Sets Definition and 1000 Threads

  1. J

    Proving the triangle inequality property of the distance between sets

    Proving the "triangle inequality" property of the distance between sets Here's the problem and how far I've gotten on it: If you are unfamiliar with that notation, S(A, B) = (A \ B) U (B \ A), which is the symmetric difference. And D(A, B) = m^*(S(A, B)), which is the outer measure of...
  2. C

    Nested Open Sets: Example & Intersection

    Homework Statement Give an example of an infinite collection of nested open sets. o_1 \supseteq o_2 \supseteq o_3 \supseteq o_4 ... Whose intersection \bigcap_{n=1}^{ \infty} O_n is closed and non empty. Homework Equations A set O \subseteq \mathbb{R} is open if for all points...
  3. S

    Proof: function of int. of family of sets and int. of function of family of sets

    Homework Statement The question is "Prove f(\bigcap_{\alpha \in \Omega} A_\alpha) \subseteq \bigcap_{\alpha \in \Omega} f(A_\alpha) where f:X \rightarrow Y and \{A_\alpha : \alpha \in \Omega\} is a collection of subsets of X. Also, prove the statement's equality when f is an injective...
  4. U

    Closure of relations betweens sets

    Hi all! I am searching for an algorithm (most likely already present in the literature) that could solve the following problem: Instance: Properties of sets of elements and relations between sets of elements Question: Find the closure of the properties and relations Possible properties...
  5. C

    A perpetual machine model that sets me thinking

    Now I'm not a PMI (perpetual machine inventor). In fact I'm quite convinced that there is no such thing as that. But a while ago, I saw the schematics of a perpetual machine that is hard to debate. Well this is how the machine worked. The inventor argued that if you have two magnets as...
  6. N

    Determining open and connected sets.

    Homework Statement Are the following regions in the plane (1) open (2) connected and (3) domains? a. the real numbers; b. the first quadrant including its boundary; c. the first quadrant excluding its boundary; d. the complement of the unit circle; f. C \ Z = {z ∈ C : z \notinZ}...
  7. J

    What is the significance of JD and how is it calculated?

    Hi, When you have to calculate the rising or setting time of a celestial body, you have to handle with hour angle and sidereal time. Sidereal time for the rising is given by T = alpha - H and by T = alpha + H for the setting (alpha = right ascension). Why - H in one hand, and + H on the...
  8. L

    Closed set as infinite intersection of open sets

    This is not a homework problem, just something I was thinking about. In a general metric space, is it true that every closed set can be expressed as the intersection of an infinite collection of open sets? I don't really know where to begin. Since the finite intersection of open sets is open...
  9. E

    Cartesian product of open sets is a open set

    Homework Statement This is not really coursework. Instead, this is some sort of curiosity and proposition formulation rush. Then the initial questions are that if this is a valid result that is worth to be proven. Let X,Y be metric spaces and X\times Y with another metric the product metric...
  10. N

    Whats an infinite intersection of open sets

    whats an infinite intersection of open sets? how is it different from finite intersection of open sets and why is it a closed set in the case of ∞ intersection but open in case of finite. To quote kingwinner, is it being defined as a limit? it really does look look like a limit in the case...
  11. T

    Help with part of my Linear Algebras project - affine sets and mappings

    ℝHomework Statement I'm a 2nd year undergraduate student, so I suppose many users here won't find this too difficult, but I've had some issues with the following questions and, of course, any help would be very much appreciated: (i) Prove f: V→W is affine (where V and W are real vector space)...
  12. 6

    Exploring Closed Sets in Metric Spaces through Infinite Intersections

    Homework Statement Find (X,d) a metric space, and a countable collection of open sets U\subsetX for i \in Z^{+} for which \bigcap^{∞}_{i=1} U_i is not open Homework Equations A set is U subset of X is closed w.r.t X if its complement X\U ={ x\inX, x\notinU} The Attempt at a Solution Well...
  13. D

    Zakon Vol 1, Ch2, Sec-6, Prob-19 : Cardinality of union of 2 sets

    Homework Statement Show by induction that if the fi nite sets A and B have m and n elements, respectively, then (i) A X B has mn elements; (ii) A has 2m subsets; (iii) If further A \cap B = \varphi, then A \cup B has m+ n elements. NOTE : I am only interested in the (iii) section of...
  14. K

    Julia Sets: Periodic and Non-Periodic Points Explained

    I'm a little bewildered when reading about these Julia sets. From the definition a Julia set is the closure of all repelling periodic points of a complex map f. However I read that a Julia set always contain periodic and non-periodic points. Wasn't the definition including only periodic points...
  15. T

    Proof about the decomposition of the reals into two sets.

    Homework Statement Let S and T be nonempty sets of real numbers such that every real number is in S or T and if s \in S and t \in T, then s < t. Prove that there is a unique real number β such that every real number less than β is in S and every real number greater than β is in T. The Attempt...
  16. Z

    How can I get a function relation with these two sets?

    Homework Statement I have these two sets: Pairwise, (1, 1) (2, 4) (3, 9) (4, 16). Clearly this is just squared. How can I get a function relation with like: (1, 1) (2, 3) (3, 9) (4, 10) or like (1, 1) (2, 5) (3, 12) (4, 22) Homework Equations The Attempt at a...
  17. C

    Limits in infinite unions of sets

    Suppose I define sets D_n = \lbrace x \in [0,1] | x has an n-digit long binary expansion \rbrace . Now consider \bigcup_{n \in \mathbb{N}} D_n. This is just the set of Dyadic rationals and therefore countable for sure. Now for the question: is this equal to \bigcup_{n = 0}^{\infty} D_n...
  18. T

    Real Analysis: countably infinite subsets of infinite sets proof

    Homework Statement Prove that every infinite subset contains a countably infinite subset. Homework Equations The Attempt at a Solution Right now, I'm working on a proof by cases. Let S be an infinite subset. Case 1: If S is countably infinite, because the set S is a subset...
  19. S

    Discrete math, sets, power sets.

    Homework Statement If s (0,1), find |P(S)|, |P(P(S))|, |P(P(P(S)))| Homework Equations The Attempt at a Solution |P(S)| = {(0), (1), (0,1), ∅} = 4 |P(P(S))| = {...} = 16. |P (P(P(S)))| = {...} = 16 ^4 ...but how? as my lecturer explained it, it come from pascals...
  20. C

    Probability of 2 equivalent random selections from integer sets

    What is the probability that a number selected from 0-9 will be the same number as one randomly selected from 0-4? Relevant equations: $$P(A \cap B) = P(A)*P(B|A)$$ I used the equation above, using A as the event that the number selected from 0-9 will be between 0 and 4, and B as the event that...
  21. B

    Clopen Sets: Closure = Interior?

    For a subset which is both closed and open (clopen) does its closure equal its interior?
  22. K

    Showing two countably infinite sets have a 1-1 correspondence

    Homework Statement Suppose A and B are both countably infinite sets. Prove there is a 1-1 correspondence between A and B. Homework Equations The Attempt at a Solution Since A is countably infinite, there exists a mapping f such that f maps ℕto A that is 1-1 and onto...
  23. T

    Real Analysis: one to one correspondence between two countably infinite sets

    Homework Statement Suppose that A and B are both countably infinite sets. Prove that there is a one to one correspondence between A and B. Homework Equations The Attempt at a Solution By definition of countably infinite, there is a one to one correspondence between Z+ and A and...
  24. B

    Abstract math prove involwing sets

    Homework Statement Let Ts denote the set of points in the x; y plane lying on the square whose vertices are (-s; s), (s; s), (s;-s), (-s;-s), but not interior to the square. For example, T1 consists of the vertices (-1; 1), (1; 1), (1;-1), (-1;-1) and the four line segments joining them...
  25. M

    [Cardinality] Prove there is no bijection between two sets

    Homework Statement prove there is no continuous bijection from the unit circle (the boundary; x^2+y^2=1) to R Homework Equations The Attempt at a Solution is this possible to show by cardinality? since if two sets have different cardinality, then there is no bijection between...
  26. T

    Understand Affine Subsets & Mappings: Research Project for Undergrads

    This research project is to help me (I'm an undergraduate) get my head around this topic. It is concerned with affine subsets of a vector space and the mappings between them. As an application, the construction of certain fractal sets in the plane is considered. It would be considered pretty...
  27. Z

    Sequences, sets and cluster points

    Hello all, I am having trouble with a homework problem. The problem is as such: Let a = {zn = (xn,yn) be a subset of ℝ2 and zn be a sequence in ℝ2 such that xn ≠ xm and yn ≠ ym for n≠m. Let Ax and Ay be the projections onto the x and y-axis (i.e. Ax = {xn} and Ay = {yn}. Assume that the...
  28. 1

    Unions and intersections of collections of sets

    My proof class just took a turn for the worst for me - I don't understand this. First, the notation is extremely confusing to me, I need help to make sure I'm getting this. If An is some set for some natural number n such as [-n, n]. Then (script A) the collection is the set of all An...
  29. Useful nucleus

    Are all open sets compact in the discrete topology?

    A subset K of a metric space X is said to be compact if every open cover of K contains a finite subcover. Does not this imply that every open set is compact. Because let F is open, then F= F \bigcup ∅. Since F and ∅ are open , we obtained a finite subcover of F. Am I missing something here?
  30. I

    Let A, B and C be sets. Prove that

    *Sorry wrong section* Let A, B and C be sets. Prove that if A\subseteqB\cupC and A\capB=∅, then A\subseteqC. My attempted solution: Assume A\subseteqB\cupC and A\capB=∅. Then \veex (x\inA\rightarrowx\inB\cupx\inc). I'm not sure where to start and how to prove this. Any help would be greatly...
  31. K

    Showing two certain sets have no elements in common

    Homework Statement Let x and y be irrational numbers such that x-y is also irrational. Let A={x+r|r is in Q} and B={y+r|r is in Q} Prove that the sets A and B have no elements in common. Homework Equations The Attempt at a Solution Since x and y are in A and B, then...
  32. K

    Finding X and Y with respect to third variable if having two sets of X and Y

    My problem At temperature 5 degrees the Y=1.00228594 for X=435 and Y=1.000986038 for X=449 and Y=0.999760292 for X=463 At temperature 7 degrees the Y=1.002094781 for X=435 and Y=1.00079709 for X=449 and Y=0.999573015 for X =463 For a new temperature of 9.6 degrees how to find the Y...
  33. I

    MHB Finding a set which is not equinumerous with series of sets

    Hi Let \( A_1=\mathbb{Z^+} \) and \( \forall n\in \mathbb{Z^+}\) let \( A_{n+1}=\mathcal{P}(A_n) \) I have to come up with an infinite set which is not equinumerous with \( A_n \) for any \( n\in \mathbb{Z^+} \). Clearly \( \mathbb{R}\) will not fit the bill since \( \mathbb{R}\;\sim\; A_2...
  34. I

    MHB Can Power Sets of Equinumerous Sets Remain Equinumerous?

    Hi Here is the problem. Let A be a set with at least two elements. Also suppose. \[ A\times A \sim A \] Then prove that \[ \mathcal{P}(A)\times \mathcal{P}(A)\sim \mathcal{P}(A) \] Let a and b be the two elements of this set. Then I want to exploit the result that \[ \mathcal{P}(A)\;\sim...
  35. S

    MHB Questions about sets and subsets

    Hi, the question goes as follows: Given two subsets X and Y of a universal set U, prove that: (refer to picture) I'm having particular trouble on D and E, if anyone can clear this up or provide some clarification for me that would be appreciated! I know a venn diagram might not be possible...
  36. N

    Generating sets based on a recursive language definition

    I've searched the internet and the forums for any help on this, but I can't seem to find a topic that details what the successive sets will contain. Here is an example question: (I have many HW problems like this, I just don't know where to start) Let L be the language over {a,b} generated by...
  37. T

    Linearly Independence and Sets of Functions

    Homework Statement The Attempt at a Solution I don't think I'm really understanding this problem. Let me tell you what I know: A set is linearly independent if a_1 A_1 +...+a_n A_n = \vec0 for a_1,...,a_n \in R forces a_1 = ...=a_n = 0. If f,g,h take any of the x_i \in S, then one of the...
  38. Deveno

    What is the role of forcing in understanding uncountable sets in set theory?

    I have recently become suspicious of the real numbers. For nearly 3 decades I accepted their axiomatic existence as a complete, ordered archimedian field. The Dedekind-cut, and Cauchy sequence, and "infinite decimal" constructions all made sense to me. And then I started reading about models...
  39. T

    Proving sets with structural induction

    Consider the set S defined recursively as follows: • 3 ∈ S, • if x,y ∈ S,then x−y∈S, • if x∈S, then 2x ∈ S, • S contains no other element. Use Structural Induction to write a detailed, carefully structured proof that ∀ x ∈ S, ∃ n ∈ Z, x = 3n. What I've got is since 3 is in the set...
  40. T

    Can Structural Induction Prove All Elements of Set T Are Powers of 2?

    Consider the set T defined recursively as follows: • 2∈T, • if x∈T and x>1,then x/2 ∈T, • if x∈T and x>1,then x^2 ∈T, • T contains no other element. Use Structural Induction to write a detailed, carefully structured proof that ∀ x ∈ T, ∃ n ∈ N, x = 2n. I'm not sure how to...
  41. H

    Find the bit string for the following sets.

    Homework Statement The a universal set: u = {1,2,3,4,5,6,7,8,9,10} 1) Find the bit string for b = {4,3,3,5,2,3,3,} 2) Find the bit string for the union of two sets. Homework Equations 1)Would I first begin this problem by realizing that set b is the same as {2,3,4,5}? The...
  42. K

    What is the Proof of Equality for Sets of Random Variables?

    Hey there. I'm asked to prove: If X1,...Xn are random variables defined on a set Ω and B1,...,Bn C R1 then prove that (X1,...,Xn)^-1 (B1x...xBn) = X1^-1B1 n X2^-1B2 n ... n Xn^-1Bn so I think I can explain the proof, but just not write it out. This is my attempt if omega is on the left hand...
  43. S

    A geometric property of a map from points to sets?

    I'm interested in the proper way to give a mathematical definition of a certain geometric property exhibited by certain maps from points to sets. Consider mappings from a n-dimensional space of real numbers P into subsets of an m-dimensional space S of real numbers. For a practical...
  44. S

    Multiple sets of linearly independent vectors

    hallo I am trying to calculate the probability to obtain 2 sets of linearly independent vectors from a set of binary vectors of length k. For example: k = 4, and therefore I have 2^k = 16 vectors to select from. I want to randomly select 7 vectors (no repetition). What is the...
  45. A

    Sets closed under complex exponentiation

    The rational (and also algebraic) elements of ℂ are closed under addition, multiplication, and rational exponentiation (the algebraic numbers, that is), but not under complex exponentiation. For instance, (-1)^i=e^{-\pi}, with is not rational, and in fact it is even transcendental. Is there any...
  46. F

    Are A and B subsets of R \ {0}?

    & means belong to and # not equal to : $ subsets of A={(t-1,1/t): t&R, t # 0} B= {(x,y) &R^2:y=1/(x+1), x#-1} i started by say A$B let x= t-1 and y=1/t so we have y= 1/(t-1)+ = Y=1/t hence A$B to prove B$A is where i am stuck- as I think I have got my first part wrong anyway and I...
  47. K

    Is There an Order of Operations for Boolean Expressions in Sets?

    Hello all. Currently working on simplifying some Boolean expressions, one of the questions is: ( A int B U C) int B I do not know how to go about simplifying the first term because there are not any parentheses within it and I have both the intersection and union symbols. Is there an...
  48. J

    Recursion Theorem and c.e. sets

    Let \varphi_e denote the p.c. function computed by the Turing Machine with code number e. Given M=\{x : \neg(y<x)[\varphi_y=\varphi_x]\} prove that M is infinite and contains no infinite c.e. subset. I have already proved that M is infinite. A necessary and sufficient condition to prove that M...
  49. A

    Closed Sets in \mathbb{C}: Showing Unclosedness by Example

    Homework Statement Show by example that an infinite union of closed sets in \mathbb{C} need not be closed. The Attempt at a Solution In \mathbb{R} I know that an infinite union of the closed sets A_{n}=[1/n,1-1/n] is open. Not sure if it works in \mathbb{C} as well.
  50. A

    Bio mechanic force/moment problem sets

    Homework Statement Hi everyone, I have 2 questions that I think are relatively easy but are frustrating in that I just can't seem to get an answer 1st -- If a lateral-to-medial load is applied to the foot, a counteracting moment is produced at the knee joint in a lateral-medial plane, which...
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