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. W

    Mathematica Local max/min of Mathematica data sets.

    Is there a way in Mathematica to find the local maxima of a set of points? I have a fairly fine data set, and I can clearly see several peaks in it that I would like to know the numerical value of (as in, the highest point- I don't need a spline approximation or anything too fancy like that). I...
  2. M

    Logical Definition of Two Disjoint Sets

    Homework Statement I'm self-studying Daniel Velleman's How to Prove It, and I'm wondering if there is some way to write that "Sets A and B are disjoint" using symbols, other than the A \cap B = \emptyset given in the book.Homework Equations The Attempt at a Solution I'm thinking that if A...
  3. J

    Uncountable union of a chain of countable sets can be uncountable?

    Let X be a non-empty set, and let S contain all countable subsets of X. Partially order S by inclusion. Let C be a totally ordered subset ("chain") of S, and let U = \cup_{E \in C} E It appears that U is not always countable: if it were, U would be an upper bound of the chain C, and U would...
  4. T

    Indexed Cartesian Product as sets of functions

    I'm going through the set theory material in the appendix of Knapp's Basic Algebra. I want to make sure that I understand what he says is the set theoretic notion of the indexed cartesian product, \prod_{x\in S}A_{x}. He says that this can be thought of as the set of all functions...
  5. Char. Limit

    Is a Number Prime if It Has No Factors Less Than Its Square Root?

    Homework Statement Prove that if a given natural number M has no factors less than or equal to M1/2, then M is prime. Homework Equations None! The Attempt at a Solution So I was wondering, say I have two sets P and Q. Furthermore, say that P \bigcup Q = \mathbb{N} and P \bigcap Q =...
  6. S

    Boundary of closed sets (Spivak's C. on M.)

    Homework Statement I have been self studying Spivak's Calculus on Manifolds, and in chapter 1, section 2 (Subsets of Euclidean Space) there's a problem in which you have to find the interior, exterior and boundary points of the set U=\{x\in R^n : |x|\leq 1\}. While it is evident that...
  7. P

    Proof concerning infinite sets

    Good evening, I was self-studying from a Discrete Mathematics book, and I ran across a question about infinite sets. Homework Statement The exercise asked to show that a set S is infinite if and only if there is a proper subset A of S such that there is a one-to-one correspondence between A...
  8. N

    Bounding the Diameter of Union of Two Sets in a Metric Space

    Hi, I am stuck with the following proofs. In metric space here, A,B,C are subset of metric space (X,d) and C is bounded Problem 1.) d(A,B) <=d(A,C)+d(B,C)+diam(C) Problem 2.)|d(b,A)-d(c,A)| <= d(b,c) where 'b' belongs to 'B' and 'c' belongs to 'C'. Problem 3)- diam(A U B)<= diam A+...
  9. S

    Dice probabilities on opposed sets of 10-sided dice

    Hi, I was wondering if anyone might want to help me with some less common dice probability. The dice mechanic is similar to the board game Risk (two sets of dice being compared), but the dice pools are varying numbers of 10-sided dice on either side (2 pools compared of a varying number)...
  10. D

    Exploring Finite Sets & Limit Points

    Is it true that finite sets don't have limit points?
  11. P

    Subsets of non countably infinite sets

    I was reading an introductory chapter on probability related to sample spaces. It had a mention that for uncountably infinite sets, ie. in sets in which 1 to 1 mapping of its elements with positive integers is not possible, the number of subsets is not 2^n. I certainly find this very...
  12. B

    Sigma Algebras generated by Sets

    Hi, All: A simple question: If A is the sigma- algebra generated by a collection C of subsets of an ambient set X. Isn't it trivial that the sigma-algebra generated by A is A itself? One definition is that the sigma algebra generated by a collection S of subsets is the...
  13. B

    What Is the Cardinality of Nested Power Sets?

    Homework Statement I'm having a little bit of trouble grasping the idea of null sets and power sets. Please check my answers to the following questions: Determine the cardinality: a. P(\oslash) b. P(P(\oslash)) c. P(P(P(\oslash)))Homework Equations n/a The Attempt at a Solution a. since...
  14. Shackleford

    How do I determine if a given set of vectors can generate a vector space?

    How do I show the vectors, polynomials, and matrices generate the given sets? A subset of a vector space generates the vector space if the span of the subset is the vector space. The span is the set of all linear combinations. For 6, do I show the vector are linearly independent and can thus...
  15. D

    Family of Sets Proof Theorem: Counterexample & Error

    Homework Statement Incorrect Theorem: Suppose F and G are familes of sets. If \bigcupF and \bigcupG are disjoint, then so are F and G a) What's wrong with the following proof of the theorem? Proof. Suppose \bigcupF and \bigcupG are disjoint. Suppose F and G are not disjoint. Then we can...
  16. W

    Open sets in R being the union of open intervals

    Hello, I know one proof of this well known theorem that assumes on the metric of R being the standard metric. Does this result generalize to arbitrary metrics on R? thank you
  17. C

    Can all open sets in R^n be expressed as countable union of open cubes?

    Hi everyone, I came across a problem that requires knowing this fact. But can any open set in R^n be expressed as the countable union of "cubes". That is subsets of the form (a_1,b_1) \times ... \times (a_n, b_n) .
  18. S

    Augmented matrices and solution sets - Please help - exam is tomorrow

    Basically there are 2 equations ; x+2y+3z = 1 2x+4y+6z=2 I put them into a matrix and row reduce to get 1 2 3 | 1 0 0 0 | 0 so we can say x = 1 - 2y -3z and let y and z = 0 to get a solution is (1,0,0) Now i need to find the nullspace to find the whole solution set; so x +...
  19. T

    Common sets of eigenfunctions in angular momentum

    Hi, I'm a second year physics undergrad currently revising quantum mechanics, and I came across a phrase about angular momentum which has confused me, so I was wondering if anyone could help. We looked at different components of angular momentum (in Cartesian) and decided that they did not...
  20. D

    Non-Lebesgue Measurable Sets: Understanding Measurement

    In analysis we were shown the existence of non-Lebesgue measurable sets (eg a choice function over the rational equivalence partition of an interval). From the proof it seems that this means you can't assign number to the Lebesgue measure of this set i.e. if you say its measure is zero it's not...
  21. C

    Add 2 Dense Sets for Non-Dense Set Result

    I can add 2 dense sets together and get a non dense set right?
  22. C

    Question about infinite sets ?

    Before i say this i think transfinite numbers and infinite sets are really cool. But could you argue that infinite sets don't exist, I mean you couldn't show me one. I apologize if this is in the wrong section. I do believe in infinite sets but I was just wondering if you could argue this...
  23. A

    Mathematica Import and plot multiple data sets in Mathematica

    I have a dat file with multiple data sets, with the following structure: # t = 0.0 , ... -10.000 0.00001 1.000001 ... -9.000 0.00002 0.900001 ... ... 10.000 0.00005 1.000001 ... # t = 0.2 , ... -10.000 0.00301 1.000203 ... -9.000 0.02222 0.900043 ... ... 10.000 0.00025 1.000551...
  24. Y

    Is X a Metric Space? Closed and Open Sets

    Assume X is a metric space, then X and the empty set are both closed and open, am I correct?
  25. H

    Showing two sets are equivalent

    Homework Statement Show that R \approx R+ , that is, the set of all real numbers is equivalent to the set of all positive real numbers Homework Equations The only relevant equation is finding one such that F:R\rightarrowR+ is a bijection. The Attempt at a Solution I've...
  26. MathWarrior

    Mathematical functions from data sets?

    I feel like I have gone pretty far in math now, but I keep finding myself asking the same question. Say I had a series of data points from like a randomly collected survey or stock stock price graph over time etc. Is there a way to take this seemingly random and scattered data and turn it into...
  27. A

    Is Transitivity Applicable in Proving Subset Relations Among Sets A, B, and C?

    Hello I'm having problems with actually proving this with some mathematics. Let A, B, C be sets. Prove that if A is a subset of B and B is a subset of C then A is a subset of C. Thanks.
  28. P

    Given sets A and B, prove that A is a subset of B (Apostol)

    Homework Statement Continuing with my Apostol efforts. From Section I 2.5: These exercises go over some of the absolute basics of sets. In 3. I'm given A = {1}, B = {1,2} and asked to decide whether some statements are true or false, proving the ones that are true. Seeing which ones are...
  29. andrewkirk

    Infinitely recursive sets that don't contradict ZF Axiom of Regularity

    Hello, I have just been reading about the Zermelo-Frankel (ZF) axioms for set theory and thinking about their consequences. I understand that the Axiom of Regularity is needed in order to prevent contradictions like Russell's Paradox arising. That axiom says that any non-empty set A must contain...
  30. S

    Is <f,g> an Inner Product on C[a,b]?

    Homework Statement show that if we define the following operation: let f=f(x) and g=g(x) be two functions in C[a,b] and define <f,g>=int(a to b) f(x)g(x)dx show that the conditions of therom are satisified with this operation. Use h=h(x) to help with part b this shows that this operation is...
  31. A

    Axiom of Choice: finite sets to infinite sets

    So if we have a finite collection of disjoint non-empty sets, one can show using ZF only(with no need of AC) there is a choice function. I understand the reason for this. My confusion is when one goes to non-finite collection of sets. For example if the index set is the Natural numbers, why do...
  32. P

    Comparing two sets of data: multiple time points

    I am trying to compare two groups (for statistical significance), a control and a treatment group across more than one time point, for a single variable. For example Control Treatment 0 sec x x 5 sec x x 10 sec x...
  33. T

    Question on a particular collection of sets

    Let {X} be a set. Let {\mathcal{G}} be a non-empty collection of subsets of {X} such that {\mathcal{G}} is closed under finite intersections. Assume that there exists a sequence {X_h \in \mathcal{G}} such that {X = \cup_h X_h} . Let {\mathcal{M}} be the smallest collection of...
  34. A

    Equivalence of sets proof assistance

    Homework Statement Suppose there exist three functions: f:A\stackrel{1-1}{\rightarrow}B g:B\stackrel{1-1}{\rightarrow}C h:C\stackrel{1-1}{\rightarrow}A Prove A\approxB\approxC Do not assume the functions map onto their codomains. Homework Equations The Attempt at a Solution I took a...
  35. J

    Compute the first and follow sets of the grammar

    Homework Statement compute the first and follow sets of the following grammar S -> ACB | CbB | Ba A -> da | BC B -> g|λ C -> h|λ The Attempt at a Solution First(S) = First(ACB) U First(CbB) U First(Ba) First(ACB) = First(A) - {λ} U First(CB) First(CbB) = First(C) - {λ} U {b} First(B) =...
  36. H

    Countable Sets: Cantor's Theorem & Galileo's Paradox

    Hi! Can You please confirm that there is no contradiction between Cantor's theorem of countably infinite power sets and Galileo's paradox? - Thanks
  37. J

    True or False: Boundary of Sets in R2

    Homework Statement True or false: Let S be any set in R2. The boundary of S is the set of points contained in S which are not in the interior of S. Homework Equations The Attempt at a Solution Common sense tells me true. I don't really understand it though, if S is an open set...
  38. Char. Limit

    Comparing Sets of Convergent Sequences and Series

    So I had this question in PF chat, but I decided this would be a better place for it. Say I have two sets, S and S'. S is the set of all convergent sequences. S' is the set of all convergent series...es. Is S larger than S', and if so, how much larger?
  39. Q

    Proving Continuity of Compositions: Sets of Measure Zero

    Homework Statement I am trying to prove that if f is continuous almost everywhere on [a,b], and if g is cont a.e. on [c,d], with f[a,b] contained in [c,d], then g composite f is cont. a.e. The Attempt at a Solution ------ Originally, my proof went something like this: f is cont...
  40. G

    Cardinality of the Union of Two Sets that have Same Cardinality as Real Numbers

    Homework Statement Let U and V both have the same cardinality as R (the real numbers). Show that U\cupV also has the same cardinality as R. Homework Equations The Attempt at a Solution Because U and V both have the same cardinality as R, I that that this means \exists f: R\rightarrowU that is...
  41. G

    [Topology]Determining compact sets of R

    Homework Statement Is A = {0} union {1/n | n \in {1,2,3,...}} compact in R? Is B = (0,1] compact in R?Homework Equations Definition of compactness, and equivalent definitions for the space R.The Attempt at a Solution A is compact, but I can't seem to find a plausible proof of it... It should...
  42. X

    Prove Set of Real Numbers Unbounded: Tips & Examples

    How to prove that set of real numbers is unbounded?
  43. K

    Basic analysis: Proof by induction on sets

    Homework Statement Prove by induction that the set [a_{n} | n_{0}\leq n \leq n_{1}] is bounded. a_{n} are the elements of the sequence (a_{n}) n \in N Homework Equations Definition of set bounded above: \forall x \in S, \exists M \in R such that x \leq M The Attempt at a Solution...
  44. N

    Sets and Logics, problem to solve a question

    Homework Statement Convert (AC3)16 to base 10. I'm new to this kind of material. I would really appreciate your help for this one. Thanks, Roy Homework Equations The Attempt at a Solution
  45. C

    Distance between two sets equals distance between the closure of the sets

    My problem is as follows: If we define d(A,B) = inf{ d(x,y) : x in A and y in B }, show that d(clos(A),clos(B)) = d(A,B), where clos(A) is the closure of A My attempt at a solution was this: Since A is a subset of the closure of A, then d(A,B) must be less than or equal to the distance...
  46. Demon117

    Understanding Zero Sets: Real Analysis Examples

    What is the definition of a zero set and what exactly does it mean? I have come across different responses on the internet, but none of them explain really what it means or give good examples, I am having a rough time with this concept in real analysis. For example, how would I determine...
  47. O

    Can a G-delta Set and a Borel Set Have the Same Lebesgue Measure?

    Homework Statement Show there's a G-delta set B, with E \subseteq B s.t. \lambda(E) = \lambda(B) Where \lambda is the Lebesgue measure and E is a Borel set. Homework Equations - G-delta set is a countable intersection of open set. - Lebesgue measure has properties: monotonicity...
  48. B

    Union of Closed Sets: Finite vs. Infinite Examples

    A Union of a FINITE collection of closed sets is closed. But if it is an infinite collection? Can someone provide an example please?
  49. M

    Finding Data Sets & Ideas for Neural Network Training

    I'm looking for advice. I have an assignment to train a neural network. I am confident in my abilities to write the code and train the neural network. The problems are: 1. I lack an idea. I'd like to do something interesting and potentially useful, obviously. 2. I need a rather large data...
  50. B

    Is the Intersection of Infinite Non-Empty Open Subsets Empty?

    If you take the intersection of non empty open subsets in Rn as n tends to infinity, such that U_1 \supseteq U_2 \supseteq U_3... Is it empty?
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