Checking whether the given Cartesian product has the least upper bound property or not .
Clash Royale CLAN TAG#URR8PPP
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Problem
a)Does $[0,1] times [0,1]$ in the dictionary order have the least upper bound property ?
b)What about $[0,1] times [0,1)$ ?
c) What about $[ 0,1)times [0,1]$?
My understanding of dictionary order -
An order relation $<$ on $Atimes B$ is defined:
$(a_1,b_1) < (a_2,b_2)$ if $a_1 <_A a_2$, or if $a_1 = a_2$ and $b_1 <_B b_2$.
It is called dictionary order relation on $Atimes B$.
How to proceed ?
$Attempt$
Let X = [0,1]×[0,1]
Suppose that A$subset$ X is a nonempty subset with upper bound.
Xo := exists yin[0,1]:(x,y)in A$ $subset$ [0,1] .
Xo has the least upper bound xo.
Yo:= $yin[0,1]$ $subset$ [0,1] .
If Yo is empty (xo,0) is the least upper bound . If
Yo is non-empty,then Yo is bounded above by 1,so (xo,yo) is the least upper bound of A.
real-analysis elementary-set-theory
asked Aug 14 at 10:20
blue boy
613311
 |Â
show 9 more comments
up vote
2
down vote
favorite
Problem
a)Does $[0,1] times [0,1]$ in the dictionary order have the least upper bound property ?
b)What about $[0,1] times [0,1)$ ?
c) What about $[ 0,1)times [0,1]$?
My understanding of dictionary order -
An order relation $<$ on $Atimes B$ is defined:
$(a_1,b_1) < (a_2,b_2)$ if $a_1 <_A a_2$, or if $a_1 = a_2$ and $b_1 <_B b_2$.
It is called dictionary order relation on $Atimes B$.
How to proceed ?
$Attempt$
Let X = [0,1]×[0,1]
Suppose that A$subset$ X is a nonempty subset with upper bound.
Xo := exists yin[0,1]:(x,y)in A$ $subset$ [0,1] .
Xo has the least upper bound xo.
Yo:= $yin[0,1]$ $subset$ [0,1] .
If Yo is empty (xo,0) is the least upper bound . If
Yo is non-empty,then Yo is bounded above by 1,so (xo,yo) is the least upper bound of A.
real-analysis elementary-set-theory
asked Aug 14 at 10:20
blue boy
613311
Why downvote?..
– blue boy
Aug 14 at 10:43
4
x @blue: Probably because it looks like you're seeking to have your homework done for you without needing to understand the material yourself. Exercises like this are an invitation to puzzle it out in your own mind, and thereby get a greater intuitive familiarity with the concepts; if you're just handed a solution you won't actually gain any experience from it.
– Henning Makholm
Aug 14 at 10:45
1
Also, out of sheer curiosity. How did you choose the tags for this question (and the other one you posted around the same time)?
– Asaf Karagila♦
Aug 14 at 11:04
Ok. I will try to show my work next time . I can see how it may seem so.
– blue boy
Aug 14 at 11:25
1
Did you perhaps spend a few minutes reading the excerpts that you get when selecting the tags? How is this a question about foundations of mathematics? Or about set theory (as indicated by the tag excerpt)? Many people worked very hard on these things. Please show some effort on your side when you choose tags. Especially when they have this kind of information with them.
– Asaf Karagila♦
Aug 14 at 11:29
 |Â
show 9 more comments
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Problem
a)Does $[0,1] times [0,1]$ in the dictionary order have the least upper bound property ?
b)What about $[0,1] times [0,1)$ ?
c) What about $[ 0,1)times [0,1]$?
My understanding of dictionary order -
An order relation $<$ on $Atimes B$ is defined:
$(a_1,b_1) < (a_2,b_2)$ if $a_1 <_A a_2$, or if $a_1 = a_2$ and $b_1 <_B b_2$.
It is called dictionary order relation on $Atimes B$.
How to proceed ?
$Attempt$
Let X = [0,1]×[0,1]
Suppose that A$subset$ X is a nonempty subset with upper bound.
Xo := exists yin[0,1]:(x,y)in A$ $subset$ [0,1] .
Xo has the least upper bound xo.
Yo:= $yin[0,1]$ $subset$ [0,1] .
If Yo is empty (xo,0) is the least upper bound . If
Yo is non-empty,then Yo is bounded above by 1,so (xo,yo) is the least upper bound of A.
real-analysis elementary-set-theory
asked Aug 14 at 10:20
blue boy
613311
Problem
a)Does $[0,1] times [0,1]$ in the dictionary order have the least upper bound property ?
b)What about $[0,1] times [0,1)$ ?
c) What about $[ 0,1)times [0,1]$?
My understanding of dictionary order -
An order relation $<$ on $Atimes B$ is defined:
$(a_1,b_1) < (a_2,b_2)$ if $a_1 <_A a_2$, or if $a_1 = a_2$ and $b_1 <_B b_2$.
It is called dictionary order relation on $Atimes B$.
How to proceed ?
$Attempt$
Let X = [0,1]×[0,1]
Suppose that A$subset$ X is a nonempty subset with upper bound.
Xo := exists yin[0,1]:(x,y)in A$ $subset$ [0,1] .
Xo has the least upper bound xo.
Yo:= $yin[0,1]$ $subset$ [0,1] .
If Yo is empty (xo,0) is the least upper bound . If
Yo is non-empty,then Yo is bounded above by 1,so (xo,yo) is the least upper bound of A.
real-analysis elementary-set-theory
asked Aug 14 at 10:20
blue boy
613311
edited Aug 14 at 15:46
asked Aug 14 at 10:20
blue boy
613311
asked Aug 14 at 10:20
blue boy
613311
asked Aug 14 at 10:20
blue boy
613311
613311
Why downvote?..
– blue boy
Aug 14 at 10:43
4
x @blue: Probably because it looks like you're seeking to have your homework done for you without needing to understand the material yourself. Exercises like this are an invitation to puzzle it out in your own mind, and thereby get a greater intuitive familiarity with the concepts; if you're just handed a solution you won't actually gain any experience from it.
– Henning Makholm
Aug 14 at 10:45
1
Also, out of sheer curiosity. How did you choose the tags for this question (and the other one you posted around the same time)?
– Asaf Karagila♦
Aug 14 at 11:04
Ok. I will try to show my work next time . I can see how it may seem so.
– blue boy
Aug 14 at 11:25
1
Did you perhaps spend a few minutes reading the excerpts that you get when selecting the tags? How is this a question about foundations of mathematics? Or about set theory (as indicated by the tag excerpt)? Many people worked very hard on these things. Please show some effort on your side when you choose tags. Especially when they have this kind of information with them.
– Asaf Karagila♦
Aug 14 at 11:29
 |Â
show 9 more comments
Why downvote?..
– blue boy
Aug 14 at 10:43
4
x @blue: Probably because it looks like you're seeking to have your homework done for you without needing to understand the material yourself. Exercises like this are an invitation to puzzle it out in your own mind, and thereby get a greater intuitive familiarity with the concepts; if you're just handed a solution you won't actually gain any experience from it.
– Henning Makholm
Aug 14 at 10:45
1
Also, out of sheer curiosity. How did you choose the tags for this question (and the other one you posted around the same time)?
– Asaf Karagila♦
Aug 14 at 11:04
Ok. I will try to show my work next time . I can see how it may seem so.
– blue boy
Aug 14 at 11:25
1
Did you perhaps spend a few minutes reading the excerpts that you get when selecting the tags? How is this a question about foundations of mathematics? Or about set theory (as indicated by the tag excerpt)? Many people worked very hard on these things. Please show some effort on your side when you choose tags. Especially when they have this kind of information with them.
– Asaf Karagila♦
Aug 14 at 11:29
Why downvote?..
– blue boy
Aug 14 at 10:43
Why downvote?..
– blue boy
Aug 14 at 10:43
4
4
x @blue: Probably because it looks like you're seeking to have your homework done for you without needing to understand the material yourself. Exercises like this are an invitation to puzzle it out in your own mind, and thereby get a greater intuitive familiarity with the concepts; if you're just handed a solution you won't actually gain any experience from it.
– Henning Makholm
Aug 14 at 10:45
x @blue: Probably because it looks like you're seeking to have your homework done for you without needing to understand the material yourself. Exercises like this are an invitation to puzzle it out in your own mind, and thereby get a greater intuitive familiarity with the concepts; if you're just handed a solution you won't actually gain any experience from it.
– Henning Makholm
Aug 14 at 10:45
1
1
Also, out of sheer curiosity. How did you choose the tags for this question (and the other one you posted around the same time)?
– Asaf Karagila♦
Aug 14 at 11:04
Also, out of sheer curiosity. How did you choose the tags for this question (and the other one you posted around the same time)?
– Asaf Karagila♦
Aug 14 at 11:04
Ok. I will try to show my work next time . I can see how it may seem so.
– blue boy
Aug 14 at 11:25
Ok. I will try to show my work next time . I can see how it may seem so.
– blue boy
Aug 14 at 11:25
1
1
Did you perhaps spend a few minutes reading the excerpts that you get when selecting the tags? How is this a question about foundations of mathematics? Or about set theory (as indicated by the tag excerpt)? Many people worked very hard on these things. Please show some effort on your side when you choose tags. Especially when they have this kind of information with them.
– Asaf Karagila♦
Aug 14 at 11:29
Did you perhaps spend a few minutes reading the excerpts that you get when selecting the tags? How is this a question about foundations of mathematics? Or about set theory (as indicated by the tag excerpt)? Many people worked very hard on these things. Please show some effort on your side when you choose tags. Especially when they have this kind of information with them.
– Asaf Karagila♦
Aug 14 at 11:29
 |Â
show 9 more comments
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Why downvote?..
– blue boy
Aug 14 at 10:43
4
x @blue: Probably because it looks like you're seeking to have your homework done for you without needing to understand the material yourself. Exercises like this are an invitation to puzzle it out in your own mind, and thereby get a greater intuitive familiarity with the concepts; if you're just handed a solution you won't actually gain any experience from it.
– Henning Makholm
Aug 14 at 10:45
1
Also, out of sheer curiosity. How did you choose the tags for this question (and the other one you posted around the same time)?
– Asaf Karagila♦
Aug 14 at 11:04
Ok. I will try to show my work next time . I can see how it may seem so.
– blue boy
Aug 14 at 11:25
1
Did you perhaps spend a few minutes reading the excerpts that you get when selecting the tags? How is this a question about foundations of mathematics? Or about set theory (as indicated by the tag excerpt)? Many people worked very hard on these things. Please show some effort on your side when you choose tags. Especially when they have this kind of information with them.
– Asaf Karagila♦
Aug 14 at 11:29