Question in Proof That Every Well-Ordering is a Total-Ordering
$begingroup$
I am trying to make sense of the following:
Theorem: Let $R$ be a relation on a set $A$.
If $R$ is a well-ordered relation on $A$, then $R$ is a total-order relation on $A$.
Proof: Suppose $x,yin A.$ Let $R={x,y}not=emptyset$. By assumption, $R$ has a least element. If it is $x$, then $xleq y$. If it is $y$, then $yleq x$.
$$text{__________________________________}$$
For a relation to be total ordered, then it must be that for each $x,yin A$, $xRy$ and $yRx$. In this case we are comparing $x , y$ where one or the other is a least element. What about if neither is a least element though? Do we then simply say that $x,y$ must relate because if we consider whatever is the least element, say $z$, then both $x,y$ can relate to $z$ and therefore must relate to each other?
elementary-set-theory relations well-orders
asked Dec 5 '18 at 15:12
M. Damon M. Damon
235
$endgroup$
|
show 6 more comments
$begingroup$
I am trying to make sense of the following:
Theorem: Let $R$ be a relation on a set $A$.
If $R$ is a well-ordered relation on $A$, then $R$ is a total-order relation on $A$.
Proof: Suppose $x,yin A.$ Let $R={x,y}not=emptyset$. By assumption, $R$ has a least element. If it is $x$, then $xleq y$. If it is $y$, then $yleq x$.
$$text{__________________________________}$$
For a relation to be total ordered, then it must be that for each $x,yin A$, $xRy$ and $yRx$. In this case we are comparing $x , y$ where one or the other is a least element. What about if neither is a least element though? Do we then simply say that $x,y$ must relate because if we consider whatever is the least element, say $z$, then both $x,y$ can relate to $z$ and therefore must relate to each other?
elementary-set-theory relations well-orders
asked Dec 5 '18 at 15:12
M. Damon M. Damon
235
$endgroup$
$begingroup$
One of them must be the least element of ${x,y}$ because there are no other elements in the set, and it has a least element.
$endgroup$
– dbx
Dec 5 '18 at 15:13
$begingroup$
Where does this proof come from? In the definitions as I've learned them, well-ordered relations are defined to be both well-founded and totally ordered, and well-founded relations need not be totally ordered. See also math.stackexchange.com/questions/116642/…
$endgroup$
– J.G.
Dec 5 '18 at 15:19
$begingroup$
The definition my book uses denotes well-ordered relation as a partially-ordered relation on $A$ such that each non-empty subset of $A$ has a least element with respect to $R$.
$endgroup$
– M. Damon
Dec 5 '18 at 15:26
$begingroup$
@J.G.: Well-founded relations have a minimal element in each non-empty set; well-orders have a minimum in each non-empty set.
$endgroup$
– Asaf Karagila♦
Dec 5 '18 at 15:27
$begingroup$
@AsafKaragila In other words, the minimal element of the non-empty set is only required to be unique in the latter case? Fair enough.
$endgroup$
– J.G.
Dec 5 '18 at 15:31
|
show 6 more comments
$begingroup$
I am trying to make sense of the following:
Theorem: Let $R$ be a relation on a set $A$.
If $R$ is a well-ordered relation on $A$, then $R$ is a total-order relation on $A$.
Proof: Suppose $x,yin A.$ Let $R={x,y}not=emptyset$. By assumption, $R$ has a least element. If it is $x$, then $xleq y$. If it is $y$, then $yleq x$.
$$text{__________________________________}$$
For a relation to be total ordered, then it must be that for each $x,yin A$, $xRy$ and $yRx$. In this case we are comparing $x , y$ where one or the other is a least element. What about if neither is a least element though? Do we then simply say that $x,y$ must relate because if we consider whatever is the least element, say $z$, then both $x,y$ can relate to $z$ and therefore must relate to each other?
elementary-set-theory relations well-orders
asked Dec 5 '18 at 15:12
M. Damon M. Damon
235
$endgroup$
I am trying to make sense of the following:
Theorem: Let $R$ be a relation on a set $A$.
If $R$ is a well-ordered relation on $A$, then $R$ is a total-order relation on $A$.
Proof: Suppose $x,yin A.$ Let $R={x,y}not=emptyset$. By assumption, $R$ has a least element. If it is $x$, then $xleq y$. If it is $y$, then $yleq x$.
$$text{__________________________________}$$
For a relation to be total ordered, then it must be that for each $x,yin A$, $xRy$ and $yRx$. In this case we are comparing $x , y$ where one or the other is a least element. What about if neither is a least element though? Do we then simply say that $x,y$ must relate because if we consider whatever is the least element, say $z$, then both $x,y$ can relate to $z$ and therefore must relate to each other?
elementary-set-theory relations well-orders
elementary-set-theory relations well-orders
asked Dec 5 '18 at 15:12
M. Damon M. Damon
235
asked Dec 5 '18 at 15:12
M. Damon M. Damon
235
asked Dec 5 '18 at 15:12
M. Damon M. Damon
235
asked Dec 5 '18 at 15:12
M. Damon M. Damon
235
asked Dec 5 '18 at 15:12
M. Damon M. Damon
235
235
$begingroup$
One of them must be the least element of ${x,y}$ because there are no other elements in the set, and it has a least element.
$endgroup$
– dbx
Dec 5 '18 at 15:13
$begingroup$
Where does this proof come from? In the definitions as I've learned them, well-ordered relations are defined to be both well-founded and totally ordered, and well-founded relations need not be totally ordered. See also math.stackexchange.com/questions/116642/…
$endgroup$
– J.G.
Dec 5 '18 at 15:19
$begingroup$
The definition my book uses denotes well-ordered relation as a partially-ordered relation on $A$ such that each non-empty subset of $A$ has a least element with respect to $R$.
$endgroup$
– M. Damon
Dec 5 '18 at 15:26
$begingroup$
@J.G.: Well-founded relations have a minimal element in each non-empty set; well-orders have a minimum in each non-empty set.
$endgroup$
– Asaf Karagila♦
Dec 5 '18 at 15:27
$begingroup$
@AsafKaragila In other words, the minimal element of the non-empty set is only required to be unique in the latter case? Fair enough.
$endgroup$
– J.G.
Dec 5 '18 at 15:31
|
show 6 more comments
$begingroup$
One of them must be the least element of ${x,y}$ because there are no other elements in the set, and it has a least element.
$endgroup$
– dbx
Dec 5 '18 at 15:13
$begingroup$
Where does this proof come from? In the definitions as I've learned them, well-ordered relations are defined to be both well-founded and totally ordered, and well-founded relations need not be totally ordered. See also math.stackexchange.com/questions/116642/…
$endgroup$
– J.G.
Dec 5 '18 at 15:19
$begingroup$
The definition my book uses denotes well-ordered relation as a partially-ordered relation on $A$ such that each non-empty subset of $A$ has a least element with respect to $R$.
$endgroup$
– M. Damon
Dec 5 '18 at 15:26
$begingroup$
@J.G.: Well-founded relations have a minimal element in each non-empty set; well-orders have a minimum in each non-empty set.
$endgroup$
– Asaf Karagila♦
Dec 5 '18 at 15:27
$begingroup$
@AsafKaragila In other words, the minimal element of the non-empty set is only required to be unique in the latter case? Fair enough.
$endgroup$
– J.G.
Dec 5 '18 at 15:31
$begingroup$
One of them must be the least element of ${x,y}$ because there are no other elements in the set, and it has a least element.
$endgroup$
– dbx
Dec 5 '18 at 15:13
$begingroup$
One of them must be the least element of ${x,y}$ because there are no other elements in the set, and it has a least element.
$endgroup$
– dbx
Dec 5 '18 at 15:13
$begingroup$
Where does this proof come from? In the definitions as I've learned them, well-ordered relations are defined to be both well-founded and totally ordered, and well-founded relations need not be totally ordered. See also math.stackexchange.com/questions/116642/…
$endgroup$
– J.G.
Dec 5 '18 at 15:19
$begingroup$
Where does this proof come from? In the definitions as I've learned them, well-ordered relations are defined to be both well-founded and totally ordered, and well-founded relations need not be totally ordered. See also math.stackexchange.com/questions/116642/…
$endgroup$
– J.G.
Dec 5 '18 at 15:19
$begingroup$
The definition my book uses denotes well-ordered relation as a partially-ordered relation on $A$ such that each non-empty subset of $A$ has a least element with respect to $R$.
$endgroup$
– M. Damon
Dec 5 '18 at 15:26
$begingroup$
The definition my book uses denotes well-ordered relation as a partially-ordered relation on $A$ such that each non-empty subset of $A$ has a least element with respect to $R$.
$endgroup$
– M. Damon
Dec 5 '18 at 15:26
$begingroup$
@J.G.: Well-founded relations have a minimal element in each non-empty set; well-orders have a minimum in each non-empty set.
$endgroup$
– Asaf Karagila♦
Dec 5 '18 at 15:27
$begingroup$
@J.G.: Well-founded relations have a minimal element in each non-empty set; well-orders have a minimum in each non-empty set.
$endgroup$
– Asaf Karagila♦
Dec 5 '18 at 15:27
$begingroup$
@AsafKaragila In other words, the minimal element of the non-empty set is only required to be unique in the latter case? Fair enough.
$endgroup$
– J.G.
Dec 5 '18 at 15:31
$begingroup$
@AsafKaragila In other words, the minimal element of the non-empty set is only required to be unique in the latter case? Fair enough.
$endgroup$
– J.G.
Dec 5 '18 at 15:31
|
show 6 more comments
1 Answer
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How could none of them be the last element? There is no other element in ${x,y}$ besides $x$ and $y$.
answered Dec 5 '18 at 15:16
José Carlos SantosJosé Carlos Santos
156k22126227
$endgroup$
$begingroup$
+1 To expand on this answer, recall that ${x,y}$ is nonempty since $x$ and $y$ are elements (it is still possible $x$ and $y$ are equal). Because $R$ is well-ordered, there must exist a $R$-least-element in ${x,y}$.
$endgroup$
– Alberto Takase
Dec 6 '18 at 10:39
add a comment |
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How could none of them be the last element? There is no other element in ${x,y}$ besides $x$ and $y$.
answered Dec 5 '18 at 15:16
José Carlos SantosJosé Carlos Santos
156k22126227
$endgroup$
$begingroup$
+1 To expand on this answer, recall that ${x,y}$ is nonempty since $x$ and $y$ are elements (it is still possible $x$ and $y$ are equal). Because $R$ is well-ordered, there must exist a $R$-least-element in ${x,y}$.
$endgroup$
– Alberto Takase
Dec 6 '18 at 10:39
add a comment |
$begingroup$
How could none of them be the last element? There is no other element in ${x,y}$ besides $x$ and $y$.
answered Dec 5 '18 at 15:16
José Carlos SantosJosé Carlos Santos
156k22126227
$endgroup$
$begingroup$
+1 To expand on this answer, recall that ${x,y}$ is nonempty since $x$ and $y$ are elements (it is still possible $x$ and $y$ are equal). Because $R$ is well-ordered, there must exist a $R$-least-element in ${x,y}$.
$endgroup$
– Alberto Takase
Dec 6 '18 at 10:39
add a comment |
$begingroup$
How could none of them be the last element? There is no other element in ${x,y}$ besides $x$ and $y$.
answered Dec 5 '18 at 15:16
José Carlos SantosJosé Carlos Santos
156k22126227
$endgroup$
How could none of them be the last element? There is no other element in ${x,y}$ besides $x$ and $y$.
answered Dec 5 '18 at 15:16
José Carlos SantosJosé Carlos Santos
156k22126227
answered Dec 5 '18 at 15:16
José Carlos SantosJosé Carlos Santos
156k22126227
answered Dec 5 '18 at 15:16
José Carlos SantosJosé Carlos Santos
156k22126227
answered Dec 5 '18 at 15:16
José Carlos SantosJosé Carlos Santos
156k22126227
156k22126227
$begingroup$
+1 To expand on this answer, recall that ${x,y}$ is nonempty since $x$ and $y$ are elements (it is still possible $x$ and $y$ are equal). Because $R$ is well-ordered, there must exist a $R$-least-element in ${x,y}$.
$endgroup$
– Alberto Takase
Dec 6 '18 at 10:39
add a comment |
$begingroup$
+1 To expand on this answer, recall that ${x,y}$ is nonempty since $x$ and $y$ are elements (it is still possible $x$ and $y$ are equal). Because $R$ is well-ordered, there must exist a $R$-least-element in ${x,y}$.
$endgroup$
– Alberto Takase
Dec 6 '18 at 10:39
$begingroup$
+1 To expand on this answer, recall that ${x,y}$ is nonempty since $x$ and $y$ are elements (it is still possible $x$ and $y$ are equal). Because $R$ is well-ordered, there must exist a $R$-least-element in ${x,y}$.
$endgroup$
– Alberto Takase
Dec 6 '18 at 10:39
$begingroup$
+1 To expand on this answer, recall that ${x,y}$ is nonempty since $x$ and $y$ are elements (it is still possible $x$ and $y$ are equal). Because $R$ is well-ordered, there must exist a $R$-least-element in ${x,y}$.
$endgroup$
– Alberto Takase
Dec 6 '18 at 10:39
add a comment |
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$begingroup$
One of them must be the least element of ${x,y}$ because there are no other elements in the set, and it has a least element.
$endgroup$
– dbx
Dec 5 '18 at 15:13
$begingroup$
Where does this proof come from? In the definitions as I've learned them, well-ordered relations are defined to be both well-founded and totally ordered, and well-founded relations need not be totally ordered. See also math.stackexchange.com/questions/116642/…
$endgroup$
– J.G.
Dec 5 '18 at 15:19
$begingroup$
The definition my book uses denotes well-ordered relation as a partially-ordered relation on $A$ such that each non-empty subset of $A$ has a least element with respect to $R$.
$endgroup$
– M. Damon
Dec 5 '18 at 15:26
$begingroup$
@J.G.: Well-founded relations have a minimal element in each non-empty set; well-orders have a minimum in each non-empty set.
$endgroup$
– Asaf Karagila♦
Dec 5 '18 at 15:27
$begingroup$
@AsafKaragila In other words, the minimal element of the non-empty set is only required to be unique in the latter case? Fair enough.
$endgroup$
– J.G.
Dec 5 '18 at 15:31