X = [1,3]
Y = (1,3]
X  Y ={xy  x ∈ X, y ∈ Y}
I have two questions regarding this:
a) Find the value of XY
b) Are sup(XY) and inf(XY) elements of XY?
In my work so far I realise that for a) I have 13=2 by (X's smallest element)  (Y's largest value).
But now for Y there is no infinum so the largest value of XY is not possible.
So it the final answer just (2)?
Also for part b) is it only that inf(XY) is not possible

ineffablemind
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 05022017 20:48

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 05022017 21:28
N.B. It's "infimum" not "infinum"
(Original post by ineffablemind)
X = [1,3]
Y = (1,3]
X  Y ={xy  x ∈ X, y ∈ Y}
I have two questions regarding this:
a) Find the value of XY
b) Are sup(XY) and inf(XY) elements of XY?
In my work so far I realise that for a) I have 13=2 by (X's smallest element)  (Y's largest value).
But now for Y there is no infinum so the largest value of XY is not possible.
The infimum is the greatest lower bound. If you take a number A that is larger than 1, but smaller than 3 i.e. then there is an element of Y that is less than A e.g. . So no number bigger than 1 is a lower bound. However for all , so 1 is a lower bound. So 1 is the greatest lower bound. 
ineffablemind
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 05022017 22:22
(Original post by atsruser)
N.B. It's "infimum" not "infinum"
Largest/smallest value of XY?
Y does indeed have an infimum; it's 1.
The infimum is the greatest lower bound. If you take a number A that is larger than 1, but smaller than 3 i.e. then there is an element of Y that is less than A e.g. . So no number bigger than 1 is a lower bound. However for all , so 1 is a lower bound. So 1 is the greatest lower bound.
Yes so for part a) I have 13=2 by (X's smallest element)  (Y's largest value) which I consider to be correct. Is the answer simply 2, or do I need to bracket it off such as (2).
But I have that for Y there is no smallest possible element as you suggested that 1 is the infimum. Hence there will be no largest element possible for XY.
For the inf(XY) would I then have to calculate inf(X) and inf(Y) separatley. And then subrtract the difference?
Thank you for clarifying that I am grateful for your help 
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 05022017 22:59
(Original post by ineffablemind)
Thank you for your reply.
Yes so for part a) I have 13=2 by (X's smallest element)  (Y's largest value) which I consider to be correct. Is the answer simply 2, or do I need to bracket it off such as (2).
But I have that for Y there is no smallest possible element as you suggested that 1 is the infimum. Hence there will be no largest element possible for XY.
For the inf(XY) would I then have to calculate inf(X) and inf(Y) separatley. And then subrtract the difference?
So using this endpoint, you will be forming elements in XY of the form . What happens when you apply that to the max/min of X? 
DFranklin
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 05022017 23:29
(Original post by atsruser)
There seems to be a word missing from your part a)  is it largest or smallest value? 
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 06022017 12:15
(Original post by DFranklin)
It could be that the answer expected is a set. It is possible to write X  Y as an interval, after all... 
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 08022017 13:37
So:
inf(A) = 1.
sup(A)=3
inf(B) = infinity
sup(B) = 3
So AB = 31 = 2?
Then inf(AB) is not possible
And
sup(AB) = 0
Have I got this correct now? 
DFranklin
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 08022017 13:40
(Original post by ineffablemind)
So:
inf(A) = 1.
sup(A)=3
inf(B) = infinity
sup(B) = 3
So AB = 31 = 2?
Then inf(AB) is not possible
And
sup(AB) = 0
Have I got this correct now? 
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 13022017 22:48
I am ineffablemind I can't remember my password omg:
Anyway I am at the stage:
X−Y = [−2;2)
So going back to my question that
Are sup(XY) and inf(XY) elements of X  Y?
inf(X) = 1 and sup(X) = 3
Then:
inf(Y) = 1 and sup(Y) = 3
inf(XY) = inf X  sup Y = 1  3 = 2 sup(XY) = sup X  inf Y = 3  1 = 2
So to solve my question:
Since inf(XY) = 2 and sup(XY) = 2. Then from the interval notation 2 is not bounded in the interval notation, whereas 2 is. So then inf(XY) is an element of XY and sup(XY) is not an element of XY.
Would people agree this is correct? 
DFranklin
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 14022017 01:03
(Original post by alexgreyx)
I am ineffablemind I can't remember my password omg:
Anyway I am at the stage:
X−Y = [−2;2)
So going back to my question that
Are sup(XY) and inf(XY) elements of X  Y?
inf(X) = 1 and sup(X) = 3
Then:
inf(Y) = 1 and sup(Y) = 3
inf(XY) = inf X  sup Y = 1  3 = 2 sup(XY) = sup X  inf Y = 3  1 = 2
So to solve my question:
Since inf(XY) = 2 and sup(XY) = 2. Then from the interval notation 2 is not bounded in the interval notation, whereas 2 is. So then inf(XY) is an element of XY and sup(XY) is not an element of XY.
Would people agree this is correct?
XY = [2, 2)
and
inf(XY) = inf X  sup Y
need justification. They are, basically, the two "meaty" parts of the question, and in both cases you've just said "this is true" without any justification.
[If you've had a proof that inf(XY) = inf X  sup Y, then that's fine, you don't need to prove it in an exercise (but you probably would need to reproduce the proof if this was an exam question).] 
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 14022017 11:06
(Original post by DFranklin)
There's nothing wrong in what you wrote, but I would say the two statements:
XY = [2, 2)
and
inf(XY) = inf X  sup Y
need justification. They are, basically, the two "meaty" parts of the question, and in both cases you've just said "this is true" without any justification.
[If you've had a proof that inf(XY) = inf X  sup Y, then that's fine, you don't need to prove it in an exercise (but you probably would need to reproduce the proof if this was an exam question).]
Thanks 
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 14022017 21:02
(Original post by DFranklin)
There's nothing wrong in what you wrote, but I would say the two statements:
XY = [2, 2)
and
inf(XY) = inf X  sup Y
need justification. They are, basically, the two "meaty" parts of the question, and in both cases you've just said "this is true" without any justification.
[If you've had a proof that inf(XY) = inf X  sup Y, then that's fine, you don't need to prove it in an exercise (but you probably would need to reproduce the proof if this was an exam question).] 
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 14022017 23:33
(Original post by alexgreyx)
hey sir can I just ask you  how would you justify how inf(XY) is an element etc. I asked my teacher about if how I was saying it like in terms of bounded, he said to express it by interval notation i.e. [2,2) = 2<= xy < 2 hence 2 is an element and 2 is not. What would you say is better
x=1 and y = 3 does the job and you know that both those values are in their respective sets.Last edited by Zacken; 14022017 at 23:35. 
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 15022017 14:18
(Original post by Zacken)
Do you mean you want to show that 2 is indeed an element of XY? If so, you just need to exhibit a value of and such that (then xy = 2 by the definition of )
x=1 and y = 3 does the job and you know that both those values are in their respective sets. 
DFranklin
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 15022017 14:30
(Original post by alexgreyx)
Yes you are correct, I have my full solution althought I didn't define the proof that you stated.
It's perhaps worth emphasiing that with questions like these, there are very few marks for getting the right answers  the marks are for proving they are right. 
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 15022017 17:08
(Original post by DFranklin)
I just feel I should make it clear that by not actually providing detail on the two points I've raised, in my opinion you'd lose almost all the marks for this question if it was actually in an exam. It's not "oh, you've missed a couple of details", it's "there are two key things I expect to see shown in a solution, and you haven't done either of them".
It's perhaps worth emphasiing that with questions like these, there are very few marks for getting the right answers  the marks are for proving they are right.
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