A few Analysis QuestionsWatch

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Thread starter 11 years ago
#1
I've been self studying using the Oxford Analysis I notes and problem sheets which I found online to give myself something to do over the holidays, and I really need some guidance with some of the questions on the second problem sheet.

The first one asks me to prove that there exists a unique real number a such that a^3 = 2. So I use the standard method of considering a set X containing elements x such that x^3 < 2 and showing that the supremum of this set must be the a we're looking for, and so exists by the Completeness Axiom.

So we discount the possibilities that sup X^3 < 2 and sup X^3 > 2 by contradiction, which is where the problem lies. It is intuitivitely clear that the supremum must be such that sup X^3 = 2, since otherwise we could add or subtract some epsilon sufficiently small that either our supremum is less than some element of X or it is not the lowest bound, but I'm not at all sure how to prove that either assumption leads to a contradiction. The notes contain a similar question but they use some obscure looking functions that I don't think I would be able to adapt to this problem.

Might it be possible to use the fact that (c-d)^3 = c^3 - 3c^2d + 3cd^2 - d^3 and let c = sup X and d = epsilon and fiddle around with that until I can show that we can make the 3 terms involving d arbitrarily small such that the whole expression is greater than 2 and factorise (and adapt that for the sup X^3 < 2 case)?

I would be grateful for a small hint to get me going in the right direction (or a confirmation that I've got the right idea, if that's the case).

The second question asks me to prove that there is no subset of the complex numbers to which the positivity axioms can be applied (i.e there's a set P that is closed under addition and multiplication, and for every complex number c either c belongs to the set, negative c belongs to the set or c = 0). I assume it means none apart from the real numbers and its subsets. I don't know where to start with this one, but I should probably look for some kind of contradiction?

I didn't realise this was going to be quite so verbose when I started writing .
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11 years ago
#2
Err, well if and then I think it is fine to just conclude that as x cannot be strictly greater and strictly less than a number at the same time.
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Thread starter 11 years ago
#3
That isn't quite what I meant. I didn't explain things very well, really.

sup x^3 must be either greater than 2, less than 2 or equal to 2. We assume it is greater than 2 and derive a contradiction (there are bounds of X less than the supremum) and assume it's less than 2 and derive a contradiction (there are elements of X greater than the supremum).
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11 years ago
#4
Is the problem deriving the contradictions, or where to go after you have the contradictions?
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Thread starter 11 years ago
#5
It's deriving the contradictions. Once I have those, I can deduce that sup X^3 = 2, so I know the cube root of two exists. Showing it's unique I can manage.
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11 years ago
#6
Okay, so let's define , and we will assume that . Then for all , . Now your goal is to find a number that is in X but greater than s.

Here's a hint:
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Thread starter 11 years ago
#7
Alright, thanks very much, Kolya. I'm going to take a look at that one again tomorrow since it's very late.

Any hints for the second question?
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11 years ago
#8
What exactly do the notes say the positivity axioms are?
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Thread starter 11 years ago
#9
That there exists a set P (the positive numbers) which is closed under addition and multiplication, and that every real number is either = 0, belongs to P, or is such that the additive inverse of that number belongs to P.

(basically a > 0 and b > 0 => a + b > 0,
a > 0 and b > 0 => ab > 0 and
one of -a > 0, a = 0, a > 0 holds)

EDIT: I'm extremely tired, so I'm now going off to bed. Thanks for your help, Kolya, and I'll see if I can handle either of those questions when I'm not so exhausted. Any advice for the second question will be more than welcome, the chances of me working it out on my own are pretty slim.
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11 years ago
#10
For this question, I think we need to see something like we can take two complex numbers say and and consider whether we can get their product to lie outside of our "positive" co-domain. Then remember that either those complex numbers, or their negatives, are in the subset, so consider the possible cases and show the axiom doesn't hold in any of them.

MAJOR spoiler:
Spoiler:
Show
(Their product is and it can have a real part less than 0. If they do have a real part less than 0 then consider the negative version of the complex numbers (we would have to consider the four cases (+ve)(+ve), (+ve)(-ve),(-ve)(+ve), (-ve)(-ve), but of course the 1st and 4th case, and the 2nd and 3rd case, are identical, so we only need to consider two cases) and if they also suggest a negative real/imaginary part then our positivity axiom doesn't work. I think this is the basic idea, although I think it might need tidying-up/clarifying in places. However, it seems reasonable at 3:30am.

.
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11 years ago
#11
To get the contradiction you can do something like the following.

Let S = {real x: x^3 < 2}

This is bounded above by 2 (say) as x^3 is increasing and is non-empty as 1 is in S. So set c = supS.

We will produce contradictions to show c^3>2 and c^3<2 can't be true. I'll do the details for the first case.

(i) Say c^3 > 2. Let e> 0. Then

(c-e)^3
= c^3 - 3 e c^2 + 3 e c - e^3
> c^3 - 3 e c^2 - e^3
> c^3 - 3 e c^2 - e [IF e<1]
> 2 [IF e < (c^3-2)/(3c^2+1) which is positive]

So if e < min{1, (c^3-2)/(3c^2+1)} then (c-e)^3 > 2 and so c-e is an upper bound for S as x^3 is increasing. But it's less than the least upper bound!
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11 years ago
#12
there is a much much much easier way of doing this - just apply IVT to f(x) = x^3-2 (oddly the proof of IVT, though the same idea to what people are doing above, is far more easy on the eye..)
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11 years ago
#13
Chewwy; not so easy unless you first prove that x^3-2 is a continuous function. (What RichE has posted is essentially a "merging" of proving that and the IVT into one thing).

Plus, of course, the objective of questions like this is to get you used to doing manipulations with supremums, not quoting theorems.
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11 years ago
#14
(Original post by Chewwy)
there is a much much much easier way of doing this - just apply IVT to f(x) = x^3-2 (oddly the proof of IVT, though the same idea to what people are doing above, is far more easy on the eye..)
This is a first-principles type question from the axioms of the real numbers. The IVT is a long way off.
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11 years ago
#15
I confess, I'm not seeing how to do the ordering one, but I'm also wondering what tacit assumptions I might be missing (obvious counterexamples seem to be P={suitable subset of R} and P = {everything} or P = {everything other than zero}) so I haven't looked too hard.
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11 years ago
#16
(Original post by DFranklin)
I confess, I'm not seeing how to do the ordering one, but I'm also wondering what tacit assumptions I might be missing (obvious counterexamples seem to be P={suitable subset of R} and P = {everything} or P = {everything other than zero}) so I haven't looked too hard.
But for any c precisely one of

c is in P,
-c is in P,
c=0

must be true. But I'm not quite sure what the above counterexamples denote.
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11 years ago
#17
(Original post by jadc)
The second question asks me to prove that there is no subset of the complex numbers to which the positivity axioms can be applied (i.e there's a set P that is closed under addition and multiplication, and for every complex number c either c belongs to the set, negative c belongs to the set or c = 0). I assume it means none apart from the real numbers and its subsets. I don't know where to start with this one, but I should probably look for some kind of contradiction?
Suppose such a subset P existed. Then precisely one of the following must be true: i =0 or i is in P or -i is in P. Show how each case leads to a contradiction
0
11 years ago
#18
(Original post by RichE)
But for any c precisely one of

c is in P,
-c is in P,
c=0

must be true.
Bah. I missed that bit.
0
11 years ago
#19
The shame 0
11 years ago
#20
(Original post by jadc)
I've been self studying using the Oxford Analysis I notes and problem sheets which I found online to give myself something to do over the holidays, and I really need some guidance with some of the questions on the second problem sheet.

The first one asks me to prove that there exists a unique real number a such that a^3 = 2. So I use the standard method of considering a set X containing elements x such that x^3 < 2 and showing that the supremum of this set must be the a we're looking for, and so exists by the Completeness Axiom.

So we discount the possibilities that sup X^3 < 2 and sup X^3 > 2 by contradiction, which is where the problem lies. It is intuitivitely clear that the supremum must be such that sup X^3 = 2, since otherwise we could add or subtract some epsilon sufficiently small that either our supremum is less than some element of X or it is not the lowest bound, but I'm not at all sure how to prove that either assumption leads to a contradiction. The notes contain a similar question but they use some obscure looking functions that I don't think I would be able to adapt to this problem.

Might it be possible to use the fact that (c-d)^3 = c^3 - 3c^2d + 3cd^2 - d^3 and let c = sup X and d = epsilon and fiddle around with that until I can show that we can make the 3 terms involving d arbitrarily small such that the whole expression is greater than 2 and factorise (and adapt that for the sup X^3 < 2 case)?

I would be grateful for a small hint to get me going in the right direction (or a confirmation that I've got the right idea, if that's the case).

The second question asks me to prove that there is no subset of the complex numbers to which the positivity axioms can be applied (i.e there's a set P that is closed under addition and multiplication, and for every complex number c either c belongs to the set, negative c belongs to the set or c = 0). I assume it means none apart from the real numbers and its subsets. I don't know where to start with this one, but I should probably look for some kind of contradiction?

I didn't realise this was going to be quite so verbose when I started writing .
Hey, if you don't mind, could you link me to that site, it looks quite interesting. I couldn't find it with a search... so thanks , you inspired me to delve deeper!
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