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. C

    Comparing two sets of data with percentages

    Hi everyone, I was wondering if someone could help with the following: I am doing my undergraduate project and have collected two sets of answers following a survey, which I would like to compare. The questions (29 of them) are mostly Likert style, some allowing for multiple responses...
  2. A

    Infinite intersection of open sets in C that is closed

    Homework Statement Find an infinite intersection of open sets in C that is closed. The Attempt at a Solution Consider the sets A_n = (-1/n,1/n). Since 0 in A_n for all n, 0 in \bigcap A_{n}. Here I'm a little stuck -- is the proof in R analogous to the proof in C, or do I need a...
  3. T

    Linear Independence and Intersections of Sets

    Homework Statement Let E' and E'' be linearly independent sets of vectors in V. Show that E' \cap E'' is linearly independent. The Attempt at a SolutionTo show a contradiction, let E' \cap E'' be linearly dependent. Also let A be all of the vectors in E' \cap E''. Thus, A \subseteq E' and A...
  4. F

    Understanding Compact Sets: Exploring the Definition and Examples

    Hello physicsforum - I recently began self-studying real analysis on a whim and I have run into some trouble understanding the idea of compactness. I have no accessible living sources near me and I have yet to find a source that has explained the matter clearly, so I shall turn here for...
  5. S

    Inquiry about proofs involving families of sets

    Homework Statement This post does not concern a particular problem or exercise, but instead a peculiarity (for me) in one genre: proofs involving families of sets (that is, sets containing sets as elements). I have noticed that in some statements of theorems which involve families of sets...
  6. T

    Union and Intersection of Sets

    Homework Statement Let A = {x\in R | |x| >1}, B = {x\in R | -2<x<3}. Find A \cup B and A\cap B The Attempt at a SolutionI thought I might attempt this via a number line. Since I don't know how to make a number line in Latex, I'll describe it. I have A as being all of R except for the region...
  7. G

    Linear Algebra - Solution Sets of Linear Systems

    See Attachment 2 for question or read below Describe and compare the solution sets of X_1 + 5 X_2 - 3 X_3 = 0 and X_1 + 5 X_2 - 3 X_3 = -2. See Attachment 1 for answer from back of book I do not understand how the answer in the back of the book answers the question or were to even begin to get...
  8. G

    Finite Intersection of Open Sets Are Always Open?

    Suppose we have non-empty A_{1} and non-empty A_{2} which are both open. By "open" I mean all points of A_{1} and A_{2} are internal points. There is an argument -- which I have seen online and in textbooks -- that A_{1} \cap A_{2} = A is open (assuming A is non-empty) since: 1. For some x...
  9. wolram

    Finding a Bargain: 2 Ponies & 2 Sets of Tack for Just a Bag of Sand!

    So this pony was up for sale it only costs a monkey, so i went to see it, it was a little beauty so i agreed to buy it, i asked how much for the tack, the guy said a pony so i bought that as well. Then i saw the other pony and asked the guy how much, he said the same a monkey, so i bought...
  10. A

    Prove combination of two sets contains an open ball

    So this was an exam question that our professor handed out ( In class. I didn't get the question right) Let E be a subset of R^n, n>= 2. Suppose that E measurable and m(E)>0. Prove that: E+E = {x+y: x in E, y in E } contains an open ball. (The text Zygmund that we used showed an...
  11. C

    Show that two sets of vectors span the same subspace

    Homework Statement Show that the two sets of vectors {A=(1,1,0), B=(0,0,1)} and {C=(1,1,1), D=(-1,-1,1)} span the same subspace of R3. Homework Equations {A=(1,1,0), B=(0,0,1)} {C=(1,1,1), D=(-1,-1,1)} The Attempt at a Solution aA+bB=(a,a,0)+(0,0,b)=(a,a,b)...
  12. C

    Determining if certain sets are vector spaces

    Homework Statement The set of all pairs of real numbers of the form (1,x) with the operations: (1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx) k being a scalar Is this a vector space?Homework Equations (1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx)The Attempt at a Solution I verified most of the axioms...
  13. D

    Sets intersection and the axiom of choice?

    I'm working on some topology in \mathbb{R}^n problem, and I run across this: Given \{F_n\} a family of subsets of \mathbb{R}^n , then if x is a point in the clausure of the union of the family, then x \in \overline{\cup F_n} wich means that for every \delta > 0 one has B(x,\delta) \cap...
  14. Ahmed Abdullah

    What is an indexed family of sets. I need a simple example

    I have looked it in the Wikipedia, but no simple example. So I am not sure. Is the indexed family of sets just power sets, indexed (indexing means labeling as I understand)? For example the indexed family of sets of set A ={1,2,3,4,5,6} is just the collection of element from power set. A sub 1...
  15. M

    Equal sets and bijective correspondence

    Homework Statement If [n] and [m] are equal, then they are bijective correspondent. I define f \subset\{(n,m)\mid n \in [n], m\in [m]\}. Suppose [n]=[m]. Let(n,m_1),(n,m_2)\in f. Because [n]=[m], then m_1=m_2. So for all n \in [n], there exists a unique m\in [m] such that f(n)=m. So f...
  16. R

    Is the following correct? (concerns sets and convergence)

    Let A = {(x,y) in R^2 | x^2 + y^2 <= 81} Let B = {((x,y) in R^2 | (x-10)^2 + (y-10)^2 <= 1} then here "A intersection B" is the empty set. Then let x_n be the sequence (0,10-(2/n)) which is a sequence in A and y_n be the sequence (10/n,10) which is a sequence in B. would |x_n - y_n| tend...
  17. S

    Real Analysis Question: Sequences and Closed Sets

    Homework Statement Let {xn} be a sequence of real numbers. Let E denote the set of all numbers z that have the property that there exists a subsequence {xnk} convergent to z. Show that E is closed. Homework Equations A closed set must contain all of its accumulation points. Sets with no...
  18. J

    Is the Intersection of an Infinite Collection of Open Sets Always Open?

    Homework Statement a) If U and V are open sets, then the intersection of U and V (written U \cap V) is an open set. b) Is this true for an infinite collection of open sets? Homework Equations Just knowledge about open sets. The Attempt at a Solution a) Let U and V be open...
  19. I

    Distance between sets (a triangle-type inequality)

    I've been reading a book called Superfractals, and I'm having trouble with a particular proof: Definitions: The distance from a point x \in X to a set B \in \mathbb{H}(X) (where \mathbb{H}(X) is the space of nonempty compact subsets of X is: D_B(x):=\mbox{min}\lbrace d(x,b):b \in B\rbrace The...
  20. R

    Mathematica Mathematica: Integrating over data sets?

    I've got a Mathematica question which might be quite basic, but I couldn't find much about it in the documentation (possibly because it's so basic) so please bear with me! I have a set of data, call it xi(ρ), which I want to integrate over some distribution function (log-normal in this case)...
  21. N

    Can a Set of Four Vectors in ℝ³ Span the Space?

    Homework Statement Consider a set of vectors: S = {v_{1}, v_{2}, v_{3}, v_{4}\subset ℝ^{3} a) Can S be a spanning set for ℝ^{3}? Give reasons for your answer. b) Will all such sets S be spanning sets? Give a reason for your answer. The Attempt at a Solution a) Yes, because a...
  22. P

    Compact Sets of Metric Spaces Which Are Also Open

    Are there any down to Earth examples besides the empty set? Edit: No discrete metric shenanigans either.
  23. J

    Proof that perfect sets in R^k are uncountable

    In Rudin's Principles of Mathematical Analysis, Theorem 2.43 is that all nonempty perfect sets in R^k are uncountable. The proof Rudin gives goes like this: Let P be a nonempty perfect set in R^k. Since P has limit points, P is infinite. Suppose P is countable and denote the points of P by...
  24. E

    Inherent negativity of seemingly symmetric finite integer sets

    Hi everyone. My first post on this great forum, keep up all the good ideas. Apologies if this is in the wrong section and for any lack of appropriate jargon in my post. I am not a mathematician. I have a theory / lemma which I would like your feedback on:- Take a finite set S of integers which...
  25. D

    Spanning sets and polynomials.

    Homework Statement Do the polynomials t^{3} + 2t + 1,t^{2} - t + 2, t^{3} +2, -t^{3} + t^{2} - 5t + 2 span P_{3}? Homework Equations N/A The Attempt at a Solution My attempt: let at^{3} + bt^{2} + ct + d be an arbitrary vector in P_{3}, then: c_{1}(t^{3} + 2t + 1) + c_{2}(t^{2} -...
  26. M

    Linear Algebra - showing sets are linearly independent/dependent

    Homework Statement Using the fact that a set S is linearly dependent if and only if at least one of the vectors, vj, can be expressed as a linear combination of the remaining vectors, obtain necessary and sufficient conditions for a set {u,v} of 2 vectors to be linearly independent. Determine...
  27. A

    Infinite union of closed sets that isn't closed?

    So I have to find an infinite union of closed sets that isn't closed. I've thought of something that might work: \bigcup[0,x] where 0\leq x<1. Then, \bigcup[0,x] = [0,1), right?
  28. A

    Is the Arbitrary Union of Open Sets in R Open?

    I have to prove that the arbitrary union of open sets (in R) is open. So this is what I have so far: Let \{A_{i\in I}\} be a collection of open sets in \mathbb{R}. I want to show that \bigcup_{i\in I}A_{i} is also open... Any ideas from here?
  29. D

    The set of all sets which are elements of themselves

    Russell's paradox concerns itself with the set S=\{x|x\notin x\}\;\;\;S\in S ? but it is supposedly solved in ZFC theory. Now, what about the set U=\{y|y\in y\}\;\;\;U\in U ? Is U an element of itself?
  30. S

    P(5 sets of twins in the SD state football tournament)

    All, I'm guessing that I'm vastly over- or under-thinking this one, but here it is. Question: In the South Dakota State High School Football Tournament, there are 12 teams competing with a total across all 12 teams of 555 players. Among the 555 players there are 5 sets of twins (i.e., 10...
  31. B

    Do tuples exist which aren't elements of a cartesian product of sets?

    Do tuples exist which aren't elements of a cartesian product of sets? Can you just write an ordered list of elements which does not necessarily have to be defined in sets? (or does every tuple need to be defined through sets in order for it to rigourously exist in mathematics?)
  32. Y

    Why is the differential being onto equivalent to it not being zero?

    I have difficulty understanding the following Theorem If U is open in ℝ^2, F: U \rightarrow ℝ is a differentiable function with Lipschitz derivative, and X_c=\{x\in U|F(x)=c\}, then X_c is a smooth curve if [\operatorname{D}F(\textbf{a})] is onto for \textbf{a}\in X_c; i.e., if \big[...
  33. P

    Point-wise continuity on all of R using compact sets

    Ok, so basically I am trying to decide whether my mathematics is valid or if there is some subtly which I am missing: Lets say I have a 1-1 strictly increasing point-wise continuous function f: R -> R, and I want to show that the inverse function g: f(R) -> R is also point-wise continuous...
  34. P

    I have two sets of deterministic numbers, collected in the two

    I have two sets of deterministic numbers, collected in the two vectors: x=[x(1),...,x(n)] and y=[y(1),...,y(n)]. My (determinstic) theory says that x(i)=y(i) for all i=1,...,n. But instead, I want to assume that the numbers x and y are stochastic. If we let f(.) be a pdf, does this mean that I...
  35. G

    Understanding Countable Sets in Measure Zero Definition

    Homework Statement While studying a book "analysis on manifolds" by munkres, I see a definition of measure zero. That is, Let A be a subset of R^{n}. We say A has measure zero in R^{n} if for every ε>0, there is a covering Q_{1},Q_{2},... of A by countably many rectangles...
  36. G

    Absolutely continuous functions and sets of measure 0.

    Homework Statement Prove that if f: [a,b] -> R is absolutely continuous, and E ∁ [a,b] has measure zero, then f(E) has measure zero. Homework Equations A function f: [a,b] -> R is absolutely continuous if for every ε > 0 there is an δ > 0 such that for every finite sequence {(xj,xj')}...
  37. T

    Shortest Distance between 2 convex sets

    Hi, I hope someone can help me out with this problem: Let set S be defined by (x in En :f(x) <=c} f: En -> E1 is convex and differentiable and gradient of f(xo) is not 0 when f(x) = c. Let xo be a point such that f(xo) = c and let d = gradient at xo. Let lamda be any positive number and...
  38. J

    (a) If A and B are disjoint closed sets in some

    (a) If A and B are disjoint closed sets in some ... Homework Statement (a) If A and B are disjoint closed sets in some metric space X, prove that they are separated. (b) Prove the same for disjoint open sets. (c) Fix p in X, ∂ >0, define A to be the set of all q in Z for which d(p,q)...
  39. S

    Apostol 1.19 - Understanding where my logic went wrong (Sets, sup, inf)

    Okay, so I'm struggling with understanding where I went wrong. The instructor feels like I don't understand the material and when she presented my explanation to a colleague, he too agreed with her. I would really appreciate if someone could tell me the first part of where I went wrong in my...
  40. D

    Discrete math - Infinite sets having the same cardinality.

    From a pdf textbook: Example (infinite sets having the same cardinality). Let f : (0, 1) → (1,∞) be defined by f(x) = 1/x. Then f is a 1-1 correspondence. (Exercise: prove it.) Therefore, |(0, 1)| = |(1,∞)|. Exercise. Show that |(0,∞)| = |(1,∞)| = |(0, 1)|. Use this result and the fact that (0,∞)...
  41. E

    Logical Proofs Regarding Sets and Subsets

    Homework Statement The following is all the information needed: Homework Equations There are, of course, all the basic rules of logic and set identities to be considered. The Attempt at a Solution Not really sure how to attempt this one, to be honest. I know that (A ⊆ B) can...
  42. N

    Understanding Connectedness in Planar Sets: A Brief Overview

    I don't get the meaning of "connected" in the chapter of planar sets. The textbook said " An open set S is said to be connected if every pair of points z1, z2 in S can be joined by a polygonal path that lies entirely in S" So do i just randomly pick 2 points in S to check if they are both in...
  43. L

    What is it that sets the speed limit of light?

    There are quite a number of threads (and FAQ's) that discuss why the speed of light is the "number" that it is, but I'm having difficulty finding some information on what is causing, or setting, the limit. So, let me ask and answer a different question as an example of the what I am tryinh to...
  44. S

    Totally Bounded Sets: Proving Closure is Also Totally Bounded

    Homework Statement Show that If S is totally bounded in ℂ, then the S closure is also totally bounded in ℂ. Homework Equations The Attempt at a Solution Assume S is totally bounded. then for very ε>0 there are finitely many discs (O=Union of finitely many discs) that covers S...
  45. L

    Cantor sets, Fat cantor sets and homeo and diffeo

    Homework Statement Ok so the first part of the problem was constructing fat cantor sets, cantor sets with some positive lesbegue measure. The second part involved proving that any fat cantor set is homeomorphic to the regular cantor set. The third part asks whether there is a diffeomorphism...
  46. T

    Bijection between products of countable sets

    Homework Statement Let S1 = {a} be a set consisting of just one element and let S2 = {b, c} be a set consisting of two elements. Show that S1 × Z is bijective to S2 × Z. Homework Equations The Attempt at a Solution So I usually prove bijectivity by showing that two sets are...
  47. K

    Showing two sets are not homeomorphic in subspace topology.

    Homework Statement The true problem is too complicated to present here, but hopefully somebody can give me a hand with this simplified version. Consider the set H = \{ (x,y) \in \mathbb R^2 : y \geq 0 \} . Denote by \partial H = \{ (x,0) \}. Let U and V be open sets (relative to H) such that...
  48. S

    Uncountable family of disjoint closed sets

    Homework Statement Determine whether the following statements are true or false a) Every pairwise disjoint family of open subsets of ℝ is countable. b) Every pairwise disjoint family of closed subsets of ℝ is countable. Homework Equations part (a) is true. we can find 1-1...
  49. S

    How can I prove the properties of points in a Cantor set?

    Homework Statement Let C be a Cantor set and let x in C be given prove that a) Every neighborhood of x contains points in C, different from x. b) Every neighborhood of x contains points not in C Homework Equations How can I start to prove? The Attempt...
  50. K

    Metric Space Diameters of Sets: Find Condition

    Homework Statement Find a condition on a metric space (X,d) that ensures that there exist subsets A,B of X with A\subset B such that diam(A)=diam(B).Homework Equations diam(A)=\sup\{d(r,s):r,s\in A\}; A\subseteq B\implies diam(A)\leq diam(B).The Attempt at a Solution Well I know examples of...
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