Subtraction in Building a Set from Nonnegative Reals
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When defining equivalence classes using elements contained by the nonnegative reals, may I use subtraction in the function that defines equivalence between those classes? My thinking is that if subtraction is defined for the reals it could be used but if my elements are strictly nonnegative, I'm not sure if that makes a difference.
elementary-set-theory relations equivalence-relations
asked Dec 7 '18 at 21:11
bblohowiakbblohowiak
1049
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add a comment |
$begingroup$
When defining equivalence classes using elements contained by the nonnegative reals, may I use subtraction in the function that defines equivalence between those classes? My thinking is that if subtraction is defined for the reals it could be used but if my elements are strictly nonnegative, I'm not sure if that makes a difference.
elementary-set-theory relations equivalence-relations
asked Dec 7 '18 at 21:11
bblohowiakbblohowiak
1049
$endgroup$
1
$begingroup$
Unless you are using this function somehow to construct the negative real numbers or to define subtraction what you suggest is perfectly OK. You can edit the question to include more information if you're still unsure.
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– Ethan Bolker
Dec 7 '18 at 21:13
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Equivalence classes under what equivalence relation?
$endgroup$
– user4894
Dec 7 '18 at 21:19
add a comment |
$begingroup$
When defining equivalence classes using elements contained by the nonnegative reals, may I use subtraction in the function that defines equivalence between those classes? My thinking is that if subtraction is defined for the reals it could be used but if my elements are strictly nonnegative, I'm not sure if that makes a difference.
elementary-set-theory relations equivalence-relations
asked Dec 7 '18 at 21:11
bblohowiakbblohowiak
1049
$endgroup$
When defining equivalence classes using elements contained by the nonnegative reals, may I use subtraction in the function that defines equivalence between those classes? My thinking is that if subtraction is defined for the reals it could be used but if my elements are strictly nonnegative, I'm not sure if that makes a difference.
elementary-set-theory relations equivalence-relations
elementary-set-theory relations equivalence-relations
asked Dec 7 '18 at 21:11
bblohowiakbblohowiak
1049
asked Dec 7 '18 at 21:11
bblohowiakbblohowiak
1049
asked Dec 7 '18 at 21:11
bblohowiakbblohowiak
1049
asked Dec 7 '18 at 21:11
bblohowiakbblohowiak
1049
asked Dec 7 '18 at 21:11
bblohowiakbblohowiak
1049
1049
1
$begingroup$
Unless you are using this function somehow to construct the negative real numbers or to define subtraction what you suggest is perfectly OK. You can edit the question to include more information if you're still unsure.
$endgroup$
– Ethan Bolker
Dec 7 '18 at 21:13
$begingroup$
Equivalence classes under what equivalence relation?
$endgroup$
– user4894
Dec 7 '18 at 21:19
add a comment |
1
$begingroup$
Unless you are using this function somehow to construct the negative real numbers or to define subtraction what you suggest is perfectly OK. You can edit the question to include more information if you're still unsure.
$endgroup$
– Ethan Bolker
Dec 7 '18 at 21:13
$begingroup$
Equivalence classes under what equivalence relation?
$endgroup$
– user4894
Dec 7 '18 at 21:19
1
1
$begingroup$
Unless you are using this function somehow to construct the negative real numbers or to define subtraction what you suggest is perfectly OK. You can edit the question to include more information if you're still unsure.
$endgroup$
– Ethan Bolker
Dec 7 '18 at 21:13
$begingroup$
Unless you are using this function somehow to construct the negative real numbers or to define subtraction what you suggest is perfectly OK. You can edit the question to include more information if you're still unsure.
$endgroup$
– Ethan Bolker
Dec 7 '18 at 21:13
$begingroup$
Equivalence classes under what equivalence relation?
$endgroup$
– user4894
Dec 7 '18 at 21:19
$begingroup$
Equivalence classes under what equivalence relation?
$endgroup$
– user4894
Dec 7 '18 at 21:19
add a comment |
1 Answer
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The equivalence classes on a set $S$ can be seen a partition of $S$ in (disjoint) subsets $S_i$. You can define those subsets, i.e. each equivalence class, in any way you wish. As long as you define them to form a partition, that is such the sets $S_i$ are disjoint subsets of $S$ and satisfying $bigcup_i S_i = S$.
answered Dec 7 '18 at 21:36
Jorge AdrianoJorge Adriano
59146
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$begingroup$
The equivalence classes on a set $S$ can be seen a partition of $S$ in (disjoint) subsets $S_i$. You can define those subsets, i.e. each equivalence class, in any way you wish. As long as you define them to form a partition, that is such the sets $S_i$ are disjoint subsets of $S$ and satisfying $bigcup_i S_i = S$.
answered Dec 7 '18 at 21:36
Jorge AdrianoJorge Adriano
59146
$endgroup$
add a comment |
$begingroup$
The equivalence classes on a set $S$ can be seen a partition of $S$ in (disjoint) subsets $S_i$. You can define those subsets, i.e. each equivalence class, in any way you wish. As long as you define them to form a partition, that is such the sets $S_i$ are disjoint subsets of $S$ and satisfying $bigcup_i S_i = S$.
answered Dec 7 '18 at 21:36
Jorge AdrianoJorge Adriano
59146
$endgroup$
add a comment |
$begingroup$
The equivalence classes on a set $S$ can be seen a partition of $S$ in (disjoint) subsets $S_i$. You can define those subsets, i.e. each equivalence class, in any way you wish. As long as you define them to form a partition, that is such the sets $S_i$ are disjoint subsets of $S$ and satisfying $bigcup_i S_i = S$.
answered Dec 7 '18 at 21:36
Jorge AdrianoJorge Adriano
59146
$endgroup$
The equivalence classes on a set $S$ can be seen a partition of $S$ in (disjoint) subsets $S_i$. You can define those subsets, i.e. each equivalence class, in any way you wish. As long as you define them to form a partition, that is such the sets $S_i$ are disjoint subsets of $S$ and satisfying $bigcup_i S_i = S$.
answered Dec 7 '18 at 21:36
Jorge AdrianoJorge Adriano
59146
answered Dec 7 '18 at 21:36
Jorge AdrianoJorge Adriano
59146
answered Dec 7 '18 at 21:36
Jorge AdrianoJorge Adriano
59146
answered Dec 7 '18 at 21:36
Jorge AdrianoJorge Adriano
59146
59146
add a comment |
add a comment |
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1
$begingroup$
Unless you are using this function somehow to construct the negative real numbers or to define subtraction what you suggest is perfectly OK. You can edit the question to include more information if you're still unsure.
$endgroup$
– Ethan Bolker
Dec 7 '18 at 21:13
$begingroup$
Equivalence classes under what equivalence relation?
$endgroup$
– user4894
Dec 7 '18 at 21:19