Prove that N is not bounded above
Hi, I was just looking through my notes for a course that is all about proofs. And in the "Fields" section I came across a definition for the "Dedekind Completeness Axiom" that says "Every non-empty subset that is bounded above has a least upper bound". The part that I can't get my head around is the idea of a least upper bound. From my understanding, this would imply that we could have more than one upper bound for a set. Which just does not make sense to me.
My gut is telling me that an upper bound only needs to be such that every element in the set is below it. So the least upper bound is the lowest of all our options for an upper bound. (Which might imply that the set converges to the least upper bound?) Although I'm not quite convinced on it.
This section follows the "Sequences" section which mainly deals with convergence and proving that a sequence converges.
I've attached an image of the section in question here
...Now that I've typed this all out, I've almost convinced myself on my way of interpreting the least upper bound (which could be a bad thing). I'll post this anyways because I'm still not 100% sure and would like another opinion.
If anyone could clear this up, it would be greatly appreciated.
Thanks
I just cannot seem to grasp this problem. I know that I need to show that there is a subset of the set M that has an upper bound but no least upper bound. I was maybe thinking that I could show the natural numbers have this property, but I am not sure if I am doing it correctly. It is easy to show that the natural numbers are bounded, but it is not apparent to me how to prove they have no least upper bound. Does anyone have any advice for doing this? Should I be considering a set other than the natural numbers? Here is a picture of my attempted proof as well as the problem statement.
https://preview.redd.it/abm03wumsyv51.png?width=1198&format=png&auto=webp&s=c194aa9271093646fb0c554632e8dac9da6a87bf
This is what I understand from the first sentence:
If there is a supremum for a set it is unique.Meaning it is only one element that can be a supremum.
This is what is confusing me:
"then M<=M' since M' is an upper bound of A and M is a least upper bound "
But if M,M' are suprema, doesn't it mean that they are already both least upper bounds?
What exactly is the difference between upper bound and least upper bound?
Sorry if this is not a good question,but I'm confused with this.Really appreciate your time,Thanks in advance.
sup?
Hey there, I was asking myself the following question while working through a book about disrete math but found myself a bit unsure about it: Suppose R is a partial order on A and B is a subset of A. Let U be the set of all upper bounds for B, then the smallest element of U is generally called the least upper bound. Is the l.u.b. always equivalent to the greatest element of B, if we understand by greatest an element b, such that, for all x in B, xRb holds? That is, can the terms l.u.b. of some set X and greatest element of X be used interchangeably?
So I want to prove that if every monotone, bounded sequence converges in R then every nonempty subset of R has a least upper bound.
So let S be a subset of R, nonempty and bounded by some real number x. Assume for contradiction it has no least upper bound. Then if x_n is an upper bound for S, there must exists x_n+1 <= x_n such that x_n+1 is also an upper bound for S.
Define the sequence (x_n ) to be the upper bounds of S with x_1 =x, the given upper bound. Then for each x_n there is an x_m such that x_m<= x_n. Thus (x_n ) is monotone decreasing.
It is also bounded since each x_n >= s for every s in S, and x_1 >= x_n for every n, then let r=d(x_1, s) for some s in S. Then the ball B_r (s) contains all of (x_n) thus it is also bounded.
By assumption this sequence converges in R, call this limit p. Then I need to show that p<= x_n for each n and that p>= s for each s in S.
Suppose there is an m, such that x_m< p. Then let r= d(p,x_m )/2. Then B_r (p) does not contain x_m or any n>= m thus the sequence can't converge to p a contradiction.
Suppose there is an s>p, then let r=d(p,s)/2. By convergence there exist x_n inside the ball B_r (p) which are then less then s, contradicting that they were upper bounds.
Thus p is the smallest upper bound of S.
Does this proof work?
The question says "Prove that the least upper bound of {x/(x+1) : x ∈ R and x ≥ 0} is equal to 1." I'm not sure how to go about proving that a value is the least upper bound of a set, so any help would be much appreciated.
Hi
I'm trying to understand the logic LP of Graham Priest, using his 'An Introduction to Non-Classical Logic' (you can find it on library genesis). On the pages 457 and 459 he discusses the quantifiers of quantified many-valued logics. I have trouble understanding these concepts of 'greatest lower bound' and 'least upper bound'.
Can someone explain it to me?
Thanks in advance!
Edit: I appreciate your help verry much, thank you. I think I've got it.
Hi, I recently started to try and teach myself real analysis and this textbook says that the least upper bound property of the set of real numbers is important and makes it different than other sets of numbers. Why is this?
Hey r/math, this part of my real analysis self-study lectures where I discuss the basics of bounded sets, least upper bounds and how to prove things through contradiction.
Check it out at https://youtu.be/GYehuVt2sG8 Let me know what you guys think!
For some context, I am a teaching assistant for my university's Real Analysis course and wanted to make a resource for anyone just starting out trying to learn real analysis.
I am trying to work out how to use the Algebra.Lattice family of Lattice *type classes*.
Firstly how do I construct a lattice ?
What I am wanting to do is to be able to construct a lattice to represent a multiple inheritance hierarchy. Then I to be able to find the Least Upper Bound of a set of classes/types. This is in order to find the type of a multiple case expression.
I am not sure if the Haskell classes are actually applicable ? but if they are how do I apply them to the following problem please ?
I'm trying to run a regression model on sports player performance data, such that the sum of the individual predictions falls between an upper bound and lower bound of a team's expected prediction distribution.
Given the current COVID situation, many star players are missing games, leading to many "no name" players getting unprecedented opportunity. The problem is that the regression model underpredicts almost all of them.
Say I know that the Dallas Cowboys have the best offensive line in football, and they're going against the worst rushing defense in football. Their star running back Ezekiel Elliot is out for the game, and hypothetically, the two running backs filling in haven't played much and are averaging 5 yards a game, and 10 yards a game, respectively.
A regression model will predict somewhere in the ballpark of 5 and 10 for those players, which implies the Dallas Cowboys will rush for 15 yards the entire game. The whole world knows that's not going to happen, and historical data says there's a 95% chance the Dallas Cowboys rush for between 75 and 125 rushing yards altogether. Is there any way to group predictions together in scikit-learn and set lower bounds and upper bounds on the team's sum?
It's almost like a multilevel model type thing where the two have to agree with each other to make logical sense. If we were predicting team rushing yards using team statistics it would predict somewhere between 75 and 125 yards. So how can we get the individual player predictions to sum up to something that makes sense.
I'm essentially trying to combine regression with constrained optimization.
Thanks so much.
Hi all, I asked an l.u.b. problem yesterday and I was able to figure it out, but this other problem has been giving me a lot of trouble. The problem states " Prove that the least upper bound of the set {x/(x^(2)+1) : x ∈ R} equals 1/2 ." I know that the 2 things I need to prove are 1. 1/2 is an upper bound to the set, and 2. For all x ∈ R, if x is an upper bound, x > 1/2, but I have been having trouble with the algebra for the first part. I cannot find a definite way to prove that for all x ∈ R, x <= 1/2. Any help would be greatly appreciated. Thank you.
Assuming there is no inflation (triple haha), is the money people can put in the stock market finite? I’m honestly not sure… like there must be an upper bound to what people can invest (in anything), so… is there a limit?
Of course the inflation will continue driving it up, but will it be the only factor in the end?
Let A be a partially ordered set and K be a subset of A. Define L = { x : x is an element in A and x is a lower bound for K } Prove that if the least upper bound of L, lub L, exists, then the greatest lower bound of K, glb K, is equal to lub L.
Here's my attempt: Assume lub L exists. Therefore, L is nonempty. Since L is nonempty and L is a subset of A, lub L is an element of A and K is nonempty.
Claim: If lub L exists, glb exists. Proof: Assume lub L exists and glb K does not exist. We know that lub L is an element of A. (Also, since glb K does not exist, there exists no y in A such that y is greater than or equal to i for all i in L.) However, lub L is greater than or equal to all i in L. Thus, if lub L exists, then glb K exists.
As a result, since we assumed that lub L exists, then glb K exists. Also, glb K is an element in A, and glb K is an element in L because glb K is a lower bound for K.
Since glb K is greater than or equal to any lower bound for K, glb K is greater than or equal to all i in L, and glb K is an upper bound for L. We know that since glb K is an element in L, glb K is less than or equal to lub L.
So we have to unique cases:
- glb K is less than or equal to lub L and glb K is not equal to lub L.
- glb K is equal to lub L
If the first case is true, since glb K is an upper bound for L, lub L is not the least upper bound of L. This cannot be true.
Thus, the second case must be true, and lub L = glb K. Q.E.D.
^^This proof falls apart at the sentence in parenthesis. I'm not sure how else to tackle this. I know that for any elements a, b, a not being less than or equal to b does not imply that b is strictly less than a, which sucks.
Just a quick question again related to my recent post. We are creating a system that suggests the fastest route to the best hospital taken by an ambulance. We utilized A* for pathfinding but the problem is we don't know the compatible/best algorithm in suggesting which hospital the patient will be admitted to.
For selecting the best hospital, the data encoded by the emergency responder (e.g. head injury specifically) should be paired to the hospital that has MRI and should be the nearest based on the calculations of the pathfinding algorithms.
Our chosen algorithm is the Upper Confidence Bound (UCB) Multi-Armed Bandit Algorithm, is this the right choice for our use case? I have also read about Rete's algorithm and I have to replace it to our current one. Thank you.
I have been revisiting some math that I studied a couple of years ago, and while reviewing partial orders, I saw this definition (emphasis mine):
In mathematics, the GLB of a subset S of a partially ordered set T is the greatest element of T that is less than or equal to all elements of S.
Alright, cool.
Then how is it that often in exams, a Hasse diagram is provided and we are supposed to find the GLB and LUB when the partial order of which it is a subset is not provided? Or questions such as this?
I am sure I am missing something very basic and straightforward, so please don't kill me :)
Well I'm posting this because I believe it's a little too advanced for posting it on the quick questions thread, but if I'm wrong, I ask the mods to state this in the reason for deleting this post. Now about my question, I'm working on a project about differentiable curves parametrized by polynomial equations, and for some basic notions about differential curves I need the first and second derivatives of the polynomial that parametrize the curve to be non zero in the whole interval where it will be definied. So in order to choose an interval that fits my needs, I must be sure that this interval does not contains roots of the polynomials definied as the first and second derivatives of the polynomial that parametrize the curve. In order to do this, I thought that there might be a way to find an upper or lower bound for the roots of a real valued polynomial. Like, given a real valued polynomial P(x), by "this" theorem I can be sure that no root of it can be greater or smaller than somo constant x0. With this I would be abble to choose an interval with only greater of lower valeus than that constant and ensure that my interval fits my needs. Does someone know about something like this? A friend of mine told me that in the field of real algebraic geometry there's something that may help me, but I know little more than the basics of real algebraic geometry and since time is precious, I wouldn't like to spend a lot of it studying real algebraic geometry just for a little part of my research.
[Currently an exam candidate]
Is it true that BITS annually increases its fees by 8%/year for the next batch? Currently annual academic fees is around 5 lacs besides travel/hostel expenses , (which is still out of reach from a lot of middle class families, and without loans, is a strict elite Class institution ), which filters a lot of capable students.
I know there's MNC, Merit basis scholarships but one can't won't be sure of his/her performance in the College .
Would a day arrive when the fees becomes unbearable even for upper middle class), say around 9lacs/annum or higher?
Would solution be a VIT based crowded model, since, I am sure that with time , and such income filter in act, a lot of better ROI Collegesike DTU,NSUT,VIT, NITs will be able to catch up.
I just wanted some clarification. For a relation R on A and a set B that is a subset of A, two sets can be constructed that contain upper bounds and lower bounds of B. The book I am using simply says that the greatest lower bound is the largest element in the set of lower bounds. Does this mean the R-largest element or just the normal usage of the word? Thanks!
Some background to my question: Earlier this year, further constraints on the tensor-to-scalar ratio (r) were proposed by M. Tristram et al., reducing the upper bounds to r < 0.044 with a 95% confidence level.
According to the article Squeezing down the Theory Space for Cosmic Inflation, most popular models predict r > 10^-4. The article makes a few good points on the positives and the negatives of these new constraints, but it appears that we are approaching a point where we may need to adapt our models to fit newly observed results (or maybe we'll get lucky and find those illusive B-modes).
The BICEP2 results in 2014 initially indicated detection of B-mode polarization anisotropy, primarily due to primordial gravitational waves, giving r ~ 0.2, which conflicted with the 2013 Planck upper bounds of r < 0.11. This has since been determined to be caused by dust.
The question: I was reading through the Planck 2015 results, which discussed the impacts of the BICEP2 findings and how they were rectified (section 6.2). In there it is stated:
> Since the Planck constraints on r are highly model dependent (and fixed mainly by lower k) it is possible to reconcile these results by introducing additional parameters, such as large tilts or strong running of the spectral indices.
This makes me think that if we do reach the point where we need to re-visit our currents models, we can at least use them as a strong basis going forward. Or is it not that simple, does 'adding additional parameters' to a cosmological model mean we would have to throw out the existing models and start from scratch?
edit: removed incorrect LaTeX markdown.
Hey guys I'm looking for help with a problem I'm almost there (Well I can start, I know what I need to show but I can't connect the dots) but I can't seem to convince myself to get from l.u.b. -> g.l.b
Question: Prove that the least upper bound property implies the following: If S is a nonempty subset of real numbers that has a lower bound, then S has a greatest lower bound.
Suppose [; S \subset \mathbb{R} ;] and [; L \subset \mathbb{R} ;]
Let [; L = {l: l \in \mathbb{R}, l \leq s, \forall s \in S } ;], Then [; S ;] is an upper bound for [;L;]
Then [; \exists! \alpha \in \mathbb{R} ;] s.t. [; \forall l \leq \alpha ;] i.e., [; \alpha = sup(L) ;]
So this is where my brain implodes I need to show alpha is actually the inf(S)
Since [; S \neq \varnothing ;] then [; \exists! \beta \in S ;] such that [; \beta = inf(S) ;]
Yet [; \forall l \leq \alpha ;] and [; \beta \leq \forall s \in S ;] then [; \alpha = \beta;]
Thus, [; \alpha = inf(S) ;]
I really can't see how to do this without [; \beta;]. Well again I do know [; \forall l, l \leq \alpha ;] then [; \alpha \leq \forall s \in S;] ... Wait is that it?
I was thinking about how you'd prove that the infinite series definition of the exponential function on R (exp(x)) is equivalent to the standard high school definition of exponents. I figured out that you can determine its infinite radius of convergence, then use uniqueness to prove that exp(a+b) = exp(a)exp(b). Well, no, that last bit came from Quora, which is why I was thinking about this. Anyway, I went about trying to show that exp(x) = e^x on R, but I'm not sure if my final step is valid. Essentially, I said that, having shown that exp(x) is defined for all x in R and that exp(x) = e^x for rational x, I could use the least upper bound property to determine that exp(x) must be the least upper bound of the set defined by exp^-1 (y) < x (since it's easy to show that exp(x) is increasing), and that since e^x is extended to R in the same way, I could conclude that they are equivalent on R. The problem is, I haven't taken analysis yet (I'm in my second semester of freshman calc) and so while that step sounds reasonable, I can't tell whether it's actually valid or not. Is it? Thanks in advance.
I just cannot seem to grasp this problem. I know that I need to show that there is a subset of the set M that has an upper bound but no least upper bound. I was maybe thinking that I could show the natural numbers have this property, but I am not sure if I am doing it correctly. It is easy to show that the natural numbers are bounded, but it is not apparent to me how to prove they have no least upper bound. Does anyone have any advice for doing this? Should I be considering a set other than the natural numbers? Here is a picture of my attempted proof as well as the problem statement.
Let A ⊆ R be bounded and let B ={(x+y)/2 | x, y ∈ A}. Prove that LUB(A) = LUB(B).
Not sure if proving that B ⊆ A suffices (I doubt it), or whether I should attempt a proof by contradiction?
