Sets Definition and 1000 Threads

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. K

    Distance between Sets and their Closures

    Homework Statement Suppose (X,d) is a metric space, and suppose that A,B\subseteq X. Show that dist(A,B)=dist(cl(A),cl(B)). Homework Equations cl(A)=\partial A\cup A. dist(A,B)=\inf \{d(a,b):a\in A,b\in B\}The Attempt at a Solution Its clear that dist(cl(A),cl(B))\leq...
  2. C

    Counting Passwords with Restrictions

    Counting Lists With Repetition Homework Statement How many ways can you create an 8 letter password using A - Z where at most 1 letter repeats? Homework Equations The Attempt at a Solution I'm not sure how to attack this problem but first I thought that A-Z considers 26 letters...
  3. S

    Algebraic Geometry Question - on ideals of algebraic sets

    Hello everyone, I was wondering if I could get some advice for the following problem: I have two algebraic sets X, X', i.e. X = V(J), Y = V(J'), and let I(X),I(Y) be the ideals of these sets, i.e. I(X) ={x \in X | f(x) = 0 for all x \in X}. I am trying to show that I(X \cap Y) is not always...
  4. F

    Proving Set Inclusion: A \subseteq A \cup B

    Homework Statement For the sets A and B, prove that A \cap B \subseteq A \subseteq A \cup B The Attempt at a Solution I am guessing I should look at only two of them first? A \subseteq A \cup B What conditions do I need?
  5. G

    Proving Compact Sets Must Be Closed

    Homework Statement Show that every compact set must be closed. I am looking for a simple proof. This is supposed to be Intro Analysis proof. Relevant equations Any compact set must be bounded. The Attempt at a Solution Suppose A is not closed, so let a be an accumulation...
  6. F

    Understanding Elements and Subsets in Set Theory

    Homework Statement Suppose A = {1,{1},{1,{1}}} Then is {{{1}}} an element of A? The Attempt at a Solution I am thinking A has the elements are only 1, {1}, {1, {1}} But {{{1}}} has only the element {{1}} While A has the element {1,{1}}, you can't just take out the...
  7. M

    A set is closed iff it equals an intersection of closed sets

    Homework Statement Let M be a metric space, A a subset of M, x a point in M. Define the metric of x to A by d(x,A) = inf d(x,y), y in A For \epsilon>0, define the sets D(A,\epsilon) = {x in M : d(x,A)<\epsilon} N(A,\epsilon) = {x in M: d(x,A)\leq\epsilon} Show that A is...
  8. E

    Do the infinite cardinals correspond to sets?

    For the infinite cardinal numbers (of which there are infinitely many), do they each necessarily correspond to some set? I mean we know that aleph-naught corresponds to N, c (aleph-one by continuum hypothesis) corresponds to R, but what about all the other infinite cardinals? Is it possible...
  9. A

    Question about sets and its closure.

    I was wondering, is S a subset of S-bar its closure? For example, if p belongs to S, does p belong to S-bar too? Does it go the other way, S-bar is a subset of S? If it is true that S is a subset of S-bar does this automatically mean that S is closed? Thanks
  10. G

    How Many Unique Outcomes are Possible in a 15-Team Soccer Tournament?

    Homework Statement In a soccer tournament of 15 teams, the top three teams are awarded gold, silver, and bronze cups, and the last three teams are dropped to a lower league. We regard two outcomes of the tournament as the same if the teams that receive the gold, silver, and bronze cups...
  11. A

    Lebesgue integration over sets of measure zero

    Is it true in general that if f is Lebesgue integrable in a measure space (X,\mathcal M,\mu) with \mu a positive measure, \mu(X) = 1, and E \in \mathcal M satisfies \mu(E) = 0, then \int_E f d\mu = 0 This is one of those things I "knew" to be true yesterday, and the day before, and the...
  12. N

    Is there a way to construct an open set whose boundary is A?

    Homework Statement Prove: a set in a topological space is closed and nowhere dense if and only if it is the boundary of an open set.Homework Equations Basic definitions of closed, nowhere dense, open and boundary.The Attempt at a Solution One direction is easy. Let A \subset X be a subset in...
  13. R

    Limit Points of Sets: Find Interior, Boundary & Open/Closed

    Homework Statement Consider the set in E^2 of points {(x,y)|(x,y)=(1/n,1-1/n), where n is a positive integar}. Find the limit points, interior points and boundary points. Determine whether this set is open or closed. Homework Equations The Attempt at a Solution I figured, 0,1 must...
  14. F

    Linear Programming - Feasible sets?

    Homework Statement Consider the following LOP K: Max z = 4x_1 + 3x_2 s.t 7x_1 + 6x_2 \leq 42 2x_1 - 3x_2 \geq -6 x_1 \geq 2, x_1 \leq 5, x_2 \geq \frac{1}{2} (a) Decide whether the solution sets are feasible i) x = (3.5, 2.5)^t ii) x = (2.1, 4)^t iii) x = (5,1/2)^t (b) Graph the...
  15. J

    Uncountable infinite sets that are not continuous

    Can you give some examples of the infinite sets that are uncountable and that are not continuous? I know the infinite sets that are countable and discrete, and I know the continuous sets, but couldn't find an example for the above situation.
  16. L

    Can open sets be written as unions of intervals?

    A theorem of real analysis states that any open set in \Re^{n} can be written as the countable union of nonoverlapping intervals, where "interval" means a parallelopiped in \Re^{n}, and nonoverlapping means the interiors of the intervals are disjoint. Well, what about something as simple as an...
  17. S

    Sketching Complex Sets Homework | Set Sketching Tips

    Homework Statement I'm having some major trouble this these two questions. Sketch the set, s, where s = {z| | z^2 - 1 | < 1 } ... z is a complex number Sketch the set, s, where s = {Z| | Z | > 2 | Z - 1 | } ... Z is a complex number 2. The attempt at a solution This is supposed to...
  18. A

    Intersection of sets with infinite number of elements

    I have to decide whether the following is true or false: If A1\supseteqA2\supseteqA3\supseteq...are all sets containing an infinite number of elements, then the intersection of those sets is infinite as well. I think I found a counterexample but I'm not sure the correct notation. I to...
  19. I

    Help indexed family sets proof

    Homework Statement Ai and Bi are indexed families of sets. Prove that Ui (Ai \bigcap Bi) \subseteq (UiAi) \bigcap (UiBi). Homework Equations The Attempt at a Solution Suppose arbitrary x. Let x \in {x l \foralli\inI(x\inAi\bigcapBi) This means x \in{x l...
  20. M

    A question about intersection of system of sets

    The book I am reading says that \bigcap \phi because every x belongs to A \in \phi(since there is no such A ) , so \bigcap S would have to be the set of all sets. now my question is why every x belongs to A \in \phi.In other word I don't completely understand what this statement mean. sorry if...
  21. E

    How can I use Matlab to plot forces and velocities from level sets?

    Homework Statement Hi, someone could help to draw the forces or the velocities in matlab to check if they are properly calculated?. Homework Equations I am adding the forces like that phi0 = phi0+dt.*force; so I do not know how to get the velocity. but I would like to get the...
  22. E

    Calculating Overall Variance & Standard Deviation for 3 Sets of Data

    Hi If I have measured the resonance frequency of three sets of resonators and calculated the mean, variance and standard deviation for each set. How do I add the three variances and standard deviations to get an overall variance and standard deviation? Well, I know that the standard...
  23. M

    Infinite Intersections of Infinite Sets

    Homework Statement Decide if the following represents a true statement about the nature of sets. If it does not, present a specific example that shows where the statement does not hold: If A_{1}\supseteqA_{2}\supseteqA_{3}\supseteqA_{4}\supseteq...A_{n} are all sets containing an...
  24. X

    Understanding Closed and Open Sets in R^d

    This is the question: Let A be an open set and B a closed set. If B ⊂ A, prove that A \ B is open. If A ⊂ B, prove that B \ A is closed. Right before this we have a theorem stated as below: In R^d, (a) the union of an arbitrary collection of open sets is open; (b) the intersection of any...
  25. E

    Finding the Union and Intersection of Indexed Collections of Sets

    Homework Statement Let I denote the interval [0,\infty). For each r \in I, define Ar = {(x,y) \in RxR : x2+y2 = r2}, Br = {(x,y) \in RxR : x2+y2 \leq r2}, Cr = {(x,y) \in RxR : x2+y2 > r2} a) Determine \bigcupr\inIAr and \bigcapr\inIAr b) Determine \bigcupr\inIBr and...
  26. S

    Question regarding algebra of sets

    Homework Statement I'm working on a proof for Intro. to Algebra, and the problem deals with proving that the operation of Symmetric Difference imposes group structure on the power set 2^x (set of all subsets) of a given set X. There is a part of my proof that I'd like to get some advice on...
  27. T

    Truth Sets for x and y in x^2 + y^2 < 50

    Homework Statement What are the truth sets of the following statements? List a few elements of the truth set if you can. c) x is a real number and 5\in{y\inR|x^{2}+y^{2}<50} The Attempt at a Solution I believe this says 5 is a member of the set of possible values for y, while y is...
  28. J

    Confused by separate definitions of sets which are bounded above

    I have been consulting different sources of analysis notes. My confusion comes from these two definitions \begin{defn} Let S be a non-empty subset of $\mathbb{R}$. \begin{enumerate} \item $S$ is Bounded above $ \Longleftrightarrow\exists\,M > 0$ s.t. $\forall\, x\in S$, $x\leq M$...
  29. N

    Demonstration with completes sets

    1. We have A\subseteq \mathcal{U}. For i_1, i_2 \in \{0,1\}and A^0 := A^c, A^1 := A. A is a complete set if A\cap A_1 ^{i1} \cap A_2^{i2} \neq \emptyset then A_1 ^{i1} \cap A_2^{i2} \subseteq A Demonstrate that A_1, A_2 are complete sets too. And if A is a complete set then A^c is a complete...
  30. K

    Is It Possible to Construct Non-Measurable Sets Without the Axiom of Choice?

    "Concrete" non-measurable sets I've had Vitali's proof of the existence of non-(Lebesgue) measurable sets branded into the side of my brain over the years. However, the proof always critically relies on evoking the axiom of choice. Has anybody every demonstrated a non-AoC construction of a...
  31. WannabeNewton

    Simple Sets Homework: Q on f[A] & f[B] General Statement

    Homework Statement All the b's in f[b] should be capitalized for the problem statement and attempt; I had it in the latex but it showed up lower case in the post I don't know why, my apologies =p. If f:X \mapsto Y and A \subset X, B \subset X, is: (a) f[A \cap B] = f[A] \cap f[B] in...
  32. F

    Sets and Algebraic Structures, help with equivalence relations

    Let Q be the group of rational numbers with respect to addition. We define a relation R on Q via aRb if and only if a − b is an even integer. Prove that this is an equivalence relation. I am very stumped with this and would welcome any help Thank you
  33. L

    Solving Supremum of Sets Homework Statement

    Homework Statement Let A be a set of real numbers that is bounded above and let B be a subset of real numbers such that A (intersect) B is non-empty. Show that sup (A(intersect)B) <= sup A The Attempt at a Solution I don't know how to start but tried this... Let C = A (intersect) B So...
  34. B

    Closed sets in Cantor Space that are not Clopen

    Hi, Is there a characterization of subsets of the Cantor space C that are closed but not open? As a totally-disconnected set/space, C has a basis of clopen sets; but I'm just curious of what the closed non-open sets are.
  35. S

    What Are the Open Sets of U(N)?

    Hi people, Let U(N) be the unitary matrices group of a positive integer N . Then, U(N) can be viewed as a subspace of \mathbb{R}^{2N^2} . I am curious what the open sets of U(N) are in this case. If it has an inherited topology from GL(N,\mathbb{C}) , what are the open sets of...
  36. P

    Creating a 'For' Loop to Calculate Answers for Sets of 11 Values

    I have a 5000x1 vector and am trying to write a function to calculate an answer for entry 1-11, then 12-22, then 23-33, etc. ... I've been trying to use a 'for' loop, basically: for i = (??) x=i+1 end Not sure what to put in the ? area. I want it to spit out answers for each set of...
  37. Fredrik

    Closed Subsets and Limits of Sequences: A Topology Book Example

    Anyone have a good example of a closed subset of a topological space that isn't closed under limits of sequences?
  38. Z

    Comparing Best-fits of different data sets that have different noise levels

    Suppose I have 2 sets of data: day1 and day2. I want to fit a model to both data sets and then compare them to each other to see which one fits the model the best (the fit is done with a computer using non-linear least squares method). The RMS of the fit would be fine except that day1 has much...
  39. N

    Does a logic with set of all sets exist?

    Hello. As I understand, in the classical logic it's impossible to "take", for example, the set of all sets. I was wondering: is it possible to create a logic where that is possible by changing some of the basic postulates by which logic works? Or is it impossible for all logics? Thank you.
  40. N

    Determining whether sets of matrices in a vectorspace are linearly independent?

    Given matrices in a vectorspace, how do you go about determining if they are independent or not? Since elements in a given vectorspace (like matrices) are vector elements of the space, I think we'd solve this the same way as we've solved for vectors in R1 -- c1u1 + c2u2 + c3u3 = 0. But I'm...
  41. D

    Open and closed sets in metric spaces

    From the definition of an open set as a set containing at least one neighborhood of each of its points, and a closed set being a set containing all its limit points, how can we show that the complement of an open set is a closed set (and vice versa)? Usually this is taken as a definition, but...
  42. Z

    Topology question: finite sets

    Homework Statement If A and B are finite, show that the set of all functions f: A --> B is finite. Homework Equations finite unions and finite caretesian products of finite sets are finite The Attempt at a Solution If f: A -> B is finite, then there exists m functions fm mapping to...
  43. S

    Product of compact sets compact in box topology?

    So Tychonoff theorem states products of compact sets are compact in the product topology. is this true for the box topology? counterexample?
  44. P

    Proving the Infinity of C\B: A Contradiction Method

    If C is an infinite set and B is a finite set then C\B is an infinite set. C\B means the complement of B relative to C Ok so I was thinking of doing this by contradiction. I have Assume C\B is a finite set. Then there exist a function\alpha that is bijective from C\B to Nk for some k...
  45. D

    Convergence of Countable Sets and the Counting Measure

    Homework Statement Suppose \Omega is an infinite set. If Q = \{x_1,x_2,...\} \subset \Omega is infinite and countable, and if B_n := \{x_1,x_2,...,x_n\}, A_n := Q - B_n , ... does A_n \downarrow \emptyset? If \mu is the counting measure on \Omega, is \lim_{n \to \infty} \mu (A_n) = 0?The...
  46. M

    Infinite intersection of indexed sets

    Every element of a set A can be written a=w.a_1a_2a_3\ldots{a_n}\ldots with w, a_n\in\mathbb{Z} and 0\leq a_n\leq9 for every n\in\mathbb{N}. If A is bounded, there exists a greatest whole part \overline{w} of the elements of A, and because any set S of elements a_n is bounded, for every n, there...
  47. M

    Infinite intersection of indexed sets

    Consider the set A_n=\{0.9, 0.99, 0.999,...\} , where the greatest element of A_n has n 9s in its decimal expansion. Then 0.999\ldots=1\in\bigcap_{n=1}^\infty{A_n}. Is this possible even though \not\exists{n}(1\in{A_n})? Edit: I see that 0.999\ldots=1\not\in\bigcap_{n=1}^\infty{A_n}...
  48. U

    Symmetric difference in sets

    Homework Statement There is a symmetric difference in sets X & Y, X Y is defined to be the sets of elements that are either X or Y but not both Prove that for any sets X,Y & Z that (X\oplusY)\oplus(Y\oplusZ) = X\oplusZ Homework Equations \oplus = symmetic difference The Attempt at...
  49. Z

    How deep Sets affect Measure Theory?

    Guys, I'm taking real analysis starting with open, close, compact sets, and neighborhoods. Now I'm addict to rely on these concepts to do my proofs. In the future I will have to take Measure Theory. Can anyone give me a percentage indication for how many percent theorems are proven by the set...
  50. W

    Examples of infinite/arbitrary unions of closed sets that remain closed.

    Hello, I am trying to think of examples of these. At the moment, I can only think of ( on R ) closed intervals being the union of single-point sets ( infinitely many, the ones inside ).. et c. I also think the cantor set is an example of this. Are there more "natural" examples? Thank you for...
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