The set of all straight lines
$begingroup$
Prove that the set of all straight lines, each of which passes through at least two point such that its two coordinates are integers, is Countable set.
If $A$ is a set of straight lines referred to in the task, then I think $$A=left{ y=ax+b, a,bin mathbb C right} cup left{ x=c, cinmathbb C right} $$ and exist at least two $a,b in mathbb Z$ or $c in mathbb Z$.
However I don't knew how I can show that $A$ is countable.
Can anyone help me?
elementary-set-theory
asked Jan 8 at 17:58
MP3129MP3129
872211
$endgroup$
add a comment |
$begingroup$
Prove that the set of all straight lines, each of which passes through at least two point such that its two coordinates are integers, is Countable set.
If $A$ is a set of straight lines referred to in the task, then I think $$A=left{ y=ax+b, a,bin mathbb C right} cup left{ x=c, cinmathbb C right} $$ and exist at least two $a,b in mathbb Z$ or $c in mathbb Z$.
However I don't knew how I can show that $A$ is countable.
Can anyone help me?
elementary-set-theory
asked Jan 8 at 17:58
MP3129MP3129
872211
$endgroup$
add a comment |
$begingroup$
Prove that the set of all straight lines, each of which passes through at least two point such that its two coordinates are integers, is Countable set.
If $A$ is a set of straight lines referred to in the task, then I think $$A=left{ y=ax+b, a,bin mathbb C right} cup left{ x=c, cinmathbb C right} $$ and exist at least two $a,b in mathbb Z$ or $c in mathbb Z$.
However I don't knew how I can show that $A$ is countable.
Can anyone help me?
elementary-set-theory
asked Jan 8 at 17:58
MP3129MP3129
872211
$endgroup$
Prove that the set of all straight lines, each of which passes through at least two point such that its two coordinates are integers, is Countable set.
If $A$ is a set of straight lines referred to in the task, then I think $$A=left{ y=ax+b, a,bin mathbb C right} cup left{ x=c, cinmathbb C right} $$ and exist at least two $a,b in mathbb Z$ or $c in mathbb Z$.
However I don't knew how I can show that $A$ is countable.
Can anyone help me?
elementary-set-theory
elementary-set-theory
asked Jan 8 at 17:58
MP3129MP3129
872211
asked Jan 8 at 17:58
MP3129MP3129
872211
asked Jan 8 at 17:58
MP3129MP3129
872211
asked Jan 8 at 17:58
MP3129MP3129
872211
asked Jan 8 at 17:58
MP3129MP3129
872211
872211
add a comment |
add a comment |
1 Answer
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The set$$S=left{bigl((a,b),(c,d)bigr)inmathbb{Z}^2timesmathbb{Z}^2,middle|,(a,b)neq(c,d)right}$$is countable. For each element of $bigl((a,b),(c,d)bigr)in S$, there is one and only one line passing through $(a,b)$ and $(c,d)$. This defines a surjective map from $S$ onto your set. So, since $S$ is countable, your set is either finite or countable. But it is clearly infinite. So…
answered Jan 8 at 18:04
José Carlos SantosJosé Carlos Santos
175k23134243
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$begingroup$
The set$$S=left{bigl((a,b),(c,d)bigr)inmathbb{Z}^2timesmathbb{Z}^2,middle|,(a,b)neq(c,d)right}$$is countable. For each element of $bigl((a,b),(c,d)bigr)in S$, there is one and only one line passing through $(a,b)$ and $(c,d)$. This defines a surjective map from $S$ onto your set. So, since $S$ is countable, your set is either finite or countable. But it is clearly infinite. So…
answered Jan 8 at 18:04
José Carlos SantosJosé Carlos Santos
175k23134243
$endgroup$
add a comment |
$begingroup$
The set$$S=left{bigl((a,b),(c,d)bigr)inmathbb{Z}^2timesmathbb{Z}^2,middle|,(a,b)neq(c,d)right}$$is countable. For each element of $bigl((a,b),(c,d)bigr)in S$, there is one and only one line passing through $(a,b)$ and $(c,d)$. This defines a surjective map from $S$ onto your set. So, since $S$ is countable, your set is either finite or countable. But it is clearly infinite. So…
answered Jan 8 at 18:04
José Carlos SantosJosé Carlos Santos
175k23134243
$endgroup$
add a comment |
$begingroup$
The set$$S=left{bigl((a,b),(c,d)bigr)inmathbb{Z}^2timesmathbb{Z}^2,middle|,(a,b)neq(c,d)right}$$is countable. For each element of $bigl((a,b),(c,d)bigr)in S$, there is one and only one line passing through $(a,b)$ and $(c,d)$. This defines a surjective map from $S$ onto your set. So, since $S$ is countable, your set is either finite or countable. But it is clearly infinite. So…
answered Jan 8 at 18:04
José Carlos SantosJosé Carlos Santos
175k23134243
$endgroup$
The set$$S=left{bigl((a,b),(c,d)bigr)inmathbb{Z}^2timesmathbb{Z}^2,middle|,(a,b)neq(c,d)right}$$is countable. For each element of $bigl((a,b),(c,d)bigr)in S$, there is one and only one line passing through $(a,b)$ and $(c,d)$. This defines a surjective map from $S$ onto your set. So, since $S$ is countable, your set is either finite or countable. But it is clearly infinite. So…
answered Jan 8 at 18:04
José Carlos SantosJosé Carlos Santos
175k23134243
answered Jan 8 at 18:04
José Carlos SantosJosé Carlos Santos
175k23134243
answered Jan 8 at 18:04
José Carlos SantosJosé Carlos Santos
175k23134243
answered Jan 8 at 18:04
José Carlos SantosJosé Carlos Santos
175k23134243
175k23134243
add a comment |
add a comment |
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