# supremum and infinum questions

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X = [1,3]

Y = (1,3]

X - Y ={x-y | x ∈ X, y ∈ Y}

I have two questions regarding this:

a) Find the value of X-Y

b) Are sup(X-Y) and inf(X-Y) elements of X-Y?

In my work so far I realise that for a) I have 1-3=-2 by (X's smallest element) - (Y's largest value).

But now for Y there is no infinum so the largest value of X-Y is not possible.

So it the final answer just (2)?

Also for part b) is it only that inf(X-Y) is not possible

Y = (1,3]

X - Y ={x-y | x ∈ X, y ∈ Y}

I have two questions regarding this:

a) Find the value of X-Y

b) Are sup(X-Y) and inf(X-Y) elements of X-Y?

In my work so far I realise that for a) I have 1-3=-2 by (X's smallest element) - (Y's largest value).

But now for Y there is no infinum so the largest value of X-Y is not possible.

So it the final answer just (2)?

Also for part b) is it only that inf(X-Y) is not possible

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#2

N.B. It's "infi

Largest/smallest value of X-Y?

b) Are sup(X-Y) and inf(X-Y) elements of X-Y?

In my work so far I realise that for a) I have 1-3=-2 by (X's smallest element) - (Y's largest value).

But now for Y there is no infinum so the largest value of X-Y is not possible.
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.

**m**um" not "infinum"
(Original post by

X = [1,3]

Y = (1,3]

X - Y ={x-y | x ∈ X, y ∈ Y}

I have two questions regarding this:

a) Find the value of X-Y

**ineffablemind**)X = [1,3]

Y = (1,3]

X - Y ={x-y | x ∈ X, y ∈ Y}

I have two questions regarding this:

a) Find the value of X-Y

b) Are sup(X-Y) and inf(X-Y) elements of X-Y?

In my work so far I realise that for a) I have 1-3=-2 by (X's smallest element) - (Y's largest value).

But now for Y there is no infinum so the largest value of X-Y 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.

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(Original post by

N.B. It's "infi

Largest/smallest value of X-Y?

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.

**atsruser**)N.B. It's "infi

**m**um" not "infinum"Largest/smallest value of X-Y?

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 1-3=-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 X-Y.

For the inf(X-Y) 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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#4

(Original post by

Thank you for your reply.

Yes so for part a) I have 1-3=-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).

**ineffablemind**)Thank you for your reply.

Yes so for part a) I have 1-3=-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 X-Y.

For the inf(X-Y) would I then have to calculate inf(X) and inf(Y) separatley. And then subrtract the difference?

For the inf(X-Y) 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 X-Y of the form . What happens when you apply that to the max/min of X?

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#5

(Original post by

There seems to be a word missing from your part a) - is it largest or smallest value?

**atsruser**)There seems to be a word missing from your part a) - is it largest or smallest value?

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#6

(Original post by

It could be that the answer expected is a set. It is possible to write X - Y as an interval, after all...

**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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So:

inf(A) = 1.

sup(A)=3

inf(B) = infinity

sup(B) = 3

So A-B = 3-1 = 2?

Then inf(A-B) is not possible

And

sup(A-B) = 0

Have I got this correct now?

inf(A) = 1.

sup(A)=3

inf(B) = infinity

sup(B) = 3

So A-B = 3-1 = 2?

Then inf(A-B) is not possible

And

sup(A-B) = 0

Have I got this correct now?

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#8

(Original post by

So:

inf(A) = 1.

sup(A)=3

inf(B) = infinity

sup(B) = 3

So A-B = 3-1 = 2?

Then inf(A-B) is not possible

And

sup(A-B) = 0

Have I got this correct now?

**ineffablemind**)So:

inf(A) = 1.

sup(A)=3

inf(B) = infinity

sup(B) = 3

So A-B = 3-1 = 2?

Then inf(A-B) is not possible

And

sup(A-B) = 0

Have I got this correct now?

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#9

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(X-Y) and inf(X-Y) elements of X - Y?

inf(X) = 1 and sup(X) = 3

Then:

inf(Y) = 1 and sup(Y) = 3

inf(X-Y) = inf X - sup Y = 1 - 3 = -2 sup(X-Y) = sup X - inf Y = 3 - 1 = 2

So to solve my question:

Since inf(X-Y) = -2 and sup(X-Y) = 2. Then from the interval notation 2 is not bounded in the interval notation, whereas -2 is. So then inf(X-Y) is an element of X-Y and sup(X-Y) is not an element of X-Y.

Would people agree this is correct?

Anyway I am at the stage:

X−Y = [−2;2)

So going back to my question that

Are sup(X-Y) and inf(X-Y) elements of X - Y?

inf(X) = 1 and sup(X) = 3

Then:

inf(Y) = 1 and sup(Y) = 3

inf(X-Y) = inf X - sup Y = 1 - 3 = -2 sup(X-Y) = sup X - inf Y = 3 - 1 = 2

So to solve my question:

Since inf(X-Y) = -2 and sup(X-Y) = 2. Then from the interval notation 2 is not bounded in the interval notation, whereas -2 is. So then inf(X-Y) is an element of X-Y and sup(X-Y) is not an element of X-Y.

Would people agree this is correct?

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#10

(Original post by

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(X-Y) and inf(X-Y) elements of X - Y?

inf(X) = 1 and sup(X) = 3

Then:

inf(Y) = 1 and sup(Y) = 3

inf(X-Y) = inf X - sup Y = 1 - 3 = -2 sup(X-Y) = sup X - inf Y = 3 - 1 = 2

So to solve my question:

Since inf(X-Y) = -2 and sup(X-Y) = 2. Then from the interval notation 2 is not bounded in the interval notation, whereas -2 is. So then inf(X-Y) is an element of X-Y and sup(X-Y) is not an element of X-Y.

Would people agree this is correct?

**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(X-Y) and inf(X-Y) elements of X - Y?

inf(X) = 1 and sup(X) = 3

Then:

inf(Y) = 1 and sup(Y) = 3

inf(X-Y) = inf X - sup Y = 1 - 3 = -2 sup(X-Y) = sup X - inf Y = 3 - 1 = 2

So to solve my question:

Since inf(X-Y) = -2 and sup(X-Y) = 2. Then from the interval notation 2 is not bounded in the interval notation, whereas -2 is. So then inf(X-Y) is an element of X-Y and sup(X-Y) is not an element of X-Y.

Would people agree this is correct?

**wrong**in what you wrote, but I would say the two statements:

X-Y = [-2, 2)

and

inf(X-Y) = 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(X-Y) = 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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#11

(Original post by

There's nothing

X-Y = [-2, 2)

and

inf(X-Y) = 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(X-Y) = 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).]

**DFranklin**)There's nothing

**wrong**in what you wrote, but I would say the two statements:X-Y = [-2, 2)

and

inf(X-Y) = 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(X-Y) = 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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#12

**DFranklin**)

There's nothing

**wrong**in what you wrote, but I would say the two statements:

X-Y = [-2, 2)

and

inf(X-Y) = 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(X-Y) = 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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#13

(Original post by

hey sir can I just ask you - how would you justify how inf(X-Y) 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<= x-y < 2 hence -2 is an element and 2 is not. What would you say is better

**alexgreyx**)hey sir can I just ask you - how would you justify how inf(X-Y) 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<= x-y < 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.

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#14

(Original post by

Do you mean you want to show that -2 is indeed an element of X-Y? If so, you just need to exhibit a value of and such that (then x-y = -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.

**Zacken**)Do you mean you want to show that -2 is indeed an element of X-Y? If so, you just need to exhibit a value of and such that (then x-y = -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.

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#15

(Original post by

Yes you are correct, I have my full solution -althought I didn't define the proof that you stated.

**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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#16

(Original post by

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

**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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