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Topic: can your CAS help proving inequalities?
Replies: 19   Last Post: Mar 11, 2013 12:00 PM

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Nasser Abbasi

Posts: 5,674
Registered: 2/7/05
Re: can your CAS help proving inequalities?
Posted: Mar 9, 2013 1:04 PM
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On 3/9/2013 6:07 AM, A N Niel wrote:

>
> In Maple, the response
> {}
> means there is no solution, while the response
>
> (that is, no response) means no solution was found. A third
> possibility is where some solutions are shown, and then a disclaimer
> that some solutions may have been lost.
>
> What CAS did you use, and what does {} mean for it?
>


Mathematica. It has a special command called

"FindInstance[expr, vars]

finds an instance of vars that makes the statement expr be True.
gives results in the same form as Solve: if an instance exists,
and {} if it does not. "

http://reference.wolfram.com/mathematica/ref/FindInstance.html

So that is what I used:

-----------------------------
Remove["Global`*"]
r[a_, b_, c_, d_] := (a - c)*(a - d)*(b - c)*(b - d)
s[a_, b_, c_, d_] := (a + b)*(c + d) - 2*(a*b + c*d) -
Abs[(a - b)*(c - d)]
t[a_, b_, c_, d_] := (a + b)*(c + d) - 2*(a*b + c*d) +
Abs[(a - b)*(c - d)]
-----------------------------

and now

----------------------
FindInstance[
Min[r[a, b, c, d], r[a, c, b, d], r[a, d, c, b]] > 0, {a, b, c, d}]

{}

FindInstance[
Min[r[a, b, c, d], r[a, c, b, d], r[a, d, c, b]] > 0, {a, b, c, d}]

{}

FindInstance[
Min[t[a, b, c, d], t[a, c, b, d], t[a, d, c, b]] > 0, {a, b, c, d}]

{}
-------------------------

But I was not sure this qualifies as "proof" that is why I asked first.

--Nasser





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