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!










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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















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!










share|cite|improve this 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













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!










share|cite|improve this question















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






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edited Nov 19 at 23:56









dantopa

6,31131741




6,31131741










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


















  • 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










1 Answer
1






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.






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  • 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


















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.






share|cite|improve this answer























  • 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















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.






share|cite|improve this answer























  • 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













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.






share|cite|improve this answer














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.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Nov 19 at 20:20

























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


















  • 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



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