mathematica, Infinity is converted to DirectedInfinity automatically,
while -Infinity is converted to DirectedInfinity[-1] (DirectedInfinity[z]
means an infinite quantity which is a positive real multiple of the
complex number z).

Therefore, it will do the following deduction:

x < Infinity (its FullForm is Less[x, Infinity]) gives x <
DirectedInfinity, which cannot be worked out further, therefore it
remains this inequality.

x - Infinity < 0 (its FullForm is Less[Plus[x, Times[-1, Infinity]], 0])
gives Less[Plus[x, DirectedInfinity[-1]], 0], and furthermore gives
Less[DirectedInfinity[-1], 0], and is worked out as True.

The crucial point is, in the latter case here it used the rule that
anything plus DirectedInfinity[-1] is DirectedInfinity[-1] (and the same
holds with DirectedInfinity too). But there is no simplification for
the expression Less[anything, DirectedInfinity].

So exactly this is the difference that should be noticed when you use the
condition /;(x<y) or /;(x-y<0), since in the latter form x-y will be
worked out first before it is compared to zero.

I hope these make the things clearer.

Mianlai Zhou
Theory Group, NIKHEF
Amsterdam

On Thu, 14 Jun 2001, Jack Goldberg wrote:

> Hi group,
>
> Can someone explain the logic of the following:
>
> 	x < Infinity 	  	returns 	x < Infinity
> while
> 	x - Infinity < 0  	returns  	True
>
> I should mention that I am aware of the fact that x - Infinity simplifies
> automatically to -Infinity which is then compared to 0 and found wanting.
> The issue I'm raising is why should a CAS that has  x-Infinity < 0 return
> True not also return True for the  x < Infinity?  One awkwardness of
> having this difference of behavior can be seen in the example,
>
> MyFunction[x_,y_]/;(x<y) :=  blah
>
> and
>
> MyFunction[x_,y_]/;(x-y<0)  := blah
>
> do not do the same thing when, say, y=Infinity.
>
> Just curious :-)
>
> Jack
>
>
>

```

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