Is $le$ defined by $langle 𝑥_1, 𝑦_1 rangle le langle 𝑥_2, 𝑦_2 rangle$, if $𝑥_1 le 𝑥_2...
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I need to tell if the relation $le$ defined by relationship $langle 𝑥_1, 𝑦_1 rangle le langle 𝑥_2, 𝑦_2 rangle$, if $𝑥_1 le 𝑥_2$ $land$ $𝑦_1 ge 𝑦_2$ linear order?
I have already proven that the relation is an order, but I need to decide if it is a linear order and why. Thanks for any help!
relations order-theory
edited yesterday
Martin Sleziak
44.6k7115269
asked Nov 19 at 19:46
Jack
213
closed as unclear what you're asking by Lord Shark the Unknown, Leucippus, user10354138, Shailesh, Chinnapparaj R Nov 20 at 2:55
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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up vote
-2
down vote
favorite
I need to tell if the relation $le$ defined by relationship $langle 𝑥_1, 𝑦_1 rangle le langle 𝑥_2, 𝑦_2 rangle$, if $𝑥_1 le 𝑥_2$ $land$ $𝑦_1 ge 𝑦_2$ linear order?
I have already proven that the relation is an order, but I need to decide if it is a linear order and why. Thanks for any help!
relations order-theory
edited yesterday
Martin Sleziak
44.6k7115269
asked Nov 19 at 19:46
Jack
213
closed as unclear what you're asking by Lord Shark the Unknown, Leucippus, user10354138, Shailesh, Chinnapparaj R Nov 20 at 2:55
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
In what set are $x1, y1, x2, y2$ elements? One can only define a linear order on a specified set.
– amWhy
Nov 19 at 22:49
add a comment |
up vote
-2
down vote
favorite
up vote
-2
down vote
favorite
I need to tell if the relation $le$ defined by relationship $langle 𝑥_1, 𝑦_1 rangle le langle 𝑥_2, 𝑦_2 rangle$, if $𝑥_1 le 𝑥_2$ $land$ $𝑦_1 ge 𝑦_2$ linear order?
I have already proven that the relation is an order, but I need to decide if it is a linear order and why. Thanks for any help!
relations order-theory
edited yesterday
Martin Sleziak
44.6k7115269
asked Nov 19 at 19:46
Jack
213
I need to tell if the relation $le$ defined by relationship $langle 𝑥_1, 𝑦_1 rangle le langle 𝑥_2, 𝑦_2 rangle$, if $𝑥_1 le 𝑥_2$ $land$ $𝑦_1 ge 𝑦_2$ linear order?
I have already proven that the relation is an order, but I need to decide if it is a linear order and why. Thanks for any help!
relations order-theory
relations order-theory
edited yesterday
Martin Sleziak
44.6k7115269
asked Nov 19 at 19:46
Jack
213
edited yesterday
Martin Sleziak
44.6k7115269
asked Nov 19 at 19:46
Jack
213
edited yesterday
Martin Sleziak
44.6k7115269
edited yesterday
Martin Sleziak
44.6k7115269
edited yesterday
Martin Sleziak
44.6k7115269
44.6k7115269
asked Nov 19 at 19:46
Jack
213
asked Nov 19 at 19:46
Jack
213
asked Nov 19 at 19:46
Jack
213
213
closed as unclear what you're asking by Lord Shark the Unknown, Leucippus, user10354138, Shailesh, Chinnapparaj R Nov 20 at 2:55
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Lord Shark the Unknown, Leucippus, user10354138, Shailesh, Chinnapparaj R Nov 20 at 2:55
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
In what set are $x1, y1, x2, y2$ elements? One can only define a linear order on a specified set.
– amWhy
Nov 19 at 22:49
add a comment |
In what set are $x1, y1, x2, y2$ elements? One can only define a linear order on a specified set.
– amWhy
Nov 19 at 22:49
In what set are $x1, y1, x2, y2$ elements? One can only define a linear order on a specified set.
– amWhy
Nov 19 at 22:49
In what set are $x1, y1, x2, y2$ elements? One can only define a linear order on a specified set.
– amWhy
Nov 19 at 22:49
add a comment |
1 Answer
1
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A linear order needs two elements to be comparable i.e. for $left<x_1,y_1right>,left<x_2,y_2right>$
we have either
$$left<x_1,y_1right>preceqleft<x_2,y_2right>$$
or $$left<x_2,y_2right>preceqleft<x_1,y_1right>.$$
If you can find two pairs where neither is the case, you have proven that the relation can't be a linear order.
answered Nov 19 at 20:01
weee
4608
so ⟨1, 2⟩ ⪯ ⟨2, 1⟩ and ⟨2, 1⟩ ⪰ ⟨1, 2⟩?
– Jack
Nov 19 at 20:16
that's the same thing.
– weee
Nov 19 at 20:20
can you help me find any pairs?
– Jack
Nov 19 at 20:21
1
try $<1,1>$ and $<2,2>$
– weee
Nov 19 at 21:52
but ⟨1, 2⟩ ⪯ ⟨2, 1⟩ is true, right?
– Jack
Nov 19 at 21:55
|
show 1 more comment
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
A linear order needs two elements to be comparable i.e. for $left<x_1,y_1right>,left<x_2,y_2right>$
we have either
$$left<x_1,y_1right>preceqleft<x_2,y_2right>$$
or $$left<x_2,y_2right>preceqleft<x_1,y_1right>.$$
If you can find two pairs where neither is the case, you have proven that the relation can't be a linear order.
answered Nov 19 at 20:01
weee
4608
so ⟨1, 2⟩ ⪯ ⟨2, 1⟩ and ⟨2, 1⟩ ⪰ ⟨1, 2⟩?
– Jack
Nov 19 at 20:16
that's the same thing.
– weee
Nov 19 at 20:20
can you help me find any pairs?
– Jack
Nov 19 at 20:21
1
try $<1,1>$ and $<2,2>$
– weee
Nov 19 at 21:52
but ⟨1, 2⟩ ⪯ ⟨2, 1⟩ is true, right?
– Jack
Nov 19 at 21:55
|
show 1 more comment
up vote
2
down vote
A linear order needs two elements to be comparable i.e. for $left<x_1,y_1right>,left<x_2,y_2right>$
we have either
$$left<x_1,y_1right>preceqleft<x_2,y_2right>$$
or $$left<x_2,y_2right>preceqleft<x_1,y_1right>.$$
If you can find two pairs where neither is the case, you have proven that the relation can't be a linear order.
answered Nov 19 at 20:01
weee
4608
so ⟨1, 2⟩ ⪯ ⟨2, 1⟩ and ⟨2, 1⟩ ⪰ ⟨1, 2⟩?
– Jack
Nov 19 at 20:16
that's the same thing.
– weee
Nov 19 at 20:20
can you help me find any pairs?
– Jack
Nov 19 at 20:21
1
try $<1,1>$ and $<2,2>$
– weee
Nov 19 at 21:52
but ⟨1, 2⟩ ⪯ ⟨2, 1⟩ is true, right?
– Jack
Nov 19 at 21:55
|
show 1 more comment
up vote
2
down vote
up vote
2
down vote
A linear order needs two elements to be comparable i.e. for $left<x_1,y_1right>,left<x_2,y_2right>$
we have either
$$left<x_1,y_1right>preceqleft<x_2,y_2right>$$
or $$left<x_2,y_2right>preceqleft<x_1,y_1right>.$$
If you can find two pairs where neither is the case, you have proven that the relation can't be a linear order.
answered Nov 19 at 20:01
weee
4608
A linear order needs two elements to be comparable i.e. for $left<x_1,y_1right>,left<x_2,y_2right>$
we have either
$$left<x_1,y_1right>preceqleft<x_2,y_2right>$$
or $$left<x_2,y_2right>preceqleft<x_1,y_1right>.$$
If you can find two pairs where neither is the case, you have proven that the relation can't be a linear order.
answered Nov 19 at 20:01
weee
4608
edited Nov 19 at 20:20
answered Nov 19 at 20:01
weee
4608
answered Nov 19 at 20:01
weee
4608
answered Nov 19 at 20:01
weee
4608
4608
so ⟨1, 2⟩ ⪯ ⟨2, 1⟩ and ⟨2, 1⟩ ⪰ ⟨1, 2⟩?
– Jack
Nov 19 at 20:16
that's the same thing.
– weee
Nov 19 at 20:20
can you help me find any pairs?
– Jack
Nov 19 at 20:21
1
try $<1,1>$ and $<2,2>$
– weee
Nov 19 at 21:52
but ⟨1, 2⟩ ⪯ ⟨2, 1⟩ is true, right?
– Jack
Nov 19 at 21:55
|
show 1 more comment
so ⟨1, 2⟩ ⪯ ⟨2, 1⟩ and ⟨2, 1⟩ ⪰ ⟨1, 2⟩?
– Jack
Nov 19 at 20:16
that's the same thing.
– weee
Nov 19 at 20:20
can you help me find any pairs?
– Jack
Nov 19 at 20:21
1
try $<1,1>$ and $<2,2>$
– weee
Nov 19 at 21:52
but ⟨1, 2⟩ ⪯ ⟨2, 1⟩ is true, right?
– Jack
Nov 19 at 21:55
so ⟨1, 2⟩ ⪯ ⟨2, 1⟩ and ⟨2, 1⟩ ⪰ ⟨1, 2⟩?
– Jack
Nov 19 at 20:16
so ⟨1, 2⟩ ⪯ ⟨2, 1⟩ and ⟨2, 1⟩ ⪰ ⟨1, 2⟩?
– Jack
Nov 19 at 20:16
that's the same thing.
– weee
Nov 19 at 20:20
that's the same thing.
– weee
Nov 19 at 20:20
can you help me find any pairs?
– Jack
Nov 19 at 20:21
can you help me find any pairs?
– Jack
Nov 19 at 20:21
1
1
try $<1,1>$ and $<2,2>$
– weee
Nov 19 at 21:52
try $<1,1>$ and $<2,2>$
– weee
Nov 19 at 21:52
but ⟨1, 2⟩ ⪯ ⟨2, 1⟩ is true, right?
– Jack
Nov 19 at 21:55
but ⟨1, 2⟩ ⪯ ⟨2, 1⟩ is true, right?
– Jack
Nov 19 at 21:55
|
show 1 more comment
In what set are $x1, y1, x2, y2$ elements? One can only define a linear order on a specified set.
– amWhy
Nov 19 at 22:49