How to define the complement of a “region” in $mathbb{R}^d$ using boxes.
$begingroup$
I have a question related to set theory, of which I am a beginner. Please add/change tags if you have better references. Other than describing my main question, I'm also highlighting with the symbol [?] the parts I'm really not sure about.
Set up of the problem
Fix $rin mathbb{N}$ and $dequiv r+binom{r}{2}$ and consider the region (plane [?])
$$
begin{aligned}
mathcal{B}equiv {(b_1,b_2,..., b_d)in mathbb{R}^{d}: text{ } & b_{r+1}=b_1-b_2, b_{r+2}=b_1-b_3, ...,b_{2r-1}=b_1-b_r, \
&b_{2r}=b_2-b_3, ..., b_{3r-3}=b_2-b_r,\
&...,\
& b_d=b_{r-1}-b_r}
end{aligned}
$$
For example, when $r=2$ ($d=3$) we have the surface
$$
begin{aligned}
mathcal{B}equiv {(b_1,b_2,b_3)in mathbb{R}^{3}: text{ } & b_3=b_1-b_2}
end{aligned}
$$
When $r=3$ ($d=6$) we have
$$
begin{aligned}
mathcal{B}equiv {(b_1,..., b_6)in mathbb{R}^{6}: text{ } & b_4=b_1-b_2, b_5=b_1-b_3, b_6=b_2-b_3}
end{aligned}
$$
My goal: I want to write down the region [?] that is complement to $mathcal{B}$ in $mathbb{R}^d$ as a union of "boxes".
This is how I thought to proceed for $r=2$
Define these two boxes given $(b_1, b_2)in mathbb{R}^2$
$$
B(b_1, b_2)equiv {(x,y,z) text{ s.t. } xleq b_1, -yleq -b_2, z> b_1-b_2}
$$
$$
Q(b_1, b_2)equiv {(x,y,z) text{ s.t. } x> b_1, -y> -b_2, zleq b_1-b_2}
$$
Then,
$$
cup_{(b_1, b_2)in mathbb{R}^2} B(b_1, b_2)
$$
should be the region (plane [?]) above $mathcal{B}$ and
$$
cup_{(b_1, b_2)in mathbb{R}^2} Q(b_1, b_2)
$$
should be the region (plane [?]) below $mathcal{B}$.
Hence,
$$
Big{cup_{(b_1, b_2)in mathbb{R}^2} B(b_1, b_2)Big} cup Big{cup_{(b_1, b_2)in mathbb{R}^2} Q(b_1, b_2)Big }
$$
is the region [?] that is complement to $mathcal{B}$ in $mathbb{R}^3$.
Is this correct? How do I generalise this to any $r$?
linear-algebra geometry elementary-set-theory analytic-geometry
asked Jan 17 at 11:09
STFSTF
571422
$endgroup$
add a comment |
$begingroup$
I have a question related to set theory, of which I am a beginner. Please add/change tags if you have better references. Other than describing my main question, I'm also highlighting with the symbol [?] the parts I'm really not sure about.
Set up of the problem
Fix $rin mathbb{N}$ and $dequiv r+binom{r}{2}$ and consider the region (plane [?])
$$
begin{aligned}
mathcal{B}equiv {(b_1,b_2,..., b_d)in mathbb{R}^{d}: text{ } & b_{r+1}=b_1-b_2, b_{r+2}=b_1-b_3, ...,b_{2r-1}=b_1-b_r, \
&b_{2r}=b_2-b_3, ..., b_{3r-3}=b_2-b_r,\
&...,\
& b_d=b_{r-1}-b_r}
end{aligned}
$$
For example, when $r=2$ ($d=3$) we have the surface
$$
begin{aligned}
mathcal{B}equiv {(b_1,b_2,b_3)in mathbb{R}^{3}: text{ } & b_3=b_1-b_2}
end{aligned}
$$
When $r=3$ ($d=6$) we have
$$
begin{aligned}
mathcal{B}equiv {(b_1,..., b_6)in mathbb{R}^{6}: text{ } & b_4=b_1-b_2, b_5=b_1-b_3, b_6=b_2-b_3}
end{aligned}
$$
My goal: I want to write down the region [?] that is complement to $mathcal{B}$ in $mathbb{R}^d$ as a union of "boxes".
This is how I thought to proceed for $r=2$
Define these two boxes given $(b_1, b_2)in mathbb{R}^2$
$$
B(b_1, b_2)equiv {(x,y,z) text{ s.t. } xleq b_1, -yleq -b_2, z> b_1-b_2}
$$
$$
Q(b_1, b_2)equiv {(x,y,z) text{ s.t. } x> b_1, -y> -b_2, zleq b_1-b_2}
$$
Then,
$$
cup_{(b_1, b_2)in mathbb{R}^2} B(b_1, b_2)
$$
should be the region (plane [?]) above $mathcal{B}$ and
$$
cup_{(b_1, b_2)in mathbb{R}^2} Q(b_1, b_2)
$$
should be the region (plane [?]) below $mathcal{B}$.
Hence,
$$
Big{cup_{(b_1, b_2)in mathbb{R}^2} B(b_1, b_2)Big} cup Big{cup_{(b_1, b_2)in mathbb{R}^2} Q(b_1, b_2)Big }
$$
is the region [?] that is complement to $mathcal{B}$ in $mathbb{R}^3$.
Is this correct? How do I generalise this to any $r$?
linear-algebra geometry elementary-set-theory analytic-geometry
asked Jan 17 at 11:09
STFSTF
571422
$endgroup$
1
$begingroup$
what is the definition of "open plane above $mathcal B$"? I am not sure if these are common mathematical objects.
$endgroup$
– supinf
Jan 17 at 11:56
$begingroup$
@supinf Thanks: that's a part I'm confused about. I'm not even sure that $mathcal{B}$ is a plane given that it is in $mathbb{R}^d$. Wikipedia defines a plane as a surface for example. en.wikipedia.org/wiki/Plane_(geometry) I really need the help of an expert to clarify every single bit of my question.
$endgroup$
– STF
Jan 17 at 12:28
$begingroup$
@supinf I've modified my question to make it (I hope) more meaningful to experts. Please help if you can, thanks.
$endgroup$
– STF
Jan 17 at 12:54
1
$begingroup$
Isn't it a purely geometrical problem? I ask since you assigned the tag "general topology".
$endgroup$
– Paul Frost
Jan 17 at 13:55
$begingroup$
Thanks: I'm happy to add that tag. I mentioned in the intro that any advise on useful tags is more than appreciated.
$endgroup$
– STF
Jan 17 at 13:57
add a comment |
$begingroup$
I have a question related to set theory, of which I am a beginner. Please add/change tags if you have better references. Other than describing my main question, I'm also highlighting with the symbol [?] the parts I'm really not sure about.
Set up of the problem
Fix $rin mathbb{N}$ and $dequiv r+binom{r}{2}$ and consider the region (plane [?])
$$
begin{aligned}
mathcal{B}equiv {(b_1,b_2,..., b_d)in mathbb{R}^{d}: text{ } & b_{r+1}=b_1-b_2, b_{r+2}=b_1-b_3, ...,b_{2r-1}=b_1-b_r, \
&b_{2r}=b_2-b_3, ..., b_{3r-3}=b_2-b_r,\
&...,\
& b_d=b_{r-1}-b_r}
end{aligned}
$$
For example, when $r=2$ ($d=3$) we have the surface
$$
begin{aligned}
mathcal{B}equiv {(b_1,b_2,b_3)in mathbb{R}^{3}: text{ } & b_3=b_1-b_2}
end{aligned}
$$
When $r=3$ ($d=6$) we have
$$
begin{aligned}
mathcal{B}equiv {(b_1,..., b_6)in mathbb{R}^{6}: text{ } & b_4=b_1-b_2, b_5=b_1-b_3, b_6=b_2-b_3}
end{aligned}
$$
My goal: I want to write down the region [?] that is complement to $mathcal{B}$ in $mathbb{R}^d$ as a union of "boxes".
This is how I thought to proceed for $r=2$
Define these two boxes given $(b_1, b_2)in mathbb{R}^2$
$$
B(b_1, b_2)equiv {(x,y,z) text{ s.t. } xleq b_1, -yleq -b_2, z> b_1-b_2}
$$
$$
Q(b_1, b_2)equiv {(x,y,z) text{ s.t. } x> b_1, -y> -b_2, zleq b_1-b_2}
$$
Then,
$$
cup_{(b_1, b_2)in mathbb{R}^2} B(b_1, b_2)
$$
should be the region (plane [?]) above $mathcal{B}$ and
$$
cup_{(b_1, b_2)in mathbb{R}^2} Q(b_1, b_2)
$$
should be the region (plane [?]) below $mathcal{B}$.
Hence,
$$
Big{cup_{(b_1, b_2)in mathbb{R}^2} B(b_1, b_2)Big} cup Big{cup_{(b_1, b_2)in mathbb{R}^2} Q(b_1, b_2)Big }
$$
is the region [?] that is complement to $mathcal{B}$ in $mathbb{R}^3$.
Is this correct? How do I generalise this to any $r$?
linear-algebra geometry elementary-set-theory analytic-geometry
asked Jan 17 at 11:09
STFSTF
571422
$endgroup$
I have a question related to set theory, of which I am a beginner. Please add/change tags if you have better references. Other than describing my main question, I'm also highlighting with the symbol [?] the parts I'm really not sure about.
Set up of the problem
Fix $rin mathbb{N}$ and $dequiv r+binom{r}{2}$ and consider the region (plane [?])
$$
begin{aligned}
mathcal{B}equiv {(b_1,b_2,..., b_d)in mathbb{R}^{d}: text{ } & b_{r+1}=b_1-b_2, b_{r+2}=b_1-b_3, ...,b_{2r-1}=b_1-b_r, \
&b_{2r}=b_2-b_3, ..., b_{3r-3}=b_2-b_r,\
&...,\
& b_d=b_{r-1}-b_r}
end{aligned}
$$
For example, when $r=2$ ($d=3$) we have the surface
$$
begin{aligned}
mathcal{B}equiv {(b_1,b_2,b_3)in mathbb{R}^{3}: text{ } & b_3=b_1-b_2}
end{aligned}
$$
When $r=3$ ($d=6$) we have
$$
begin{aligned}
mathcal{B}equiv {(b_1,..., b_6)in mathbb{R}^{6}: text{ } & b_4=b_1-b_2, b_5=b_1-b_3, b_6=b_2-b_3}
end{aligned}
$$
My goal: I want to write down the region [?] that is complement to $mathcal{B}$ in $mathbb{R}^d$ as a union of "boxes".
This is how I thought to proceed for $r=2$
Define these two boxes given $(b_1, b_2)in mathbb{R}^2$
$$
B(b_1, b_2)equiv {(x,y,z) text{ s.t. } xleq b_1, -yleq -b_2, z> b_1-b_2}
$$
$$
Q(b_1, b_2)equiv {(x,y,z) text{ s.t. } x> b_1, -y> -b_2, zleq b_1-b_2}
$$
Then,
$$
cup_{(b_1, b_2)in mathbb{R}^2} B(b_1, b_2)
$$
should be the region (plane [?]) above $mathcal{B}$ and
$$
cup_{(b_1, b_2)in mathbb{R}^2} Q(b_1, b_2)
$$
should be the region (plane [?]) below $mathcal{B}$.
Hence,
$$
Big{cup_{(b_1, b_2)in mathbb{R}^2} B(b_1, b_2)Big} cup Big{cup_{(b_1, b_2)in mathbb{R}^2} Q(b_1, b_2)Big }
$$
is the region [?] that is complement to $mathcal{B}$ in $mathbb{R}^3$.
Is this correct? How do I generalise this to any $r$?
linear-algebra geometry elementary-set-theory analytic-geometry
linear-algebra geometry elementary-set-theory analytic-geometry
asked Jan 17 at 11:09
STFSTF
571422
asked Jan 17 at 11:09
STFSTF
571422
edited Jan 17 at 16:19
STF
asked Jan 17 at 11:09
STFSTF
571422
asked Jan 17 at 11:09
STFSTF
571422
asked Jan 17 at 11:09
STFSTF
571422
571422
1
$begingroup$
what is the definition of "open plane above $mathcal B$"? I am not sure if these are common mathematical objects.
$endgroup$
– supinf
Jan 17 at 11:56
$begingroup$
@supinf Thanks: that's a part I'm confused about. I'm not even sure that $mathcal{B}$ is a plane given that it is in $mathbb{R}^d$. Wikipedia defines a plane as a surface for example. en.wikipedia.org/wiki/Plane_(geometry) I really need the help of an expert to clarify every single bit of my question.
$endgroup$
– STF
Jan 17 at 12:28
$begingroup$
@supinf I've modified my question to make it (I hope) more meaningful to experts. Please help if you can, thanks.
$endgroup$
– STF
Jan 17 at 12:54
1
$begingroup$
Isn't it a purely geometrical problem? I ask since you assigned the tag "general topology".
$endgroup$
– Paul Frost
Jan 17 at 13:55
$begingroup$
Thanks: I'm happy to add that tag. I mentioned in the intro that any advise on useful tags is more than appreciated.
$endgroup$
– STF
Jan 17 at 13:57
add a comment |
1
$begingroup$
what is the definition of "open plane above $mathcal B$"? I am not sure if these are common mathematical objects.
$endgroup$
– supinf
Jan 17 at 11:56
$begingroup$
@supinf Thanks: that's a part I'm confused about. I'm not even sure that $mathcal{B}$ is a plane given that it is in $mathbb{R}^d$. Wikipedia defines a plane as a surface for example. en.wikipedia.org/wiki/Plane_(geometry) I really need the help of an expert to clarify every single bit of my question.
$endgroup$
– STF
Jan 17 at 12:28
$begingroup$
@supinf I've modified my question to make it (I hope) more meaningful to experts. Please help if you can, thanks.
$endgroup$
– STF
Jan 17 at 12:54
1
$begingroup$
Isn't it a purely geometrical problem? I ask since you assigned the tag "general topology".
$endgroup$
– Paul Frost
Jan 17 at 13:55
$begingroup$
Thanks: I'm happy to add that tag. I mentioned in the intro that any advise on useful tags is more than appreciated.
$endgroup$
– STF
Jan 17 at 13:57
1
1
$begingroup$
what is the definition of "open plane above $mathcal B$"? I am not sure if these are common mathematical objects.
$endgroup$
– supinf
Jan 17 at 11:56
$begingroup$
what is the definition of "open plane above $mathcal B$"? I am not sure if these are common mathematical objects.
$endgroup$
– supinf
Jan 17 at 11:56
$begingroup$
@supinf Thanks: that's a part I'm confused about. I'm not even sure that $mathcal{B}$ is a plane given that it is in $mathbb{R}^d$. Wikipedia defines a plane as a surface for example. en.wikipedia.org/wiki/Plane_(geometry) I really need the help of an expert to clarify every single bit of my question.
$endgroup$
– STF
Jan 17 at 12:28
$begingroup$
@supinf Thanks: that's a part I'm confused about. I'm not even sure that $mathcal{B}$ is a plane given that it is in $mathbb{R}^d$. Wikipedia defines a plane as a surface for example. en.wikipedia.org/wiki/Plane_(geometry) I really need the help of an expert to clarify every single bit of my question.
$endgroup$
– STF
Jan 17 at 12:28
$begingroup$
@supinf I've modified my question to make it (I hope) more meaningful to experts. Please help if you can, thanks.
$endgroup$
– STF
Jan 17 at 12:54
$begingroup$
@supinf I've modified my question to make it (I hope) more meaningful to experts. Please help if you can, thanks.
$endgroup$
– STF
Jan 17 at 12:54
1
1
$begingroup$
Isn't it a purely geometrical problem? I ask since you assigned the tag "general topology".
$endgroup$
– Paul Frost
Jan 17 at 13:55
$begingroup$
Isn't it a purely geometrical problem? I ask since you assigned the tag "general topology".
$endgroup$
– Paul Frost
Jan 17 at 13:55
$begingroup$
Thanks: I'm happy to add that tag. I mentioned in the intro that any advise on useful tags is more than appreciated.
$endgroup$
– STF
Jan 17 at 13:57
$begingroup$
Thanks: I'm happy to add that tag. I mentioned in the intro that any advise on useful tags is more than appreciated.
$endgroup$
– STF
Jan 17 at 13:57
add a comment |
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1
$begingroup$
what is the definition of "open plane above $mathcal B$"? I am not sure if these are common mathematical objects.
$endgroup$
– supinf
Jan 17 at 11:56
$begingroup$
@supinf Thanks: that's a part I'm confused about. I'm not even sure that $mathcal{B}$ is a plane given that it is in $mathbb{R}^d$. Wikipedia defines a plane as a surface for example. en.wikipedia.org/wiki/Plane_(geometry) I really need the help of an expert to clarify every single bit of my question.
$endgroup$
– STF
Jan 17 at 12:28
$begingroup$
@supinf I've modified my question to make it (I hope) more meaningful to experts. Please help if you can, thanks.
$endgroup$
– STF
Jan 17 at 12:54
1
$begingroup$
Isn't it a purely geometrical problem? I ask since you assigned the tag "general topology".
$endgroup$
– Paul Frost
Jan 17 at 13:55
$begingroup$
Thanks: I'm happy to add that tag. I mentioned in the intro that any advise on useful tags is more than appreciated.
$endgroup$
– STF
Jan 17 at 13:57