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I am trying to show general cases of symmetry of regions.

However, for the following, Reduce does not finish:

Reduce[ForAll[x, x > 0, ! (RegionMember[ImplicitRegion[(0 < x < 1 || -2 < x < 0 || 2 < x < 3), {x}], {x}] \[Xor] RegionMember[ImplicitRegion[(0 < x < 1 || -2 < x < 0 || 2 < x < 3), {x}], {-x}])], x, Reals]

OTOH, if I remove any one of the three conditions (from both parts), e.g.:

Reduce[ForAll[x, x > 0, ! (RegionMember[ImplicitRegion[(-2 < x < 0 || 2 < x < 3), {x}], {x}] \[Xor] RegionMember[ImplicitRegion[(-2 < x < 0 || 2 < x < 3), {x}], {-x}])], x, Reals]

it return False immediately.

What's happening?

(I know there are other ways to do this, e.g., with FindInstance.)

Thanx.

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  • $\begingroup$ It results in False in several minutes. I'm using 12.1.1.0 on Windows 10. $\endgroup$ Commented Dec 11, 2020 at 12:27
  • $\begingroup$ Timing it with AbsoluteTiming. I obtain {438.211, False} on a fresh kernel of my comp. $\endgroup$ Commented Dec 11, 2020 at 12:37
  • $\begingroup$ It also cost me 10 min in Linux to get the result. $\endgroup$ Commented Dec 11, 2020 at 12:39

2 Answers 2

Reset to default
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Maybe other way.

reg1 = ImplicitRegion[(0 < x < 1 || -2 < x < 0 || 2 < x < 3), {x}] reg2 = ImplicitRegion[(0 < x < 1 || -2 < x < 0 || 2 < x < 3) /. x -> -x, {x}] RegionEqual[reg1, reg2]

False

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  • $\begingroup$ Thanx! This is helpful. I just noticed that if I pull the exclamation point to outside of the Reduce, it also works. But I still would like to know what causes the slowdown with going from two inequalities to three. $\endgroup$ Commented Dec 11, 2020 at 11:54
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There is a workaround:

Reduce[ForAll[x, x > 0, ! (RegionMember[ ImplicitRegion[(0 < x < 1 || -2 < x < 0 || 2 < x < 3), {x}], {x}] \[Xor] RegionMember[ ImplicitRegion[(0 < x < 1 || -2 < x < 0 || 2 < x < 3), {x}], {-x}]) // Simplify], x, Reals] (*False*)
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  • $\begingroup$ Nice -- thanx! (Still wondering why without the Simplify it is stymied.) $\endgroup$ Commented Dec 11, 2020 at 12:08
  • $\begingroup$ I found other regions, e.g., -2 < x < 0 -> -2 < x < -1, which unfortunately were not solved with this Simplify so I will vote in the other answer. $\endgroup$ Commented Dec 18, 2020 at 13:05

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