Prove countability of letter combinations
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A sentence written in 'Fakelang' is made up of letters from a finite set X. These letters follow certain syntax rules. The task is to prove that the number of sentences possible in Fakelang is countable. How do you prove this?
My thinking: X is countable. A sentence made up of letters from X will be countable. No matter how long the sentence is, as long as it is finite in length, the possible combinations will also be finite (combinations will necessarily be less than $|X|^n$ due to the syntax rules). Since each subset of 'total combinations' is countable and finite, the 'total combinations' set must also be countable.
Is my thinking wrong? What is lacking for this to be considered a proof?
algebra-precalculus elementary-set-theory
asked Dec 11 '18 at 12:39
Adam JenssenAdam Jenssen
51
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add a comment |
$begingroup$
A sentence written in 'Fakelang' is made up of letters from a finite set X. These letters follow certain syntax rules. The task is to prove that the number of sentences possible in Fakelang is countable. How do you prove this?
My thinking: X is countable. A sentence made up of letters from X will be countable. No matter how long the sentence is, as long as it is finite in length, the possible combinations will also be finite (combinations will necessarily be less than $|X|^n$ due to the syntax rules). Since each subset of 'total combinations' is countable and finite, the 'total combinations' set must also be countable.
Is my thinking wrong? What is lacking for this to be considered a proof?
algebra-precalculus elementary-set-theory
asked Dec 11 '18 at 12:39
Adam JenssenAdam Jenssen
51
$endgroup$
add a comment |
$begingroup$
A sentence written in 'Fakelang' is made up of letters from a finite set X. These letters follow certain syntax rules. The task is to prove that the number of sentences possible in Fakelang is countable. How do you prove this?
My thinking: X is countable. A sentence made up of letters from X will be countable. No matter how long the sentence is, as long as it is finite in length, the possible combinations will also be finite (combinations will necessarily be less than $|X|^n$ due to the syntax rules). Since each subset of 'total combinations' is countable and finite, the 'total combinations' set must also be countable.
Is my thinking wrong? What is lacking for this to be considered a proof?
algebra-precalculus elementary-set-theory
asked Dec 11 '18 at 12:39
Adam JenssenAdam Jenssen
51
$endgroup$
A sentence written in 'Fakelang' is made up of letters from a finite set X. These letters follow certain syntax rules. The task is to prove that the number of sentences possible in Fakelang is countable. How do you prove this?
My thinking: X is countable. A sentence made up of letters from X will be countable. No matter how long the sentence is, as long as it is finite in length, the possible combinations will also be finite (combinations will necessarily be less than $|X|^n$ due to the syntax rules). Since each subset of 'total combinations' is countable and finite, the 'total combinations' set must also be countable.
Is my thinking wrong? What is lacking for this to be considered a proof?
algebra-precalculus elementary-set-theory
algebra-precalculus elementary-set-theory
asked Dec 11 '18 at 12:39
Adam JenssenAdam Jenssen
51
asked Dec 11 '18 at 12:39
Adam JenssenAdam Jenssen
51
asked Dec 11 '18 at 12:39
Adam JenssenAdam Jenssen
51
asked Dec 11 '18 at 12:39
Adam JenssenAdam Jenssen
51
asked Dec 11 '18 at 12:39
Adam JenssenAdam Jenssen
51
51
add a comment |
add a comment |
1 Answer
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You are on the right track.
A bit more formal:
Let $X$, the set of letters, be finite.
Now consider a sentence of length $n$, i.e.
$n$ spaces.
Let $P_n$ be the set of all permutations of length $n$ with elements (letters , spaces , and what not) in $X$. $P_n$ is finite.
Consider the set $P$:
$P:= displaystyle{ cup}_{n in mathbb{N}} P_n$.
$P$ as the countable union of countable (actually finite) sets is countable.
Let $S$ be the set of sentences in Fakelang of finite length, then
$S subset P$, hence countable .
answered Dec 11 '18 at 13:13
Peter SzilasPeter Szilas
11.6k2822
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$begingroup$
Thanks a lot! Additional question: if the sentence has to be of finite length, will the number of possible sentences still be infinite? Considering the fact that your can always construct a sentence longer the previous one?
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– Adam Jenssen
Dec 11 '18 at 21:29
$begingroup$
Adam.A pleasure.We are considering sentences of length, say n= 10,100, 1000,... etc.Every sentence has length n(finite).Then S is countable infinite.Is this your question?
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– Peter Szilas
Dec 11 '18 at 22:56
$begingroup$
Yes, that was the question. Thanks again!
$endgroup$
– Adam Jenssen
Dec 12 '18 at 5:36
add a comment |
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1 Answer
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1 Answer
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active
oldest
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$begingroup$
You are on the right track.
A bit more formal:
Let $X$, the set of letters, be finite.
Now consider a sentence of length $n$, i.e.
$n$ spaces.
Let $P_n$ be the set of all permutations of length $n$ with elements (letters , spaces , and what not) in $X$. $P_n$ is finite.
Consider the set $P$:
$P:= displaystyle{ cup}_{n in mathbb{N}} P_n$.
$P$ as the countable union of countable (actually finite) sets is countable.
Let $S$ be the set of sentences in Fakelang of finite length, then
$S subset P$, hence countable .
answered Dec 11 '18 at 13:13
Peter SzilasPeter Szilas
11.6k2822
$endgroup$
$begingroup$
Thanks a lot! Additional question: if the sentence has to be of finite length, will the number of possible sentences still be infinite? Considering the fact that your can always construct a sentence longer the previous one?
$endgroup$
– Adam Jenssen
Dec 11 '18 at 21:29
$begingroup$
Adam.A pleasure.We are considering sentences of length, say n= 10,100, 1000,... etc.Every sentence has length n(finite).Then S is countable infinite.Is this your question?
$endgroup$
– Peter Szilas
Dec 11 '18 at 22:56
$begingroup$
Yes, that was the question. Thanks again!
$endgroup$
– Adam Jenssen
Dec 12 '18 at 5:36
add a comment |
$begingroup$
You are on the right track.
A bit more formal:
Let $X$, the set of letters, be finite.
Now consider a sentence of length $n$, i.e.
$n$ spaces.
Let $P_n$ be the set of all permutations of length $n$ with elements (letters , spaces , and what not) in $X$. $P_n$ is finite.
Consider the set $P$:
$P:= displaystyle{ cup}_{n in mathbb{N}} P_n$.
$P$ as the countable union of countable (actually finite) sets is countable.
Let $S$ be the set of sentences in Fakelang of finite length, then
$S subset P$, hence countable .
answered Dec 11 '18 at 13:13
Peter SzilasPeter Szilas
11.6k2822
$endgroup$
$begingroup$
Thanks a lot! Additional question: if the sentence has to be of finite length, will the number of possible sentences still be infinite? Considering the fact that your can always construct a sentence longer the previous one?
$endgroup$
– Adam Jenssen
Dec 11 '18 at 21:29
$begingroup$
Adam.A pleasure.We are considering sentences of length, say n= 10,100, 1000,... etc.Every sentence has length n(finite).Then S is countable infinite.Is this your question?
$endgroup$
– Peter Szilas
Dec 11 '18 at 22:56
$begingroup$
Yes, that was the question. Thanks again!
$endgroup$
– Adam Jenssen
Dec 12 '18 at 5:36
add a comment |
$begingroup$
You are on the right track.
A bit more formal:
Let $X$, the set of letters, be finite.
Now consider a sentence of length $n$, i.e.
$n$ spaces.
Let $P_n$ be the set of all permutations of length $n$ with elements (letters , spaces , and what not) in $X$. $P_n$ is finite.
Consider the set $P$:
$P:= displaystyle{ cup}_{n in mathbb{N}} P_n$.
$P$ as the countable union of countable (actually finite) sets is countable.
Let $S$ be the set of sentences in Fakelang of finite length, then
$S subset P$, hence countable .
answered Dec 11 '18 at 13:13
Peter SzilasPeter Szilas
11.6k2822
$endgroup$
You are on the right track.
A bit more formal:
Let $X$, the set of letters, be finite.
Now consider a sentence of length $n$, i.e.
$n$ spaces.
Let $P_n$ be the set of all permutations of length $n$ with elements (letters , spaces , and what not) in $X$. $P_n$ is finite.
Consider the set $P$:
$P:= displaystyle{ cup}_{n in mathbb{N}} P_n$.
$P$ as the countable union of countable (actually finite) sets is countable.
Let $S$ be the set of sentences in Fakelang of finite length, then
$S subset P$, hence countable .
answered Dec 11 '18 at 13:13
Peter SzilasPeter Szilas
11.6k2822
edited Dec 11 '18 at 13:19
answered Dec 11 '18 at 13:13
Peter SzilasPeter Szilas
11.6k2822
answered Dec 11 '18 at 13:13
Peter SzilasPeter Szilas
11.6k2822
answered Dec 11 '18 at 13:13
Peter SzilasPeter Szilas
11.6k2822
11.6k2822
$begingroup$
Thanks a lot! Additional question: if the sentence has to be of finite length, will the number of possible sentences still be infinite? Considering the fact that your can always construct a sentence longer the previous one?
$endgroup$
– Adam Jenssen
Dec 11 '18 at 21:29
$begingroup$
Adam.A pleasure.We are considering sentences of length, say n= 10,100, 1000,... etc.Every sentence has length n(finite).Then S is countable infinite.Is this your question?
$endgroup$
– Peter Szilas
Dec 11 '18 at 22:56
$begingroup$
Yes, that was the question. Thanks again!
$endgroup$
– Adam Jenssen
Dec 12 '18 at 5:36
add a comment |
$begingroup$
Thanks a lot! Additional question: if the sentence has to be of finite length, will the number of possible sentences still be infinite? Considering the fact that your can always construct a sentence longer the previous one?
$endgroup$
– Adam Jenssen
Dec 11 '18 at 21:29
$begingroup$
Adam.A pleasure.We are considering sentences of length, say n= 10,100, 1000,... etc.Every sentence has length n(finite).Then S is countable infinite.Is this your question?
$endgroup$
– Peter Szilas
Dec 11 '18 at 22:56
$begingroup$
Yes, that was the question. Thanks again!
$endgroup$
– Adam Jenssen
Dec 12 '18 at 5:36
$begingroup$
Thanks a lot! Additional question: if the sentence has to be of finite length, will the number of possible sentences still be infinite? Considering the fact that your can always construct a sentence longer the previous one?
$endgroup$
– Adam Jenssen
Dec 11 '18 at 21:29
$begingroup$
Thanks a lot! Additional question: if the sentence has to be of finite length, will the number of possible sentences still be infinite? Considering the fact that your can always construct a sentence longer the previous one?
$endgroup$
– Adam Jenssen
Dec 11 '18 at 21:29
$begingroup$
Adam.A pleasure.We are considering sentences of length, say n= 10,100, 1000,... etc.Every sentence has length n(finite).Then S is countable infinite.Is this your question?
$endgroup$
– Peter Szilas
Dec 11 '18 at 22:56
$begingroup$
Adam.A pleasure.We are considering sentences of length, say n= 10,100, 1000,... etc.Every sentence has length n(finite).Then S is countable infinite.Is this your question?
$endgroup$
– Peter Szilas
Dec 11 '18 at 22:56
$begingroup$
Yes, that was the question. Thanks again!
$endgroup$
– Adam Jenssen
Dec 12 '18 at 5:36
$begingroup$
Yes, that was the question. Thanks again!
$endgroup$
– Adam Jenssen
Dec 12 '18 at 5:36
add a comment |
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