What is Upper bound: Definition and 113 Discussions

In mathematics, particularly in order theory, an upper bound or majorant of a subset S of some preordered set (K, ≤) is an element of K which is greater than or equal to every element of S.Dually, a lower bound or minorant of S is defined to be an element of K which is less than or equal to every element of S.
A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound.
The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.

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  1. B

    MHB Finding an Upper Bound for ln(x) in [0,1]

    Hii everyone, Can anyone tell me a decent upper bound of Ln[x](which can mimic Ln[x]) where x is in [0,1]regards, Bincy
  2. T

    Finding an Upper Bound for e^(-x^2) for Easy Integration

    Can anyone suggest an upper bound for e^{-x^2} that can be integrated easily?
  3. J

    About definition of 'Bounded above' and 'Least Upper Bound Property'

    The definition of 'Bounded above' states that: If E⊂S and S is an ordered set, there exists a β∈S such that x≤β for all x∈E. Then E is bounded above. The 'Least Upper Bound Property' states that: If E⊂S, S be an ordered set, E≠Φ (empty set) and E is bounded above, then supE (Least Upper...
  4. J

    Least Upper Bound and the Density of the Irrationals Theorem

    Homework Statement For the following set if it has an upper bound, find two different upper bounds as well as the least upper bound (LUB), justifying your answer. If the set has no upper bound, state this and justify your answer. {x | 1 < x < √(7) and x is irrational} (a proof requires the...
  5. C

    Upper bound of random variable

    Dears, If a random variable is generated with the pdf of p(f) = 1/(f^x), how can I derive the upper bound or lower bound of the random variable? Thanks,
  6. C

    Upper bound of random variable

    Dears, If a random variable is generated with the pdf of p(f) = 1/(f^x), how can I derive the upper bound or lower bound of the random variable? Thanks,
  7. P

    Is there a theoretical upper bound for density?

    Is there a theoretical upper bound for the density of matter? Given a proton (already almost as dense as a black hole, according to Wikipedia), can you theoretically compress it into any non-zero volume? Has any work been done that shows that matter can or cannot achieve certain levels of...
  8. C

    Rational numbers - bounded subset with no least upper bound

    Homework Statement Give an example of a bounded subset of Q which has no least upper bound in Q. Explain why your answer has this property. Homework Equations The Attempt at a Solution [1/8, 1/4, 3/8, 1/2, 5/8, 3/4...infinity] is this correct?
  9. B

    2-norm Pseudoinverse Upper Bound

    Hello I'm trying to show that the following upper bound on the matrix 2-norm is true: \left\|(AB)^+\right\|_2\leq\left\|A^+\right\|_2 \left\|B^+\right\|_2 where + is the matrix pseudoinverse and A\in\Re^{n\times m} and B\in\Re^{m\times p} are full-rank matrices with n\geq m\geq p...
  10. M

    Understanding Upper Bound & Sup in Theorem Proving

    This calc book that I am reading uses words like "upper bound" and "sup" a lot when proving theorems. I have never heared these terms before so it makes it hard for me to understand the proofs. I think it has to deal with max's values of a graph: For example given a set S of all elements c in...
  11. A

    Greatest lower bound/least upper bound in Q

    I have the following question: Let n\in\mathbb{Z}^{+} st. n is not a perfect square. Let A=\{x\in\mathbb{Q}|x^{2}<n\}. Show that A is bounded in \mathbb{Q} but has neither a greatest lower bound or a least upper bound in \mathbb{Q}. To show that A is bounded in \mathbb{Q} I have to show...
  12. B

    Taylor's Upper Bound: f(x) 2x Diff. Function (0,∞)

    upper bound of taylor! f(x) is two times diff. function on (0, \infty) . \lim\limits_{x\rightarrow \infty}f(x) = 0 satisfy. M=\sup\limits_{x>0}\vert f^{\prime \prime} (x) \vert satisfy . for each integer L , g(L) = \sup\limits_{x\geq L} \vert f(x) \vert, and h(L) = \sup\limits_{x\geq L} \vert...
  13. S

    How Does Subset Relation Affect Least Upper Bounds in Partial Orders?

    Homework Statement Suppose that R is a partial order on A, B1 ⊆ A, B2 ⊆ A, x1 is the least upper bound of B1, and x2 is the least upper bound of B2. Prove that if B1 ⊆ B2, then (x1,x2) ∈ R.Homework Equations The Attempt at a Solution This problem has been stumping me. After assuming B1 ⊆ B2...
  14. S

    Least upper bound proof (again)

    Homework Statement Okay, this is essentially the same question I had in an earlier thread, but i am trying to make my questions and uncertainties more clear for more accurate assistance: Suppose R is a partial order on A and B ⊆ A. Let U be the set of all upper bounds for B. a) Prove...
  15. S

    Least upper bound/ greatest lower bound proof

    Homework Statement Suppose R is a partial order on A and B ⊆ A. Let U be the set of all upper bounds for B. a) Prove that every element of B is a lower bound for U. b) Prove that if x is the greatest lower bound of U, then x is the least upper bound of B. Homework Equations The...
  16. P

    Proving no upper bound in A, where A = {x in Q | x^2 < 2}

    Homework Statement Prove that there is no upper bound in A, where A = {x in Q | x2 < 2} The Attempt at a Solution My attempt has been to assume that there is an upper bound p in A and then I have been trying to find a way to show that there is a number that is larger than p but still in A...
  17. S

    Can everybody suggest a better upper bound?

    Hi, I have sent this question a couple of days ago, but it seems that its latex form had problem. So, I decide to send it again. I will thank If somebody help me solving this problem. Consider a random variable k_1 with the given pmf as: Pr[k_1=l]=\sum_{l_1+2l_2=l}...
  18. S

    Finding an upper bound for a probability

    Hi, I will thank If somebody help me solving this problem. Consider a random variable k_1 with the given pmf as: Pr[k_1=l]=\sum_{l_1+2l_2=l} \frac{N!}{(N-l_1-l_2)!l_1!l_2!}p_1^{l_1} p_2^{l_2} (1-(p_1+p_2))^{N-l_1-l_2}where l_1,l_2 \in [0,1,...,l] . but we don't have p_1 and p_2 separately...
  19. S

    Dictionary order and least upper bound property

    Homework Statement Does [0,1] \times [0,1] in the dictionary order have the least upper bound property?Homework Equations Dictionary Order. (on \mathbb{R}^2) Let x , y \in \mathbb{R}^2 such that x=(x_1 , x_2) and y = (y_1 , y_2). We say that x < y if x_1 < y_1, or if x_1 = y_1 and x_2 < y_2...
  20. S

    An upper bound for a conditional probability

    Hi everyone, The problem: Is this relation true? If so, how (or maybe where) it could be proved?P(A│B∪C)≤P(A│B)+P(A│C)-P(A|BC) and what about its possible generalization? thanks a lot in advance.
  21. M

    Traffic Flow - Junction, Delay, Webster, Greatest Upper Bound

    [PLAIN]http://img96.imageshack.us/img96/7816/12530747.jpg Hopefully this will post successfully... Erm its the first part I'm not sure on, after that it's easy. I'm just not understanding the wording. I need to work out the effective green time during the cycle
  22. N

    Upper bound turning into supremum

    i proved that sin (1/x)<1/x prove that sup{xsin (1/x)|x>0}=1 if we say that A={xsin (1/x)|x>0} xsin (1/x)<x(1/x)=1 so one is upper bound now i need to prove that there is no smaller upper bound so that 1 is the supremum suppose that "t" is our smaller upper bound t<1 and...
  23. M

    Least upper bound property of an ordered field

    I am trying to understand the following theorem: An ordered field has the least upper bound property iff it has the greatest lower bound property. Before I try going through the proof, I have to understand the porblem. The problem is, I don't see why this would be true in the first...
  24. L

    You're welcome. I'm glad I could help.

    Homework Statement Find an upper bound M for f(x) = abs ( x+2 / x-8 ) if abs(x-7) < 1/2Homework Equations The Attempt at a Solution i first found set of x values using abs(x-7) < 1/2 which is 13/2 < x < 15/2. Now, i believe i have to find other set of x values to compare to find upper...
  25. F

    Proof of the least upper bound

    Homework Statement LEt S is supset of real numbers and suppose that there is X0 is member of S such that x0>=x for all x which is member of S(i.e. x0 is the maximum of S). show that x0=supS Homework Equations The Attempt at a Solution Not: this seems too easy question but i...
  26. Simfish

    Upper bound to 2-norm of a matrix

    Homework Statement Using the fact that ||A||_2 = \sqrt { \rho ( A^* A )}, prove that ||A||_2 \leq \sqrt { ||A||_1 ||A||_\infty }. This is an easy estimate to find in practice for an upper bound on ||A||_2. Homework Equations The Attempt at a Solution Or, in other words, the...
  27. R

    Basic Analysis - Proof Bolzano Wierestrass by Least Upper Bound

    Homework Statement Let (an) be a boundedd sequence, and define the set S= {x\in R : x < a_n for infinitely many terms a_n\} Show that there exists a subsequence (a_n_k)converging to s = sup S Homework Equations This is supposed to be a direct proof of BW using the LUB property, so no...
  28. ╔(σ_σ)╝

    Prove that the least upper bound of a set of a set of integers is

    Problem Statement: Prove that the least upper bound of a set of integers is an integer. Attempt: Using well ordered principle this is very trivial. However, is there another way? ANY comments or ideas relating to the topic would be highly appreciated. It is assumed that the set...
  29. A

    Is f(n) an Upper or Lower Bound of g(n)?

    Homework Statement 1. f(n) = n - 100 g(n) = n - 200 2. f(n) = log(2n) g(n) = log(3n) n >= 0 in all cases Find out if f(n) is an upperbound, lowerbound or both of g(n) Homework Equations The Attempt at a Solution in case of 1, f(n) has to be an upperbound of g(n) because...
  30. X

    Definite Integration with Upper bound as another integral.

    Hi guys, just got owned by my calc prof with a final exam question. Very very weird. Attempted it and different approach apparently gets u different answers. I have no idea what's going on.. I have attached the question as a word document. Too much integration to type and I cannot really use...
  31. C

    Upper bound problem in real analysis

    Homework Statement Let \mathcal{F} \subset C(\mathbb{R}) be a set of continuous functions such that for each x \in \mathbb{R} there is an M_x > 0 such that |f(x)| \leq M_x for all f \in \mathcal{F}. Homework Equations Prove that there is a nonempty open subset Y \subseteq X and an M...
  32. C

    How do we find the least upper bound and greatest lower bound?

    Homework Statement 1. (a) Solve the following inequalities and express the solutions first in interval notation, then express those intervals in set builder notation. (i) x3 + x2 > 2x (ii) \left|(2-x)\right| \leq 4 . (b) For each of the solution sets in part (a), state the least upper...
  33. T

    Find Upper Bound for \| A \| - Homework Statement

    Homework Statement assume that x and y are vectors, and A is a matrix. can anyone kindly help me to find an upper bound C w.r.t \| A \| s.t. \| x-Ay \| \leq C \cdot \| x-y\|
  34. K

    What is the difference between adherent points and accumulation/limit points?

    1) "Least upper bound axiom: Every non-empty set of real numbers that has an upper bound, has a least upper bound." Why does it have to be non-empty? Is there an upper bound for the empty set? 2) "It can be proved by induction that: every natural number "a" is of the form 2b or 2b+1 for...
  35. S

    Proving the Least Upper Bound Property: A Mathematical Inquiry

    Least Upper Bound proof... Homework Statement Suppose A is a nonempty set that has x as an upper bound. Prove that x is the least upper bound of the set A iff for any E>0 there exists a y in A such that y>x-E Homework Equations None The Attempt at a Solution The forward where you...
  36. A

    Proving f(x)=0 by Least Upper Bound on [a,b]

    if f is continuous on [a,b] with f(a)<0<f(b), show that there is a largest x in [a,b] with f(x)=0 i think it can be done by least upper bounds, but i dun know wat is the exact prove.
  37. Z

    Find an upper bound to the limit of a function

    Homework Statement Find an upper bound M for f(x) = |x-2 / x+(1/2)| if |x+1| < 1/4 Homework Equations The Attempt at a Solution I'm confused about this |x+1| < 1/4. Does this mean that |x-1| < 1/4? |x-2/x+(1/2)| = x-2/(2x+1)/2 = 2(x-2)/(2x+1) = 2x - 4/2x + 1 = x-2/x+(1/2) <...
  38. D

    Greatest upper bound analysis proof

    Homework Statement Use the completeness axiom to prove that to prove that every non-empty subset of real numbers, which is bounded below, has a greatest lower bound. Homework Equations N/A The Attempt at a Solution Assume A is a nonempty subset of real numbers which is bounded...
  39. D

    Least upper bound analysis proof

    Homework Statement Assume that A and B are nonempty sets, that A is bounded above, and that B is contained in A. Prove that B is bounded above and that the least upper bound of B is less than or equal to the least upper bound of A. Homework Equations Definition: Least Upper bound: Let...
  40. J

    What is the Difference Between Almost Upper Bounds and Upper Bounds in Calculus?

    The author of my calculus book defines an "almost upper bound" as follows: A number x is an almost upper bound for the set A if there are only finitely many number y \in A with y \geq x. He then asks the reader to prove that if A is a bounded infinite set, then the set B of all almost upper...
  41. K

    Finding the upper bound

    Say I have a container with room for B balls. I know that there are black and white balls but I don't know the ratio between them. Say I pick P balls, and R% are black. How can I use this information to establish an upper bound on the number of white balls, with C% certainty? To give a...
  42. S

    Least Upper Bound: Definition & Subbase

    I read the following: "If {T_i} is a non empty family of topologies on our set X, then the least upper bound of this family is precisely the topology generated by the class \bigcup T_i; that is, the class \bigcup T_i is an open subbase for the least upper bound of the family {T_i} ." I...
  43. O

    How Does Choosing Ball Subsets Affect Their Weight Growth in an Algorithm?

    there are n balls of weight 1/n. an opponent choose each time a subset of balls that each one has weight less than 1. then each ball in this set, its weight is multiplied by 1+\frac{1}{|S|} where S is the set of balls that the opponent chose. I need to show that for each choice of subsets...
  44. fluidistic

    An upper bound in temperature?

    I'm wondering if a gas in which all its molecules are moving very close to the speed of light has a finite temperature. More precisely, if we take the limit of the speed of the particles to be exactly the speed of light (I know it's impossible to reach, but as I'm calculating an upper bound I...
  45. Z

    Find Upper Bound for abs(f(4)(x)) of f(x)=sin(sin(x))

    help me, please if f(X) = sin(sin(x)), use a graph to find a upper bound for abs(f(4)(x)) Thanks
  46. P

    Proving (x^3)=2 using least upper bound

    Okay, my homework is "Prove that there exists a positive real number x such that (x^3)=2." and I have no clue how I can solve it. sigh. Is there anyone who can me to solve it using least upper bound property?? Thank you !
  47. S

    Supremum is the least upper bound

    Homework Statement Prove that the supremum is the least upper bound Homework Equations The Attempt at a Solution Proof: let x be an upper bound of a set S then x>=supS (by definition). If there exists an upper bound y and y<=SupS then y is not an upper bound (contradiction)...
  48. T

    Least Upper Bound to Java Printing

    I change the link. http://i359.photobucket.com/albums/oo31/tanzl/JavaPrinting-2.jpg Thanks SNOOTCHIEBOOCHEE
  49. B

    What is the Least Upper Bound Problem in Subset Inclusion?

    Homework Statement Find subsets E\subsetS1\subsetS2\subsetS3\subsetQ such that E has a least upper bound in S1, but does not have any least upper bound in S2, yet does have a least upper bound in S3. Homework Equations The Attempt at a Solution I got totally stuck with it. If...
  50. Fra

    Upper bound for K-L divergence on discrete prob. space

    Does anyone know of any analytical expression for the upper bound on the Kullback–Leibler divergence for a discrete random variable? What I am looking for is the bound expressed as 0 <= S_KL <= f(k) Where k is the number of distinguishable outcomes. Ultimately I am also looking for...
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