Mixing
Bijection between nested-tuples and map-tuples
Clash Royale CLAN TAG#URR8PPP
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Tuples are usually defined as indexed sets:
$$
langle x_1ldots x_nrangle_mathsfM := (x_i)_iin1ldots n
$$
However, this definition does not work with n-ary Cartesian products.
Perhaps, an element of $left(prod_i=1^n X_iright)$, assuming left associativity, is better thought as:
beginalign*
langle x_1ldots x_nrangle_mathsfN :=
begincases
x_1 &text if n=1\
langlelangle x_1ldots x_n-1rangle_mathsfN, x_nrangle_mathsfK &text else
endcases
endalign*
where $langlerangle_mathsfK$ denotes the Kuratowski pair and
$langle x_nldots x_nrangle_mathsfN$ and $langle x_nldots x_n+1rangle_mathsfN$ are usually abbreviated with $langle x_nrangle_mathsfN$ and $langle x_n, x_n+1rangle_mathsfN$.
I am not sure whether I read there is canonical bijection between the two definitions. One could write it as:
$$
langle x_1ldots x_nrangle_mathsfM
mapsto
langle x_1ldots x_nrangle_mathsfN
$$
but upon thinking, given the triple $langle a,b,crangle_*$,
$$
langle a,b,crangle_mathsfM =
langle 1, arangle_mathsfK,
langle 2, arangle_mathsfK,
langle 3, arangle_mathsfK
$$
while:
beginalign*
langle a,b,crangle_mathsfN &=
biglanglelangle a,brangle,cbigrangle_mathsfK\
&=leftlanglebiga,a,bbig,crightrangle_mathsfK\
&=leftleftbiga,a,bbigright,;
leftbiga,a,bbig,crightright
endalign*
So the map-tuple has three elements $langle i, *rangle_mathsfK$ (with $langlerangle_mathsfK$'s further expansible), while the nested version has two elements:
$leftbiga,a,bbigright$
and
$leftbiga,a,bbig,cright$ and the different cardinality prevents a bijection.
Summing up, usually textbooks define tuples like $langle rangle_mathsfM$ above. That does not match with tuples generated by ordinary operations such as $Atimes Btimes C$ or the common $mathbbR^n$.
How do we reconcile the two notions?
elementary-set-theory
asked Jul 22 at 16:32
antonio
207210
add a comment |Â
up vote
0
down vote
favorite
Tuples are usually defined as indexed sets:
$$
langle x_1ldots x_nrangle_mathsfM := (x_i)_iin1ldots n
$$
However, this definition does not work with n-ary Cartesian products.
Perhaps, an element of $left(prod_i=1^n X_iright)$, assuming left associativity, is better thought as:
beginalign*
langle x_1ldots x_nrangle_mathsfN :=
begincases
x_1 &text if n=1\
langlelangle x_1ldots x_n-1rangle_mathsfN, x_nrangle_mathsfK &text else
endcases
endalign*
where $langlerangle_mathsfK$ denotes the Kuratowski pair and
$langle x_nldots x_nrangle_mathsfN$ and $langle x_nldots x_n+1rangle_mathsfN$ are usually abbreviated with $langle x_nrangle_mathsfN$ and $langle x_n, x_n+1rangle_mathsfN$.
I am not sure whether I read there is canonical bijection between the two definitions. One could write it as:
$$
langle x_1ldots x_nrangle_mathsfM
mapsto
langle x_1ldots x_nrangle_mathsfN
$$
but upon thinking, given the triple $langle a,b,crangle_*$,
$$
langle a,b,crangle_mathsfM =
langle 1, arangle_mathsfK,
langle 2, arangle_mathsfK,
langle 3, arangle_mathsfK
$$
while:
beginalign*
langle a,b,crangle_mathsfN &=
biglanglelangle a,brangle,cbigrangle_mathsfK\
&=leftlanglebiga,a,bbig,crightrangle_mathsfK\
&=leftleftbiga,a,bbigright,;
leftbiga,a,bbig,crightright
endalign*
So the map-tuple has three elements $langle i, *rangle_mathsfK$ (with $langlerangle_mathsfK$'s further expansible), while the nested version has two elements:
$leftbiga,a,bbigright$
and
$leftbiga,a,bbig,cright$ and the different cardinality prevents a bijection.
Summing up, usually textbooks define tuples like $langle rangle_mathsfM$ above. That does not match with tuples generated by ordinary operations such as $Atimes Btimes C$ or the common $mathbbR^n$.
How do we reconcile the two notions?
elementary-set-theory
asked Jul 22 at 16:32
antonio
207210
Show by induction there is a bijection between finite map tuples and set tuples.
– William Elliot
Jul 22 at 22:46
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Tuples are usually defined as indexed sets:
$$
langle x_1ldots x_nrangle_mathsfM := (x_i)_iin1ldots n
$$
However, this definition does not work with n-ary Cartesian products.
Perhaps, an element of $left(prod_i=1^n X_iright)$, assuming left associativity, is better thought as:
beginalign*
langle x_1ldots x_nrangle_mathsfN :=
begincases
x_1 &text if n=1\
langlelangle x_1ldots x_n-1rangle_mathsfN, x_nrangle_mathsfK &text else
endcases
endalign*
where $langlerangle_mathsfK$ denotes the Kuratowski pair and
$langle x_nldots x_nrangle_mathsfN$ and $langle x_nldots x_n+1rangle_mathsfN$ are usually abbreviated with $langle x_nrangle_mathsfN$ and $langle x_n, x_n+1rangle_mathsfN$.
I am not sure whether I read there is canonical bijection between the two definitions. One could write it as:
$$
langle x_1ldots x_nrangle_mathsfM
mapsto
langle x_1ldots x_nrangle_mathsfN
$$
but upon thinking, given the triple $langle a,b,crangle_*$,
$$
langle a,b,crangle_mathsfM =
langle 1, arangle_mathsfK,
langle 2, arangle_mathsfK,
langle 3, arangle_mathsfK
$$
while:
beginalign*
langle a,b,crangle_mathsfN &=
biglanglelangle a,brangle,cbigrangle_mathsfK\
&=leftlanglebiga,a,bbig,crightrangle_mathsfK\
&=leftleftbiga,a,bbigright,;
leftbiga,a,bbig,crightright
endalign*
So the map-tuple has three elements $langle i, *rangle_mathsfK$ (with $langlerangle_mathsfK$'s further expansible), while the nested version has two elements:
$leftbiga,a,bbigright$
and
$leftbiga,a,bbig,cright$ and the different cardinality prevents a bijection.
Summing up, usually textbooks define tuples like $langle rangle_mathsfM$ above. That does not match with tuples generated by ordinary operations such as $Atimes Btimes C$ or the common $mathbbR^n$.
How do we reconcile the two notions?
elementary-set-theory
asked Jul 22 at 16:32
antonio
207210
Tuples are usually defined as indexed sets:
$$
langle x_1ldots x_nrangle_mathsfM := (x_i)_iin1ldots n
$$
However, this definition does not work with n-ary Cartesian products.
Perhaps, an element of $left(prod_i=1^n X_iright)$, assuming left associativity, is better thought as:
beginalign*
langle x_1ldots x_nrangle_mathsfN :=
begincases
x_1 &text if n=1\
langlelangle x_1ldots x_n-1rangle_mathsfN, x_nrangle_mathsfK &text else
endcases
endalign*
where $langlerangle_mathsfK$ denotes the Kuratowski pair and
$langle x_nldots x_nrangle_mathsfN$ and $langle x_nldots x_n+1rangle_mathsfN$ are usually abbreviated with $langle x_nrangle_mathsfN$ and $langle x_n, x_n+1rangle_mathsfN$.
I am not sure whether I read there is canonical bijection between the two definitions. One could write it as:
$$
langle x_1ldots x_nrangle_mathsfM
mapsto
langle x_1ldots x_nrangle_mathsfN
$$
but upon thinking, given the triple $langle a,b,crangle_*$,
$$
langle a,b,crangle_mathsfM =
langle 1, arangle_mathsfK,
langle 2, arangle_mathsfK,
langle 3, arangle_mathsfK
$$
while:
beginalign*
langle a,b,crangle_mathsfN &=
biglanglelangle a,brangle,cbigrangle_mathsfK\
&=leftlanglebiga,a,bbig,crightrangle_mathsfK\
&=leftleftbiga,a,bbigright,;
leftbiga,a,bbig,crightright
endalign*
So the map-tuple has three elements $langle i, *rangle_mathsfK$ (with $langlerangle_mathsfK$'s further expansible), while the nested version has two elements:
$leftbiga,a,bbigright$
and
$leftbiga,a,bbig,cright$ and the different cardinality prevents a bijection.
Summing up, usually textbooks define tuples like $langle rangle_mathsfM$ above. That does not match with tuples generated by ordinary operations such as $Atimes Btimes C$ or the common $mathbbR^n$.
How do we reconcile the two notions?
elementary-set-theory
asked Jul 22 at 16:32
antonio
207210
asked Jul 22 at 16:32
antonio
207210
asked Jul 22 at 16:32
antonio
207210
asked Jul 22 at 16:32
antonio
207210
207210
Show by induction there is a bijection between finite map tuples and set tuples.
– William Elliot
Jul 22 at 22:46
add a comment |Â
Show by induction there is a bijection between finite map tuples and set tuples.
– William Elliot
Jul 22 at 22:46
Show by induction there is a bijection between finite map tuples and set tuples.
– William Elliot
Jul 22 at 22:46
Show by induction there is a bijection between finite map tuples and set tuples.
– William Elliot
Jul 22 at 22:46
add a comment |Â
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Show by induction there is a bijection between finite map tuples and set tuples.
– William Elliot
Jul 22 at 22:46