What is Sets: Definition and 1000 Discussions

In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.
For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of more than two sets is called disjoint if any two distinct sets of the collection are disjoint.

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  1. T

    Orthagonal Sets Homework: Gram-Schmidt Algorithm in R4

    Homework Statement Use the Gram-Schmidt algorithm to convert the set S={x1, x2, x3} to an orthagonal set, given x1 = [1 1 1 1]T, x[SUB]2 = [6 0 0 2]T, x[SUB]3 = [-1 -1 2 4]T. Homework Equations The Attempt at a Solution I've used the algorithm to come up with the set of vectors...
  2. R

    Connectedness and product of sets

    Homework Statement If M is connected and N is connected, prove that MXN must be connected. Homework Equations A set M is connected if it has no separation. A separation is 2 nonempty open sets A and B such that A union B is M and A intersection B is the empty set. If X is connected...
  3. M

    Parenthsis operator for sets or probability

    It's been more than a couple of years since my formal math classes and I ran across this document: http://sysbio.oxfordjournals.org/content/45/3/380.full.pdf The problem I'm having is that I can't seem to remember what the parenthetical operator does in equation 16. Thanks!
  4. T

    Proving M1 x M1 ⊆ M2 using Lebesgue Measure

    This is in the context of a homework problem but not directly related. If Mn is the collection of measurable sets of Rn under Lebesgue measure, what would be the first step in showing that M1 x M1 ⊆ M2. I'm quite convinced it's true, but my knowledge of and ability to work with the Lebesgue...
  5. A

    Interiors of sets in topological vector spaces

    In Rudin's book Functional Analysis, he makes the following claim about the interior A^\circ of a subset A of a topological vector space X: If 0 < |\alpha| \leq 1 for \alpha \in \mathbb C, it follows that \alpha A^\circ = (\alpha A)^\circ, since scalar multiplicaiton (the mapping f_\alpha: X...
  6. X

    Two set theory questions (isomorphism and countable sets)

    First Homework Statement Let X be a finite non-empty set and Y a countable set. Prove that XxY (X cross Y) is countable Homework Equations X isomorphic {1, 2, ..., n} for some n\inN, N is natural numbers Y isomorphic to natural numbers The Attempt at a Solution I wasn't able to...
  7. K

    Prove Sums of Cantor Sets in [0,2]

    I'm supposed to show that the sum C+C ={x+y,x,y in C}=[0,2] a) Show there exist x1,y1 in C1 for which x1+y1=s. Show in general for any arbitrary n in the naturals, we can always find xn, yn in Cn for which xn+yn=s. b) Keeping in mind that the sequences xn and yn do not necessarily converge...
  8. mheslep

    NIF Laser sets records for neutron yield

    http://www.physorg.com/news/2010-11-laser-neutron-yield-energy.html DT produces 17 MeV per fusion, so that's an output of about 800 joules; a factor of about 150X short of break even for energy applied to the target, and perhaps ~1000X short of the gain required for a practical fusion power...
  9. M

    Analysis: continuous function and open sets

    Homework Statement Let (X, p) be a metric space, and let A and B be nonempty, closed, disjoint subsets of X. define d(x,A) = inf{p(x, a)|a in A} h(x) = d(x, A)/[d(x, A) + d(x, B)] defines a continuous function h: X -> [0,1]. h(x) = 0 iff x is in A, and h(x) = 1 iff x is in B. Infer that there...
  10. K

    Countable Union/Intersection of Open/Closed sets

    Homework Statement A set A is called a F set if it can be written as the countable union of closed sets. A set B is called a G set if it can be written as the countable intersection of open sets. a) Show that a closed interval [a,b] is a G set b) Show that the half-open interval (a,b] is...
  11. T

    Ring Theory - Quaternions and sets of inverses

    Homework Statement Two questions really, the first is about the ring of quaternions H and the second about a set of maps. a) Find an element c in H such that the evaluation phi_c : C[x]-->H is not a ring homomorphism. In words that is: "the evaluation phi sub c from the ring of complex...
  12. G

    Defining Closed, Open, and Compact Sets in R^n

    Homework Statement How to define closed,, open and compact sets?Are they bounded or not? Homework Equations For example {x,y:1<x<2} The Attempt at a Solution It's is opened as all points are inner Can you please say the rule for defining the type of the set? Like for example...
  13. M

    Proof involving surjective/onto and image/preimage of sets

    EXERCISE: Suppose f is surjective, and B is a subset of Y. Prove that f(f^-1(B))=B. SOLUTION: We must show that f(f^-1(B)) is a subset of B and that B is a subset of f(f^-1(B)). I have already proven that f(f^-1(B)) is a subset of B. Now I must prove that B is a subset of f(f^-1(B)) when f is...
  14. Janus

    Why Would a Smoke Detector Go Off Without Smoke?

    Okay, I'm at work and I get a phone call from my daughter. Our smoke detector had gone off for no apparent reason. Unfortunately, it is the type that runs off of house current and the only way to turn it off is to turn the breaker off. However, that breaker also supplies power to the whole...
  15. D

    Proving the union of two open sets is open

    Homework Statement Prove if S1 and S2 are both open then S1 \capS2 is also open Homework Equations S1 is open means boundary(S1) \subset S1c Same for S2 pThe Attempt at a Solution We want to prove boundary(S1\capS2) \subset (S1 (intersection) S2)c Then idunno how to...
  16. M

    Understanding Sets: Simplifying a Confusing Explanation and Question

    I'm not quite understanding the explanation, or what the question is even asking. Anyone care to politely put it in simpler terms? Or in words or in a way that would make it easy for me to see?
  17. marcus

    Causal Sets lectures have started (video online)

    The first lecture of the series (18 October) is already online. http://pirsa.org/C10020 The series is called Invitation to Causal Sets.
  18. C

    No of blade sets in reaction turbines

    why do we need to have many sets of blades to utilize the gas pressure in reaction turbines properly? why not have just a set of comparatively wider blades?
  19. M

    Is coutnable unions of finite sets an infinite set?

    Hiya. :) While doing an assignment I ran into this little problem. We are working in the set of natural numbers \mathbb{N}. If i collect each natural number in a set S_1 = \{1\}, S_2 = \{2\},\ldots, S_n = \{n\},\ldots What happens when I take the countable union of all these? S =...
  20. B

    On the sums of elements of uncountable sets

    I want to prove the following proposition: Given any uncountable set of real numbers S, there exists a countable sub-collection of numbers in S, whose sum is infinite. Please point me in the right direction.
  21. mnb96

    Triangle inequaility for sets of same cardinality

    Hello, given a set \Omega, we consider all its subsets A_1,A_2,A_3,\ldots with same cardinality k. Do you have some hint in order to prove the following: \forall A_x,A_y,A_z\subseteq \Omega such that |A_x|=|A_y|=|A_z|=k |A_x-A_z| \leq |A_x-A_y|+ |A_y-A_z| Thanks
  22. S

    Basic Set Theory (Indexed Collection of Sets)

    Homework Statement Give an example of an indexed collection of sets {A_{\alpha} : \alpha\in\Delta} such that each A_{\alpha}\subseteq(0,1) , and for all \alpha and \beta\in\Delta, A_{\alpha}\cap A_{\beta}\neq \emptyset but \bigcap_{\alpha\in\Delta}A_{\alpha} = \emptyset.Homework Equations...
  23. P

    Comparing Worm Sets vs Spur Gears

    I'm doing a comparison on worm sets over spur gear sets and I came across a question that seems rather simple, but I can't quite resolve. Here it is: [PLAIN]http://img403.imageshack.us/img403/4848/42786849.png I do know that worm sets typically achieve high torque and a low speed gear...
  24. J

    Comparing Non-Uniform Data Sets

    I'm not sure my title is very descriptive, but I tried my best. I also hope I am posting this in the right forum. If not, please let me know. (I thought it might be better posted in the social sciences forum.) I have a project where I am analyzing the results of multiple reviewers on a set of...
  25. S

    Help me graph manually, for level sets

    Homework Statement Characterize and Sketch several level sets for the function:: z=x2+y2/2(x+y) Homework Equations N/A The Attempt at a Solution i tried to set a z, for example say for z=0.5, then i get in the form of:: x+y = x2+y2 now its the difficult part, graphing...
  26. D

    Exploring the Relationship Between Open and Closed Sets in Topology

    Homework Statement Prove that if S is open and Sc is open then boundary of S must be empty The Attempt at a Solution S is open means boundary of S is a subset of Sc Sc is open means boundary of Sc is a subset of S (By taking complement of both sides from the definition ?) This means that they...
  27. B

    Compact Sets: Need help understanding

    My professor proved the following: If C subset of X is a compact and A subset of C is closed then A is compact. Proof: Let U_alpha be an open cover of A. A subset of X is closed implies that U_0 = X\A is open. C is a subset of (U_0) U (U_alpha) and covers X. In particular they cover C...
  28. K

    Determining Boundaries and Interior Points for Rational and Irrational Numbers

    I hate to do this, but I've actually answered the questions. It's just that it seems strange to ask the student questions with such answers, as well as giving so much space to answer something so simple, I feel like I've done something wrong. Homework Statement Let set Q represent all rational...
  29. B

    Closed and Open sets in R (or 'clopen')

    I'm sure this has been asked before, but the proofs I've seen use the fact R is connected or continuous functions is some way. I'm trying to prove it with the things that have only been presented in the book so far (Mathematical Analysis by Apostol). So, let A be a subset of R which is both...
  30. S

    Solving a Problem with Sets: x+y <xy, then y>0

    Hi, I'm having a lot of trouble with the following question: Homework Statement (a) Let x,y ∈ Z. Prove that if x>0 and x+y <xy, then y>0 Homework Equations x+y <xy, then y>0 The Attempt at a Solution I am very confused with this problem, and am not even sure on how to start...
  31. J

    Does AB Equal G When |A| + |B| Exceeds |G|?

    If A and B are mere SUBSETS of G and |A| + |B| > |G|, then AB = G. My thought is that a "weakest case" would be where A has |G|/2 + 1 elements (if |A| + |B| > |G|, then one of A or B must have over half of G's elements) and |G|/2 of them form a subgroup of G. Then taking B= the subgroup, it is...
  32. marcus

    Lecture series on Causal Sets approach to QG

    I heard that Fay Dowker (Imperial College London) is scheduled to give a series of introductory lectures on CS-quantum gravity at Perimeter in October. They've been posting video of talks like this on PIRSA fairly consistently. So I expect Dowker's talks will be online--a short introductory...
  33. A

    Mapping intervals to sets which contain them

    I have recently been extremely bothered by the fact that we can construct a bijection from [0,1] onto the entire two-dimensional plane which itself contains [0,1]. Similarly, I have been bothered by the fact that we can construct a bijection from (0,1) to all real numbers. Indeed we do so...
  34. M

    Countable sets | If k:A->N is 1-to-1, then A is

    Countable sets | If k:A-->N is 1-to-1, then A is... Homework Statement Suppose we found a 1-to-1 function k that maps the set A to the set N, where N is the set of natural numbers. What can we say about the set A? Homework Equations The Attempt at a Solution The answer is 'A is...
  35. M

    Closure of Sets Proof Homework | Equations & Solution Attempt

    Homework Statement See attachment Homework Equations The Attempt at a Solution I am not sure how I should approach this first off. I have tried this 3 ways but I always decide they don't work. Click on the other attachment to see my work, It's only the first part of the first...
  36. radou

    Closures on Ordered Sets: Are the Endpoints Always Included?

    Homework Statement OK, this question may sound somewhat trivial, but I'd still like to check my answer. It's from Munkres. Let X be an ordered set with the order topology. One needs to prove that Cl(<a, b>) is a subset of [a, b] and investigate the conditions necessary for equality...
  37. A

    Is the Set of All Algebraic Numbers Countable?

    Homework Statement A complex number z is said to be algebraic if there are integers a0; a1...; an not all zero such that z is a root of the polynomial, Prove that the set of all algebraic numbers is countable. Homework Equations The Attempt at a Solution For every natural...
  38. N

    Inverse Images and Sets (union & intersection)

    Homework Statement Suppose f is a function with sets A and B. 1. Show that: I_{f} \left(A \cap B\right) = I_{f} \left(A\right) \cap I_{f} \left(B\right) Inverse Image of F (A intersects B) = Inverse Image of F (A) intersects Inverse Image of B. 2. Show by giving a counter example that...
  39. T

    Proving Compactness of K ∩ F Using Convergent Sequences

    Homework Statement Show that if K is compact and F is closed, then K n F is compact. Homework Equations A subset K of R is compact if every sequence in K has a subsequence that converges to a limit that is also in K. The Attempt at a Solution I know that closed sets can be...
  40. A

    Proving Sets and Functions Homework: f(f^-1(C)) = [C Intersection Im(f)]

    Homework Statement f: A -> B is a function with C a subset of B. Prove that f(f^-1(C)) = [C intersection Im(f)]. (f^-1(c) = f inverse of C) Homework Equations The Attempt at a Solution Please let me know how to approach to the solution (not using venn diagrams). Also if...
  41. X

    Understanding Event Spaces and Probability Domains: Solving Problems A, B, and C

    Homework Statement I have some problems that I don't know the answer to. A) Does there exist an event space, i.e., probability domain that has exactly one set? B) Is there an event space, i.e. probability domain that has exactly three sets? Is there one that has exactly four sets...
  42. V

    What is the relationship between sup of unbounded sets in real numbers?

    [b]1. sup (empty set) = -infinity, and if V is not bounded above, then sup V = +infinity. Prove if V\subseteqW\subseteqReal Numbers then sup V is lessthan/equalto supW [b]3. I used a proof by contrapositive, but I'm not sure if it is completely valid...
  43. P

    (Real Analysis) Find sets E\F and f(E)\f(F)

    Homework Statement The problem #11. The Attempt at a Solution My partial answer is attached. There, I found E\F. I still don't understand what is f(E) and f(F) and how to derive them from E and F.
  44. E

    Does Godel's Incompleteness Theorem Apply to Fuzzy Sets?

    Hello all Does Godel's incompleteness theorem still hold true for fuzzy sets? My feeling is that it doesn't since the http://en.wikipedia.org/wiki/Law_of_excluded_middle" no longer applies. Is this reasoning correct?
  45. M

    Infinite intersection of open sets

    I understand that the finite intersection of open set is open, but is it true that the infinite intersection of open set is closed? or is it possible for it to be open as well? Thank you, M
  46. A

    Prove: Set of rational numbers cannot be expressed as intersection of open sets

    Homework Statement Show that the set of rational numbers in the interval (0, 1) cannot be expressed as the intersection of a countable collection of open sets. Homework Equations The Attempt at a Solution This sounds like something requiring proof by contradiction. There must be...
  47. H

    Given 2 Sets of Data, Find 3rd (Velocity, Acceleration?)

    Homework Statement Falling 20 yards above the ground takes 6 seconds to reach the ground. Falling 100 yards above the ground takes 9 seconds to reach the ground. How long would it take to reach the ground from 70 yards?Homework Equations v=d/t ?I'm completely clueless. Any help with this...
  48. G

    What is a non-rectifiable bounded closed set in \mathbb{R}?

    Homework Statement I am trying to work my way through Analysis on manifolds by Munkres. Question 14.5 has me stumped. Any hints on how to tackle it would be appreciated. The question is: Find a bounded closed set in \mathbb{R} that is not rectifiable Homework Equations A subset S of...
  49. C

    Non-convergence written with sets

    Hey, everyone. I'm trying to prove the following: f_n and f_n are real-valued function in \Omega \{\omega: f_n(\omega) \nrightarrow f(\omega) \} = \\ \bigcup^{\infty}_{k=1} \bigcap^{\infty}_{N=1} \bigcup^{\infty}_{n=1} \{ \omega : | f_n(\omega) - f(\omega) |...
  50. ╔(σ_σ)╝

    Show that the union of countable sets is countable

    Homework Statement Show that if A_{1}, A_{2},... are countable sets, so is A_{1}\cup A_{2}\cup... Homework Equations The Attempt at a Solution Part one of the question is okay, I would like to believe I can handle that but, part B, I am not so sure. My solution is as follows ( using the...
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