Partitioning an infinite set into fixed number of sets
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Suppose we have a set of size $kappa$, and want to partition it into $mu$ sets, where $kappa$ is an infinite cardinal, and $1<muleqkappa$. I am aware that it can be done in $2^kappa$ ways (in the standard ZFC theory), but I cannot find a reference for this, nor a simple explanation.
reference-request infinitary-combinatorics
asked Jan 17 at 13:06
newbienewbie
32
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add a comment |
$begingroup$
Suppose we have a set of size $kappa$, and want to partition it into $mu$ sets, where $kappa$ is an infinite cardinal, and $1<muleqkappa$. I am aware that it can be done in $2^kappa$ ways (in the standard ZFC theory), but I cannot find a reference for this, nor a simple explanation.
reference-request infinitary-combinatorics
asked Jan 17 at 13:06
newbienewbie
32
$endgroup$
add a comment |
$begingroup$
Suppose we have a set of size $kappa$, and want to partition it into $mu$ sets, where $kappa$ is an infinite cardinal, and $1<muleqkappa$. I am aware that it can be done in $2^kappa$ ways (in the standard ZFC theory), but I cannot find a reference for this, nor a simple explanation.
reference-request infinitary-combinatorics
asked Jan 17 at 13:06
newbienewbie
32
$endgroup$
Suppose we have a set of size $kappa$, and want to partition it into $mu$ sets, where $kappa$ is an infinite cardinal, and $1<muleqkappa$. I am aware that it can be done in $2^kappa$ ways (in the standard ZFC theory), but I cannot find a reference for this, nor a simple explanation.
reference-request infinitary-combinatorics
reference-request infinitary-combinatorics
asked Jan 17 at 13:06
newbienewbie
32
asked Jan 17 at 13:06
newbienewbie
32
asked Jan 17 at 13:06
newbienewbie
32
asked Jan 17 at 13:06
newbienewbie
32
asked Jan 17 at 13:06
newbienewbie
32
32
add a comment |
add a comment |
1 Answer
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There are $2^kappa$ ways of dividing $kappa$ into $2$ sets. To see why, note that ${A,kappasetminus A}$ is a partition of $kappa$, and the function mapping $Amapsto{A,kappasetminus A}$ is a $2$-to-$1$ function. By standard cardinal arithmetic, this means that the image of the function is of size $2^kappa$.
On the other hand, if we have a partition of $kappa$, then we can represent it as a function $fcolonkappatokappa$. In fact, as many functions of that form. But there are only $kappa^kappa=2^kappa$ such functions, which gives us the upper bound of $2^kappa$. Therefore, there are exactly $2^kappa$ partitions, even to $kappa$ parts.
So in particular, partitioning $kappa$ into $mu$ parts has $2^kappa$ to go about that.
answered Jan 17 at 13:09
Asaf Karagila♦Asaf Karagila
308k33441775
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$begingroup$
There are $2^kappa$ ways of dividing $kappa$ into $2$ sets. To see why, note that ${A,kappasetminus A}$ is a partition of $kappa$, and the function mapping $Amapsto{A,kappasetminus A}$ is a $2$-to-$1$ function. By standard cardinal arithmetic, this means that the image of the function is of size $2^kappa$.
On the other hand, if we have a partition of $kappa$, then we can represent it as a function $fcolonkappatokappa$. In fact, as many functions of that form. But there are only $kappa^kappa=2^kappa$ such functions, which gives us the upper bound of $2^kappa$. Therefore, there are exactly $2^kappa$ partitions, even to $kappa$ parts.
So in particular, partitioning $kappa$ into $mu$ parts has $2^kappa$ to go about that.
answered Jan 17 at 13:09
Asaf Karagila♦Asaf Karagila
308k33441775
$endgroup$
add a comment |
$begingroup$
There are $2^kappa$ ways of dividing $kappa$ into $2$ sets. To see why, note that ${A,kappasetminus A}$ is a partition of $kappa$, and the function mapping $Amapsto{A,kappasetminus A}$ is a $2$-to-$1$ function. By standard cardinal arithmetic, this means that the image of the function is of size $2^kappa$.
On the other hand, if we have a partition of $kappa$, then we can represent it as a function $fcolonkappatokappa$. In fact, as many functions of that form. But there are only $kappa^kappa=2^kappa$ such functions, which gives us the upper bound of $2^kappa$. Therefore, there are exactly $2^kappa$ partitions, even to $kappa$ parts.
So in particular, partitioning $kappa$ into $mu$ parts has $2^kappa$ to go about that.
answered Jan 17 at 13:09
Asaf Karagila♦Asaf Karagila
308k33441775
$endgroup$
add a comment |
$begingroup$
There are $2^kappa$ ways of dividing $kappa$ into $2$ sets. To see why, note that ${A,kappasetminus A}$ is a partition of $kappa$, and the function mapping $Amapsto{A,kappasetminus A}$ is a $2$-to-$1$ function. By standard cardinal arithmetic, this means that the image of the function is of size $2^kappa$.
On the other hand, if we have a partition of $kappa$, then we can represent it as a function $fcolonkappatokappa$. In fact, as many functions of that form. But there are only $kappa^kappa=2^kappa$ such functions, which gives us the upper bound of $2^kappa$. Therefore, there are exactly $2^kappa$ partitions, even to $kappa$ parts.
So in particular, partitioning $kappa$ into $mu$ parts has $2^kappa$ to go about that.
answered Jan 17 at 13:09
Asaf Karagila♦Asaf Karagila
308k33441775
$endgroup$
There are $2^kappa$ ways of dividing $kappa$ into $2$ sets. To see why, note that ${A,kappasetminus A}$ is a partition of $kappa$, and the function mapping $Amapsto{A,kappasetminus A}$ is a $2$-to-$1$ function. By standard cardinal arithmetic, this means that the image of the function is of size $2^kappa$.
On the other hand, if we have a partition of $kappa$, then we can represent it as a function $fcolonkappatokappa$. In fact, as many functions of that form. But there are only $kappa^kappa=2^kappa$ such functions, which gives us the upper bound of $2^kappa$. Therefore, there are exactly $2^kappa$ partitions, even to $kappa$ parts.
So in particular, partitioning $kappa$ into $mu$ parts has $2^kappa$ to go about that.
answered Jan 17 at 13:09
Asaf Karagila♦Asaf Karagila
308k33441775
edited Jan 17 at 14:02
answered Jan 17 at 13:09
Asaf Karagila♦Asaf Karagila
308k33441775
answered Jan 17 at 13:09
Asaf Karagila♦Asaf Karagila
308k33441775
answered Jan 17 at 13:09
Asaf Karagila♦Asaf Karagila
308k33441775
308k33441775
add a comment |
add a comment |
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