Mixing
Finding how many distinct equivalence classes there are.
$begingroup$
Define a relation R on the set of all integers Z by xRy (x related to y) if and only if x-y=3k for some integer k.
I have already verified that this is in fact an equivalence relation. But now I need to find how many distinct equivalence classes there are.
I am confused on how to find how many distinct equivalence classes there are.
analysis relations equivalence-relations
asked Feb 1 at 3:04
AnneAnne
105
$endgroup$
add a comment |
$begingroup$
Define a relation R on the set of all integers Z by xRy (x related to y) if and only if x-y=3k for some integer k.
I have already verified that this is in fact an equivalence relation. But now I need to find how many distinct equivalence classes there are.
I am confused on how to find how many distinct equivalence classes there are.
analysis relations equivalence-relations
asked Feb 1 at 3:04
AnneAnne
105
$endgroup$
2
$begingroup$
Hint: take any integer, say $0$. Find all the elements equivalent to $0$. They form an equivalence class. Take any integer not in this equivalence class and repeat. Keep going until there are no more "unused" integers.
$endgroup$
– David
Feb 1 at 3:08
$begingroup$
Write $x=3a+r, y=3b+s$ where $r,s$ are the remainders when dividing $x,y$ by 3.
$endgroup$
– Jens Schwaiger
Feb 1 at 3:09
add a comment |
$begingroup$
Define a relation R on the set of all integers Z by xRy (x related to y) if and only if x-y=3k for some integer k.
I have already verified that this is in fact an equivalence relation. But now I need to find how many distinct equivalence classes there are.
I am confused on how to find how many distinct equivalence classes there are.
analysis relations equivalence-relations
asked Feb 1 at 3:04
AnneAnne
105
$endgroup$
Define a relation R on the set of all integers Z by xRy (x related to y) if and only if x-y=3k for some integer k.
I have already verified that this is in fact an equivalence relation. But now I need to find how many distinct equivalence classes there are.
I am confused on how to find how many distinct equivalence classes there are.
analysis relations equivalence-relations
analysis relations equivalence-relations
asked Feb 1 at 3:04
AnneAnne
105
asked Feb 1 at 3:04
AnneAnne
105
asked Feb 1 at 3:04
AnneAnne
105
asked Feb 1 at 3:04
AnneAnne
105
asked Feb 1 at 3:04
AnneAnne
105
105
2
$begingroup$
Hint: take any integer, say $0$. Find all the elements equivalent to $0$. They form an equivalence class. Take any integer not in this equivalence class and repeat. Keep going until there are no more "unused" integers.
$endgroup$
– David
Feb 1 at 3:08
$begingroup$
Write $x=3a+r, y=3b+s$ where $r,s$ are the remainders when dividing $x,y$ by 3.
$endgroup$
– Jens Schwaiger
Feb 1 at 3:09
add a comment |
2
$begingroup$
Hint: take any integer, say $0$. Find all the elements equivalent to $0$. They form an equivalence class. Take any integer not in this equivalence class and repeat. Keep going until there are no more "unused" integers.
$endgroup$
– David
Feb 1 at 3:08
$begingroup$
Write $x=3a+r, y=3b+s$ where $r,s$ are the remainders when dividing $x,y$ by 3.
$endgroup$
– Jens Schwaiger
Feb 1 at 3:09
2
2
$begingroup$
Hint: take any integer, say $0$. Find all the elements equivalent to $0$. They form an equivalence class. Take any integer not in this equivalence class and repeat. Keep going until there are no more "unused" integers.
$endgroup$
– David
Feb 1 at 3:08
$begingroup$
Hint: take any integer, say $0$. Find all the elements equivalent to $0$. They form an equivalence class. Take any integer not in this equivalence class and repeat. Keep going until there are no more "unused" integers.
$endgroup$
– David
Feb 1 at 3:08
$begingroup$
Write $x=3a+r, y=3b+s$ where $r,s$ are the remainders when dividing $x,y$ by 3.
$endgroup$
– Jens Schwaiger
Feb 1 at 3:09
$begingroup$
Write $x=3a+r, y=3b+s$ where $r,s$ are the remainders when dividing $x,y$ by 3.
$endgroup$
– Jens Schwaiger
Feb 1 at 3:09
add a comment |
1 Answer
1
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oldest
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$begingroup$
All equivalence classes are congruent classes of modulo 3. $Cl(0)={0, 3, -3, 6, -6,ldots}$ $Cl(1) = {1, -2, 4, -5, 7, ldots}$ $Cl(2)={2, -1, 5, -4,ldots}$
answered Feb 1 at 3:12
OfflawOfflaw
3189
$endgroup$
$begingroup$
Can you explain a little more on what you mean? Are equivalence classes the remainders then?
$endgroup$
– Anne
Feb 1 at 3:22
$begingroup$
To construct, start with $Cl(0) = {min mathbb{Z}:m-0=3k}$ and continue the process.
$endgroup$
– Offlaw
Feb 1 at 3:24
add a comment |
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1 Answer
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$begingroup$
All equivalence classes are congruent classes of modulo 3. $Cl(0)={0, 3, -3, 6, -6,ldots}$ $Cl(1) = {1, -2, 4, -5, 7, ldots}$ $Cl(2)={2, -1, 5, -4,ldots}$
answered Feb 1 at 3:12
OfflawOfflaw
3189
$endgroup$
$begingroup$
Can you explain a little more on what you mean? Are equivalence classes the remainders then?
$endgroup$
– Anne
Feb 1 at 3:22
$begingroup$
To construct, start with $Cl(0) = {min mathbb{Z}:m-0=3k}$ and continue the process.
$endgroup$
– Offlaw
Feb 1 at 3:24
add a comment |
$begingroup$
All equivalence classes are congruent classes of modulo 3. $Cl(0)={0, 3, -3, 6, -6,ldots}$ $Cl(1) = {1, -2, 4, -5, 7, ldots}$ $Cl(2)={2, -1, 5, -4,ldots}$
answered Feb 1 at 3:12
OfflawOfflaw
3189
$endgroup$
$begingroup$
Can you explain a little more on what you mean? Are equivalence classes the remainders then?
$endgroup$
– Anne
Feb 1 at 3:22
$begingroup$
To construct, start with $Cl(0) = {min mathbb{Z}:m-0=3k}$ and continue the process.
$endgroup$
– Offlaw
Feb 1 at 3:24
add a comment |
$begingroup$
All equivalence classes are congruent classes of modulo 3. $Cl(0)={0, 3, -3, 6, -6,ldots}$ $Cl(1) = {1, -2, 4, -5, 7, ldots}$ $Cl(2)={2, -1, 5, -4,ldots}$
answered Feb 1 at 3:12
OfflawOfflaw
3189
$endgroup$
All equivalence classes are congruent classes of modulo 3. $Cl(0)={0, 3, -3, 6, -6,ldots}$ $Cl(1) = {1, -2, 4, -5, 7, ldots}$ $Cl(2)={2, -1, 5, -4,ldots}$
answered Feb 1 at 3:12
OfflawOfflaw
3189
answered Feb 1 at 3:12
OfflawOfflaw
3189
answered Feb 1 at 3:12
OfflawOfflaw
3189
answered Feb 1 at 3:12
OfflawOfflaw
3189
3189
$begingroup$
Can you explain a little more on what you mean? Are equivalence classes the remainders then?
$endgroup$
– Anne
Feb 1 at 3:22
$begingroup$
To construct, start with $Cl(0) = {min mathbb{Z}:m-0=3k}$ and continue the process.
$endgroup$
– Offlaw
Feb 1 at 3:24
add a comment |
$begingroup$
Can you explain a little more on what you mean? Are equivalence classes the remainders then?
$endgroup$
– Anne
Feb 1 at 3:22
$begingroup$
To construct, start with $Cl(0) = {min mathbb{Z}:m-0=3k}$ and continue the process.
$endgroup$
– Offlaw
Feb 1 at 3:24
$begingroup$
Can you explain a little more on what you mean? Are equivalence classes the remainders then?
$endgroup$
– Anne
Feb 1 at 3:22
$begingroup$
Can you explain a little more on what you mean? Are equivalence classes the remainders then?
$endgroup$
– Anne
Feb 1 at 3:22
$begingroup$
To construct, start with $Cl(0) = {min mathbb{Z}:m-0=3k}$ and continue the process.
$endgroup$
– Offlaw
Feb 1 at 3:24
$begingroup$
To construct, start with $Cl(0) = {min mathbb{Z}:m-0=3k}$ and continue the process.
$endgroup$
– Offlaw
Feb 1 at 3:24
add a comment |
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$begingroup$
Hint: take any integer, say $0$. Find all the elements equivalent to $0$. They form an equivalence class. Take any integer not in this equivalence class and repeat. Keep going until there are no more "unused" integers.
$endgroup$
– David
Feb 1 at 3:08
$begingroup$
Write $x=3a+r, y=3b+s$ where $r,s$ are the remainders when dividing $x,y$ by 3.
$endgroup$
– Jens Schwaiger
Feb 1 at 3:09