FindInstance question

• To: mathgroup at smc.vnet.net
• Subject: [mg55736] FindInstance question
• From: János <janos.lobb at yale.edu>
• Date: Tue, 5 Apr 2005 03:20:55 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

If I am looking for the solution set of x < x^2, Reduce gives me the

In[14]:=
Reduce[x < x^2, x]
Out[14]=
x < 0 || x > 1

I tried to use FindInstance and ListPlot to visualize it for my 7th
grade son, based upon the example on the Book Section 3.4.8.

In[15]:=
FindInstance[x < x^2, x,
Reals, 50]
Out[15]=
{{x -> -4375}, {x -> -4304},
{x -> -4089}, {x -> -3945},
{x -> -3707}, {x -> -3682},
{x -> -3486}, {x -> -3414},
{x -> -3331}, {x -> -3286},
{x -> -3248}, {x -> -3018},
{x -> -2974}, {x -> -2953},
{x -> -2941}, {x -> -2877},
{x -> -2864}, {x -> -2687},
{x -> -2525}, {x -> -2373},
{x -> -2108}, {x -> -2074},
{x -> -1773}, {x -> -1403},
{x -> -1402}, {x -> -1348},
{x -> -1092}, {x -> -566},
{x -> -51}, {x -> 77},
{x -> 195}, {x -> 1181},
{x -> 1216}, {x -> 1547},
{x -> 1687}, {x -> 1765},
{x -> 1876}, {x -> 2040},
{x -> 2713}, {x -> 2734},
{x -> 3018}, {x -> 3312},
{x -> 3455}, {x -> 3503},
{x -> 3704}, {x -> 3927},
{x -> 3974}, {x -> 3985},
{x -> 4349}, {x -> 4944}}

Well, the closest values to 0 and 1 are -51 and 77.  It is not
terrible useful to ListPlot them.  If I select 500 points instead of
50 it just gets worse.

My question is:

1,  Is it possible to suggest FindInstance to use values "much
closer" to the boundaries of 0 and 1 or

2 If it is not possible, what methods others would use for this
situation to visualize ?

I tried

In[18]:=
FindInstance[x < x^2 &&
x > -2 && x < 2, x, Reals,
50]

but that is not the original inequality :)
I also tried to replace the Reals domain with something - logical
looking - else, but Mathematica really wants a built in domain there.

In[32]:=
FindInstance[x < x^2, x,
x > -2 && x < 2, 50]

or

In[37]:=
FindInstance[x < x^2, x,
{-2, 2}, 50]

and it complained:

FindInstance::"bddom":"Value \!\(\(\(x > \(\(-2\)\)\)\) && \(\(x < 2\)
\)\) of \
the domain argument should be Complexes, Reals, Algebraics, Rationals, \
Integers, Primes, Booleans, or Automatic. \
\", \
ButtonFrame->None, ButtonData:>\"FindInstance::bddom\"]\)"

A  non existent "user defined Domain" would be handy here.  I am
thinking of a

myDomain[x_Real]:=UserDefinedDomain[x > -2 && x < 3]

or similar construction to put into FindInstance.  Is that too much