Why the inequality does not take into account the domain?

Question

I have this inequality:

Reduce[(4000-1000k)/(k-4) < 0]

and the answer is

k ∈ Reals

I would expect k != 4.

Perhaps because the lim k-> 4 of this is -1000? – Sjoerd C. de Vries Oct 07 '13 at 17:41 — Sjoerd C. de Vries, Oct 07 '13 at 17:41
This does not mean it is mathematically correct. – enzotib Oct 07 '13 at 18:51 — enzotib, Oct 07 '13 at 18:51

score 5 · Accepted Answer · edited Apr 13 '17 at 12:55

5

The following is a kind of remedy for the problem at hand:

Reduce[# < 0 && Denominator[#] != 0]&[ (4000 - 1000 k)/(k - 4)]

k < 4 || k > 4

Even though the issues in the OP could be easily resolved nonetheless they are not mathematically correct and for this reason one could consider them as bugs, this is a similar problem
( 4 ∈ Complexes as well as 4 ∈ Reals but 0/0 is Indeterminate thus
TrueQ[Indeterminate ∈ Reals] yields False):

Reduce[(4000 - 1000 k)/(k - 4) ∈ Reals, k]

True

while this one is correct:

Reduce[(4000 - 1000 k)/(k - 4) ∈ Reals, k, Reals]

k < 4 || k > 4

Similar issue one can find here: Issue with NSolve.

Therefore we can conclude

Reduce[(4000 - 1000 k)/(k - 4) < 0, k]
Reduce[(4000 - 1000 k)/(k - 4) < 0, k, Reals]
Reduce[(4000 - 1000 k)/(k - 4) < 0, k, Complexes]

k ∈ Reals
True
True

yield results which appear to be incorrect in special cases, they are only generically correct. All of these above work as with a simplification technique:

Simplify[(4000 - 1000 k)/(k - 4)]

-1000

but from strictly mathematical point of view they shouldn't. Nevermind what the documentation says (genericity et consortes) this is of course a bug:

Reduce[(4000 - 1000 k)/(k - 4) < 0 && 3 < k < 5, k, Integers]

k == 4

edited Apr 13 '17 at 12:55

Community

1

answered Oct 07 '13 at 18:23

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I would not tag this as a bug as it seems to be by design. Also, I'm not sure that from a practical point of view ConditionalExpression[-1000, k != 4] would be a better (more useful/usable) return value for that Simplify. The function FunctionDomain is coming, which is a sort of practical solution to this problem. – Szabolcs Mar 22 '14 at 16:06
@Szabolcs See the last line of the code, that is a bug. One cannot argue that's not since the domain Integers and the condition 3 < k < 5 selects only k == 4 but for such a number (4000 - 1000 k)/(k - 4) < 0 cannot be satisfied. – Artes Mar 22 '14 at 21:02
@Szabolcs By the way, do you know when the new version of Mathematica appears? Seems like it should be in a few days. – Artes Mar 22 '14 at 21:28
I don't know when it will appear. The new documentation has been online for quite a while. Are there any other signs that it's coming soon? I wasn't expecting it just yet. – Szabolcs Mar 22 '14 at 23:06
@Szabolcs I mean this question Information on Mma v10. – Artes Mar 22 '14 at 23:36

score 2 · Answer 2 · edited Oct 08 '13 at 16:50

2

Please see this tutorial on the results given by symbolic functions. In short, the results returned by them are generically correct:

http://reference.wolfram.com/mathematica/tutorial/GenericAndNonGenericCases.html

edited Oct 08 '13 at 16:50

Artes

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answered Oct 07 '13 at 17:41

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score 0 · Answer 3 · answered Jun 28 '14 at 00:47

0

In version 10, one could use this workaround:

Reduce[FunctionDomain[(4000 - 1000 k)/(k - 4), k] && (4000 - 1000 k)/(k - 4) < 0]

FunctionDomain alone returns k < 4 || k > 4 and can be used to restrict the domain of k within Reduce ...

answered Jun 28 '14 at 00:47

Szabolcs

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Why the inequality does not take into account the domain?

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