# Definition of an implicit region with Resolve [closed]

I am trying to define a region so I can calculate its volume with Volume[region].

Here is an example of the problem I am having (my real function f is much more complicated).

function[x_, y_, z_] :=
Resolve[Exists[i, i ∈ Integers && i > 0 && i < 3 && -i < 0 && x < i]]
ImplicitRegion[function[x, y, z] && x^2 + y^2 + z^2 <= 1, {x, y, z}]


It seems my region is not correctly specified. Where did I make an error in syntax of ImplicitRegion?

## closed as off-topic by Szabolcs, corey979, m_goldberg, Feyre, MarcoBDec 16 '16 at 23:57

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question arises due to a simple mistake such as a trivial syntax error, incorrect capitalization, spelling mistake, or other typographical error and is unlikely to help any future visitors, or else it is easily found in the documentation." – Szabolcs, corey979, m_goldberg, Feyre, MarcoB
If this question can be reworded to fit the rules in the help center, please edit the question.

• Why TrueQ? Of course it gives False; consider TrueQ[a < 0] - no assumptions on a. Moreover, why Resolve? Well, I guess it's an artifact of your bigger problem. Nevertheless: region = ImplicitRegion[ function[x, y, z] && x^2 + y^2 + z^2 <= 1, {x, y, z}] and Volume[region] yields 1.52494 – corey979 Dec 16 '16 at 13:58
• I have changed the code so that it better corresponds to my real problem. – Agnieszka Dec 16 '16 at 14:50
• TrueQ is for programming, not for representing mathematical ideas. TrueQ[anything] is False. Only TrueQ[True] is True. Why did you use TrueQ here? – Szabolcs Dec 16 '16 at 14:51
• Sorry, I have made some changes again. – Agnieszka Dec 16 '16 at 14:52
• But TrueQ of Resolve[...] should not simply give Resolve[...]? – Agnieszka Dec 16 '16 at 14:54

## 1 Answer

The expression

Exists[i, i ∈ Integers && i > 0 && i < 3 && -i < 0 && x < i]


is equivalent to

Exists[i, i ∈ Integers && 0 < i < 3 && x < i]


which is clearly equivalent to

x < 2


That's the half space of all points {x, y, z} with x < 2. This half space contains the unit sphere, the other component of your region, so the volume of their intersection is the volume of the sphere. That is, 4 π/3.

• The above functin f was only some example, which should only show that by such a syntax I get an error from Mathematica (ImplicitRegion ... is not a correctly specified region.), which I do not understand. How could I avoid the error? – Agnieszka Dec 19 '16 at 6:35
• @Agnieszka. Your example is trivial and is not a good model for you real problem. We can't help you when you give us a trivial example. Without your real code or something close to it, there is no way for anyone to know what corrections to might work. – m_goldberg Dec 19 '16 at 16:14