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. S

    Is the statement on the UNION and INTERSECTION of Indexed Sets always true?

    Homework Statement Can't quite figure out the LaTeX for Indexed Sets, so bear with me: From "Book of Proof" Section 1.8 #11 http://www.people.vcu.edu/~rhammack/BookOfProof/index.html Is the UNION of Aa, where a is in I, a subset of the INTERSECTION of Aa always true for any collection of sets...
  2. S

    Rationale of infinitely countable sets

    What is the nature of countable infinitity? (warning: pseudo-philosophical question follows) I can illustrate the question better in specific context: Axiom for the existence of successor operation that is capable of "counting" the set of natural numbers. Is there some phenomenon that...
  3. pairofstrings

    How Do Sets A and B Relate Through Equivalence Properties?

    How to define relationship between two sets (mathematics)? Please, let me make an assumption here. Assumption : We have a set 'A' and it has a equivalent relation with another set 'B'. Now, remember that for a set to have an equivalent relation with itself or another set it should satisfy...
  4. F

    Identifying Compact Sets in the Slitted and Moore Planes: What's the Method?

    Is there any easy way to find all the compact sets of 1) the slitted plane and 2) the Moore plane? 1) defined as the topology generated by a base consisting of z\cup A where A is a disc about z with finitely many lines deleted. I believe the compact sets in this topology coincide with the...
  5. T

    Lower Limit Topology Clopen Sets

    Homework Statement Let T be the lower-limit topology on R. Is (R,T) connected? Prove your answer. The Attempt at a Solution Since there exists a proper subset V of R such that V is both open and closed (since all intervals of the form [a,b) are open and closed), then (R,T) is...
  6. J

    There exists a set of all finite sets?

    Homework Statement There exists a set of all finite sets? Prove your answer. Homework Equations The Attempt at a Solution 1. Assume there is a set A of all finite sets 2. Take the power of set of A This is where i get stuck...what I'm thinking is that there exists a element...
  7. J

    Proving E is Measurable with Compact Sets

    Homework Statement Prove that E is measurable if and only if E \bigcap K is measurable for every compact set K. Homework Equations E is measurable if for each \epsilon < 0 we can find a closed set F and an open set G with F \subset E \subset G such that m*(G\F) < \epsilon. Corollary...
  8. C

    Understanding Convex Analysis: Solving a Sequence of Sets

    Homework Statement Hello! I'm having some trouble trying to understand basic concepts of Convex Analysis (I study it independently). In particular, I have a book (Convex Analysis and Optimization - Bertsekas) which gives a definition for the convergence of a sequence of sets: Homework...
  9. srfriggen

    Proving Well-Ordered Subsets of Real Numbers

    Homework Statement "Prove that if A is any well-ordered set of real numbers and B is a nonempty subset of A, then B is also well-ordered" Homework Equations The Attempt at a Solution If B \subseteq A, then B\subseteq{x1, x2, x3...xn}. Since B\subseteqA, the smallest...
  10. R

    Family Of Sets Intersecting once

    Homework Statement This isn't exactly a homework question, but I saw it in a book, and I think it is interesting: Let S_1,\ldots,S_n be a collection of subsets of [tex]\{1,\ldots,n\}[/itex] such that |S_i \cap S_j| \leq 1 whenever i\neq j. Then the total number of elements in all of the lists...
  11. nomadreid

    Real-valued measurable cardinals versus Vitali sets

    If there exists a real-valued measurable cardinal, then there is a countably additive extension of Lebesgue measure to all sets of real numbers. This would include then the Vitali sets, which are an example of sets that are not Lebesgue measurable for weaker assumptions than the existence of a...
  12. 0

    How Can a Finite Set of Natural Numbers Be an Open Set in Topology?

    So I want to start learning about topological spaces, however I couldn't even get past the definition. My book states: DEFINITION (Open sets, neighborhoods) Let E be an arbitrary set. A topology on E is the data of a set O of subsets of E, called the open subsets of E (for the given topology)...
  13. K

    Linearly Independent Sets After Subtraction

    Homework Statement Here is a really simple lin.alg problem that for some reason I'm having trouble doing. Assume that \left\{ v_i \right\} is a set of linearly independent vectors. Take w to be a non-zero vector that can be written as a linear combination of the v_i . Show that \left\{ v_i...
  14. N

    What is the Impact of Functions on Sets?

    Homework Statement It is located in the pdf or the following link: http://www.scribd.com/doc/50331847/hw". There seems to be 2 typos: "1. There should not be an initial "If" at the beginning. 2. The intersection on the right side of the equation should be a union, I believe." Homework...
  15. mnb96

    Cardinality of sets of functions

    Hello, let's consider the set \Omega of all the continuous and integrable functions f:R \to R. Suppose we now take two subsets A and B, where: - A is the subset of all the gaussian functions centered at the origin: \exp(-ax^2) , where a>0 - B is the set of all the even functions...
  16. G

    How Do You Prove Subset Relationships Within Intersecting Indexed Sets?

    Homework Statement Show that the intersection of Ai (for all i in I = {1, 2, 3, ... n } = A1. Ai is a subset of Aj whenever i <= j.Homework Equations The Attempt at a Solution Show: ***I'm having trouble showing part 1***1. that the intersection of Ai is a subset of A1, and 2. A1 is a subset of...
  17. M

    Real Analysis(open nor closed sets)

    Hello, I am having trouble finding an example of a set in R^2 that is neither open nor closed. I have already shown the half open interval [0,1) is neither open nor closed, but I can't seem to find any other examples. Can someone push me in the right direction? Would x^2+ y^2<1 be open nor...
  18. S

    Prove that the intersection of any collection of closed sets in a topological space X

    Prove that the intersection of any collection of closed sets in a topological space X is closed. Homework Statement Homework Equations The Attempt at a Solution
  19. K

    Intersection of axiomatizable sets

    Homework Statement Suppose that T and F are both axiomatizable, complete, consistent theories. Is T\cap F axiomatizable? Homework Equations A theory T is a set of sentences such that if yo can deduce a sentence a from T, then a is in T. I have already proved that T\cap F is a theory...
  20. S

    Tic: Relations & Sets: A Subset Possibility?

    Hello guys, I am new to this forum. I have a question: A relation can be subset of some other relation? For example? I have the relations X: A <---> B Y: B <---> C Z: A <---> C X...
  21. S

    Combinatorics: Choosing Sets of Distinct Digits

    Homework Statement How many integers between 1000 and 10000 are there with distinct digits (no leading zeros) and at least one of 2 and 4 must appear? Homework Equations None The Attempt at a Solution I am discounting the case of 10,000 since that has repeating digits. Thus...
  22. A

    Matrix Elements of Operators & Orthonormal Basis Sets

    So, the rule for finding the matrix elements of an operator is: \langle b_i|O|b_j\rangle Where the "b's" are vector of the basis set. Does this rule work if the basis is not orthonormal? Because I was checking this with regular linear algebra (in R3) (finding matrix elements of linear...
  23. L

    Expanding and Simplifying Two Sets of Double Brackets (most likely really easy)

    Homework Statement Expand and Simplify 6(x-1)(x+2)-(1-x)(2+x) Homework Equations The answer in the book is 7x^2+7x-14 The Attempt at a Solution I've tried several ways, but all of them give the wrong answer. I'm not sure what to do with the negative in the middle. I know it...
  24. I

    Proving Union of Countable Sets is Countable

    Homework Statement If S is a countable set and {Ax}(s element S) is an indexed family of countable sets, then U(s element S) As is a countable set. Homework Equations The Attempt at a Solution S is countable means it is finite or countable infinite ( S equivalent to J set of...
  25. R

    Proving Finite Convex Sets Intersection is Convex

    Homework Statement Prove that the intersection of a number of finite convex sets is also a convex set Homework Equations I have a set is convex if there exists x, y in the convex S then f(ax + (1-a)y< af(x) + (1-a)y where 0<a<1The Attempt at a Solution i can prove that f(ax + (1-a)y) <...
  26. T

    Archimedean property for unbounded sets

    Does the Archimedean property work for unbounded sets? My book does a proof of the Archimedean property relying on the existence of sup which relies on the existence of a bound.
  27. A

    Question about nowhere dense sets

    Suppose you know k_0A, for some set A \subset X (where X is a metric space) and some constant k_0, has nonempty interior. Do you then know that A has nonempty interior, and/or that k A has nonempty interior for any constant k?
  28. S

    What Constitutes an Algebraic Structure in Set Theory?

    I just found out that the words "algebraic structure" have a precise definition and that this notion is not just common language! Then below this definition they give another (equivalent) definition: So based on this I have three questions: 1: How is this concept explained in terms of...
  29. S

    Countable union of countable sets vs countable product of countable sets

    I know that a countable union of countable sets is countable, and that a finite product of countable sets is countable, but even a countably infinite product of countable sets may not be countable. Let X be a countable set. Then X^{n} is countable for each n \in N. Now it should also be true...
  30. G

    Proving Bounded Open Sets Union of Disjoint Open Intervals

    I am asked to prove that any bounded open subset of R is the union of disjoint open intervals. If S = open interval (a,b), I don't really see how this could be the case (there will always be points in S that are not in the union of the disjoint sets).
  31. E

    Every locally path connected space has a basis consisting of path connected sets

    Homework Statement The definition for local path connectedness is the following: let x be in X. Then for each open subset U of X such that x is in U, there exists an open V contained in U such that x is in V and the map induced by inclusion from the path components of V to the path components...
  32. T

    What Are the Properties of Binary Relations in Sets?

    Hi, I'm struggling about with binary relations in sets. Can somebody check over and answer my questions about these sets: Given set A = {1,2,3} Provide one example each of a relation with the following properties where the cardinality of the relationship should be at least one in all...
  33. A

    Cardinality of Sets Homework: Find Subsets of Natural Numbers

    Homework Statement The problems are to find the cardinality of several sets, a proof is not required, but there must be a decent argument. a) What is the cardinality of the set of all subsets of the natural numbers that contain up to 5 elements? b) What is the cardinality of the set of all...
  34. K

    Unatisfiable union of sets- sentential logic

    Homework Statement Prove that if A and B are two sets of well-formed formulas (logical statements, abv. wff) such that A union B is not satisfiable, then there exists a wff k such that A tautologically implies k and B tautologically implies not k. Homework Equations This question is in...
  35. T

    Topology Proof (Closed/Open Sets)

    Homework Statement Let (X,T) be a topological space, let C be a closed subset of X, let U be an open subset of X. Prove that C - U is closed and U - C is open. The Attempt at a Solution I was trying to do this by 4 cases: Case 1: Let U be a proper subset of C. Then U - C = empty...
  36. S

    Measurable Sets: Proving Open Subsets of Closed Unit Square are Measurable

    Problem. Let E be the closed unit square. Prove that every open subset of E is measurable. I know that one way to show that a set, say A, is measurable is to show that its outer and inner measure coincide; another way is to exibit an elementary set B such that \mu(A\Delta B)< \epsilon...
  37. G

    Proving Transitivity of Ordinals and V_a Sets

    Homework Statement show every ordinal is a transitive set show that every level V_a of the cumulative hierarchy is a transitive set Homework Equations The Attempt at a Solution I understand that these are transitive sets, I'm just not sure how to show this. I feel like the...
  38. C

    Finite intersection of closed sets is not necessarily closed

    Hi everyone, I'm reading Rudin's Analysis and in the topology section, he implies that the finite intersection of closed sets is not necessarily closed. (pg. 34) Can someone give an example of this? I can't seem to find one.
  39. R

    Linear equations, solution sets and inner products

    Homework Statement Let W be the subspace of R4 such that W is the solution set to the following system of equations: x1-4x2+2x3-x4=0 3x1-13x2+7x3-2x4=0 Let U be subspace of R4 such that U is the set of vectors in R4 such the inner product <u,w>=0 for every w in W. Find a 2 by 4...
  40. P

    Linearly Independent Sets and Bases

    Homework Statement V is a subspace of Rn and S={v1,...,vk} is a set of linearly independent vector in V. I have to prove that any list of linearly independent vectors can be extended to a basis for V. Homework Equations None that I can think of. The Attempt at a Solution So to be...
  41. H

    Simple problem about borel and measurable sets

    Show, that Y(x(B)) = xY(B) (Y is Lebesgue_measure ) for every borel set B and x>0. Show that also for measurable sets. I don't know how to prove anything for neither borelian or measurable sets, so I'm asking someone for doing this problem, so i can do other problems with borelian and...
  42. C

    How Do Connected Subsets Prove the Union Is Connected?

    Hi, I'm having trouble understanding this proof. Theorem. Let \{ S_{i} \} _{i \in I} be a collection of connected subsets of a metric space E. Suppose there exists i_{0} \in I such that for each i \in I, S_{i} \cap S_{i_{0}} \neq \emptyset. Then \cup_{i \in I} S_{i} is connected. Proof...
  43. H

    Proving Properties of Countable Sets & Probability Spaces

    1.prove that for any set X: |X|<c <=> in P(X) exist such countable set family F, that sigma algebra generated by F contains all points. 2.let (X,E,u) be probability space and A_1,...,A_2009 in E have property u(A_i)>=1/2. Prove that there exist x such is in A_i for atleast 1005 different i. i...
  44. S

    Sets Proof (A ⊆ λB) ⇔ (B ⊆ λA)

    Just doing some of the set theory questions at the start of a calculus book & I'm kind of confused about how to prove the following: (A ⊆ λB) ⇔ (B ⊆ λA) (Note: λB denotes the complement relative to the universal set, as with & λA) I'm trying to get used to proving this as if I'm...
  45. T

    Is this formula applicable for defining max{u*,v*} as an outer measure on X?

    let \mu^{}* , v^{}* outer measura on X . Show that max{\mu^{}* , v^{}*} is an outer measure on X ?
  46. C

    Proof of Closed Sets: Cluster Points & Int. Pts

    [b]1. Prove that a set is closed if and only if it contains all of its cluster points. [b]2. Can I use part of the Lemma here that states: Every interior point of A is a Cluster point. Also what exactly is the definition of a closed set other than a set is closed if its compliment is...
  47. P

    Where can I find a bank of convergence and Taylor series problems and solutions?

    does anyone know where i can find a decent bank of convergence and taylor series problems and solutions? my calc book is a bit lacking, and i don't have the solutions manual.
  48. redtree

    Published data sets for distance modulus versus redshift

    Does anyone have a recommendation for a paper/source for data relating distance modulus and redshift data over a large range of redshift, such as 0 < z <= 5? I have found that combining data from different publications can be difficult given the variety of ways that distance modulus can be...
  49. chemisttree

    NASA NASA Sets News Conference on Astrobiology Discovery

    The LGM hunters have something to say! http://www.nasa.gov/home/hqnews/2010/nov/HQ_M10-167_Astrobiology.html
  50. chemisttree

    NASA Sets News Conference on Astrobiology Discovery

    Pay attention on Thursday at 2pm ET! http://www.nasa.gov/home/hqnews/2010/nov/HQ_M10-167_Astrobiology.html
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