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. C

    Prove the diameter of a union of sets is finite

    Homework Statement Prove that the union of a collection of indexed sets has finite diameter if the intersection of the collection is non-empty, and every set in the collection is bounded by a constant A. The Attempt at a Solution The picture I have is if they all intersect (and assuming...
  2. caters

    MHB Discovering Intersections of Infinite Primes Sets

    We have this set of primes which is infinite. This has lots of different subsets. Here is the list of subsets: Real Eisenstein primes: 3x + 2 Pythagorean primes: 4x + 1 Real Gaussian primes: 4x + 3 Landau primes: x^2 + 1 Central polygonal primes: x^2 - x + 1 Centered triangular primes: 1/2(3x^2...
  3. W

    Dividing Sets in Contact Structures, and Induced Orientations

    Hi everyone, a couple of technical questions : 1) Definition: Anyone know the definition of the induced orientation of a submanifold S of an orientable manifold M? 2)Dividing sets in contact manifolds: We have a contact 3-manifold (M3,ζ ). We define a surface S embedded in M3 to be a convex...
  4. E

    MHB About open sets in a metric space.

    Let (E=]-1,0]\cup\left\{1\right\},d) metric space with d metric given by d(x,y)=|x-y|, and ||absolute value. How I can find open sets of E explicitly? Thanks in advance.
  5. C

    Cartesian product of index family of sets

    Cartesian product of indexed family of sets The definition of a Cartesian product of an indexed family of sets (X_i)_{i\in I} is \Pi_{i\in I}X_i=\left\{f:I \rightarrow \bigcup_{i \in I} \right\} So if I understand correctly, it's a function that maps every index i to an element f(i) such...
  6. M

    Measure defined on Borel sets that it is finite on compact sets

    The problem statement Let ##\mu## be a measure defined on the Borel sets of ##\mathbb R^n## such that ##\mu## is finite on the compact sets. Let ##\mathcal H## be the class of Borel sets ##E## such that: a)##\mu(E)=inf\{\mu(G), E \subset G\}##, where ##G## is open...
  7. A

    Show that the set of sets {An} has n elements

    We define by recursion the set of sets {An:n∈ℕ} this way: A_0=∅ A_n+1=A_n ∪ {A_n}. I want to prove by induction that for all n∈ℕ, the set A_n has n elements and that A_n is transitive (i.e. if x∈y∈A_n, then x∈A_n). My thoughts: for n=0, A_1 = ∅∪ {∅} = {∅} then, for n+1: A_n+2...
  8. S

    Number of onto functions from 2 sets

    Homework Statement I have 2 sets, one with 5 elements (A) and the other two(B). How many onto functions can be made from A to B? Homework Equations The Attempt at a Solution My first thought is that it should be something like ##\frac{5!}{2!}=60##. I don't know if this is correct...
  9. E

    Understanding Double Quantifiers and Sets with Epsilon

    Homework Statement Determine if the sets A, B, C, and D satisfy the following proposition (p) for the set S: Homework Equations p: for all ε > 0, ∃ x \in S such that x < ε A = {1/n : n \in Z+} B = {n : n ε Z+} C = A \cup B D = {-1} The Attempt at a Solution I am...
  10. S

    MHB Two sets generate the same vector space

    Show that the sets \{a,b\} and \{a, b, a-b\} of real vectors generate the same vector space. How to proceed with it? I guess the following expression is helpful. c1*a+c2*b+c3*(a-b)=(c1+c3)*a+(c2-c3)*b=k1*a+k2*b
  11. G

    MHB Sets Intersection Proof: G and {F(n)} Family of Sets

    If $G$ and $\left\{F(n): n \in \mathbb{K}\right\}$ are a family of sets, show that $\displaystyle G \cap \cap_{n \in \mathbb{K}}F(n) = \cap_{n \in \mathbb{K}}(F(n) \cap B).$ I said if $b$ is an ement of $\displaystyle G \cap \cap_{n \in \mathbb{K}}F(n)$ then $b$ is in both $G$ and $F(n)$ for...
  12. E

    Sets - Relations - proof involving transitivity

    I'm having trouble with the following: Let R be a relation on A. Prove that if Dom(R) \bigcap Range(R) = ø, then R is transitive. I took the negation of the "R is transitive" to try proof by contrapositive (as the professor suggested), and have the following: \exists x,y,z \in A s.t. (x,y)...
  13. C

    Power sets and Cartesian products.

    Homework Statement For every pair of sets (A,B) we have P(AxB)=P(A)xP(B) Prove or disprove the above statement. Homework Equations The Attempt at a Solution I have attempted solving this using A={1,2} and B={a,b} AxB={(1,a),(1,b),(2,a),(2,b)}...
  14. A

    How to Reverse a Proof for an Identity with Sets?

    Homework Statement Homework Equations The Attempt at a Solution $$(A-B)\cup (C-B)=(A\cup C)-B\\ (A\cap B^{ C })\cup (C\cap B^{ C })=(A\cup C)\cap B^{ C }\\ (A\cup C)\cap B^{ C }=(A\cup C)\cap B^{ C }\\$$ I know for algebraic proofs, proofs like these are accepted if they are reversed. But...
  15. K

    Determine Union of Sets Belonging to Interval

    Let ##I## denote the interval ## [0, \infty )## . For each r ## \in I ## define: ##A_{r} = \{ (x,y), \in ##R x R : ## x^{2} +y^{2} = r^{2} \}## ##B_{r} = \{ (x,y), \in ##R x R : ## x^{2} +y^{2} \leq r^{2} \}## ##C_{r} = \{ ## ... ## : ... > r^{2} \} ## a.)...
  16. N

    Propagating Uncertainty for sets of data?

    I'm writing the paper on this experiment I just did. Basically I took sets of data for two variables (x,y) and I fit the points to a line in Origin to extract the value that I was trying to measure (J). *Using generic variables here* I found a value for J where J = [8*∏*x*a(b+c)] / y I...
  17. S

    Can open sets and closures intersect in a topological space?

    Suppose A and B are open sets in a topological Hausdorff space X.Suppose A intersection B is an empty set. Can we prove that A intersection with closure of B is also empty? Is "Hausdorff" condition necessary for that? Please help.
  18. C

    Sets and functions, theoretical calc homework?

    Homework Statement Let A,B, and C be subsets of universal set U. Prove the following A. If U=A union B and intersection of A and B is not an empty set, then A= U\B B. A\(B intersection C) = (A\B) union (A\C) Homework Equations no relevant equations required The Attempt at a...
  19. P

    Solving Set Theory Homework: Sets, Tuples, etc

    Homework Statement So if a question asks you for a pair set, with some criteria, is it enough to just say S = {a,b} or do you need something extra? Also this is another question that is from my set theory class, if the question defines a function and to solve over the Real numbers, ex...
  20. 9

    Bounded sets: x = [1, 2] U [3, 4] c R

    Hello. Please look over my answers! Homework Statement a) Prove that this set is not convex: x = [1, 2] U [3, 4] c R b) Prove the intersection of two bounded sets is bounded Homework Equations for a) x = [1, 2] U [3, 4] c R The Attempt at a Solution a) A convex set is where...
  21. A

    MHB Equinumerous and infinite sets

    Suppose A is a set with at least two elements and A\times A\sim A. Then \mathcal{P}(A)\times\mathcal{P}(A)\sim\mathcal{P}(A). My attempt: I know that \mathcal{P}((A\times A)\cup A)\sim\mathcal{P}(A\times A)\times\mathcal{P}(A)\sim\mathcal{P}(A) \times \mathcal{P}(A). How to prove that...
  22. B

    Convex sets - How do we get (1−t)x+ty

    In definition 2.17 of Rudin's text, he says that a set E is convex if for any two points x and y belonging to E, (1−t)x+ty belongs to E when 0<t<1. I learned that this means the point is between x and y. But I'm not able to see this intuitively. Can anyone help me "see" this?
  23. A

    MHB Equinumerous Sets: Defining D_x

    For any sets A,\ B,\ C, ^{(A\times B)}C\sim ^A\left(^BC\right). First I could not find a formula for the required function. Then I defined the set D_x=\{(y,z)\in B\times C\mid f(x,y)=z\}, where f\colon A\times B\to C. What do you think?
  24. F

    MHB Sets: A,B,C - Intersection & Difference

    Am I right in thinking that if we have 3 sets A,B,C, then with A intersect B represented as AB, we have: A(B\C)=(AB)\C=B(A\C)?
  25. Math Amateur

    MHB Subscripts actually under a union of sets sign

    I recently made a post on Linear and Abstract Algebra and used the following symbol {\bigcup}_{\Omega \subseteq \Gamma , | \Omega | \lt \infty} However, I really wanted (for neatness and clarity) to have the term {\Omega \subseteq \Gamma , | \Omega | \lt \infty} actually under the set...
  26. N

    MHB What are the formulas for solving sets A and B?

    Which formula's would you use to solve each set and please show the actual formulas please thank you for the help I really appreciate it as I need to know this for my final exam tomorrow, thank you! a) solving for neither set A and neither set B b) Solving for set A or Set B, but not bothc)...
  27. J

    How Can You Approximate a Borel Set with an Open Set in Measure Theory?

    Homework Statement We're given the measure space (X,A,μ) with X=\bigcup_{i=1}^{\infty} X_i where X_i⊂X_{i+1}⊂..., X_i are open for all i and μ(X_i)<+∞ for all i. Show that for every Borel set B there exists an open set U where μ(B\U)<ϵ. Homework Equations measures are subadditive The...
  28. E

    How to compare two data sets with statistics?

    I have two questions: I have a set of data, a measured spectrum. When I model the spectrum with a function, I calculate r2=1-(\sum(y-ymodel)2/\sum(y-yavg)2). Q1) However, I have reference data now, which is what the spectrum should be. So is it right to use the same calculation on it for...
  29. A

    Determining two sets of boundary conditions for a double integral prob

    Homework Statement Determining two sets of boundary conditions for a double integral problem in the polar coordinate system. Is the below correct? Homework Equations The Attempt at a Solution There are two sets of boundary conditions that you can use to solve this problem in the polar...
  30. 3

    Equivalence Relations, Cardinality and Finite Sets.

    Hey everyone, I have three problems that I'm working on that are review questions for my Math Final. Homework Statement First Question: Determine if R is an equivalence relation: R = {(x,y) \in Z x Z | x - y =5} and find the equivalence classes. Is Z | R a partition? Homework...
  31. Math Amateur

    MHB Algebraic Geometry - Affine Algebraic Sets

    Dummit and Foote, Exercise 20, Section 15.1 reads as follows: If f and g are irreducible polynomials in k[x,y] that are not associates (do not divide each other), show that \mathcal{Z} (f,g) is either the empty set or a finite set in \mathbb{A}^2 . I am somewhat overwhelmed by this...
  32. F

    MHB Measurability of Open Sets in [0,1]

    I am trying to show that an open set in [0,1] is measurable, given that [0,x] is measurable set for each x in [0,1]. So I need to show (a,b) is measurable. Using the fact that measurable sets form a sigma algebra, I have managed to show that (a,b] is measurable. So (a,b+t] is measurable for any...
  33. Superposed_Cat

    MATLAB MATLAB-Plot two sets of data on the same graph with custom increments

    Hi, I was trying to plot two sets of data on the same graph and this was my solution but this generates a graph with the x-axis in 0.5 increments where I wanted 1. How do I plot two sets of data on the same graph and customize the increments. Thanks in advance. Also, how would I make MATLAB...
  34. K

    MHB Natural Numbers ⊆/⊄ Rationals: Infinite & Uncountable Sets

    Question 1) Write ⊆ or ⊄: {x/(x+1) : x∈N} ________ QNOTE: ⊆ means SUBSET ⊄ means NOT A SUBSET ∈ means ELEMENT N means Natural Numbers Q means Rational Numbers Question 2) Which of the following sets are infinite and uncountable? R - Q {n∈N: gcd(n,15) = 3} (-2,2) N*N {1,2,9,16,...} i.e...
  35. V

    Questions about the definition of open sets

    I am currently reading Munkres' book on topology, in it he defines an open sets as follows: "If X is a topological space with topology T, we say that a subset U of X is an open set of X if U belongs to the collection T." Firstly, are the open sets a property of the set X or the topological...
  36. R

    MHB Approximation property with F sigma and G delta Sets to show a set is measurable

    Prove that a set $A\subset\mathbb{R}^n$ is (Lebesgue) measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$ such that $B\subset A\subset C$ and $C$~$B$ (C without B) is a null set. $F_{\sigma}$ is a countable union of closed sets, and...
  37. A

    Real Analysis: closed sets and limit points

    For the following example:(if possible give example or just state impossible 1) a bounded subset A of R for which sup A is not a limit point of A. An example is (0,1) union {7}. will this work? 2) a finite subset A of R that is not closed I think it is not possible. Please give some hints...
  38. S

    Solving Probability Integrals with Monotone Convergence Theorem

    I'm having trouble working out a few details from my probability book. It says if P(An) goes to zero, then the integral of X over An goes to zero as well. My book says its because of the monotone convergence theorem, but this confuses me because I thought that has to do with Xn converging to X...
  39. alyafey22

    MHB Mapping and inverse mapping of open sets and their complements

    Assume that f: E \to Y \,\,\, , E \subset X then can we say that f(E^c)=f(E)^c what about the inverse mapping f^{-1}: V \to X \,\,\, , V\subset Y do we have to have some restrictions on f and its inverse ? My immediate answer is that we have to have a bijection in order to conclude that but I...
  40. Math Amateur

    Algebriac Geometry - Morphisms of Algebraic Sets

    I am reading Dummit and Foote: Section 15.2 Radicals and Affine Varieties. On page 678, Proposition 16 reads as follows: (see attachment, page 678) --------------------------------------------------------------------------------------- Proposition 16. Suppose \phi \ : \ V...
  41. Math Amateur

    MHB Algebraic Geometry - Morphisms of Algebraic Sets - Proposition 16 (D&F)

    I am reading Dummit and Foote: Section 15.2 Radicals and Affine Varieties. On page 678, Proposition 16 reads as follows: (see attachment, page 678) --------------------------------------------------------------------------------------- Proposition 16. Suppose \phi \ : \ V \longrightarrow W...
  42. Math Amateur

    Algebriac Geometry - Morphisms of Algebraic Sets

    I am reading Dummit and Foote (D&F) Section 15.1 on Affine Algebraic Sets. On page 662 (see attached) D&F define a morphism or polynomial map of algebraic sets as follows: ---------------------------------------------------------------------------------------------- Definition. A map...
  43. Math Amateur

    MHB Algebriac Geometry - Morphisms of Algebraic Sets

    I am reading Dummit and Foote (D&F) Section 15.1 on Affine Algebraic Sets. On page 662 (see attached) D&F define a morphism or polynomial map of algebraic sets as follows: ----------------------------------------------------------------------------------------------------- Definition. A map...
  44. Math Amateur

    Affine Algebraic Sets - D&F Chapter 15, Section 15.1 - Example 2

    I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, Example 2 on page 660 reads as follows: (see attachment) ----------------------------------------------------------------------------------------------...
  45. Math Amateur

    Affine Algebraic Sets - D&F Chapter 15, Section 15.1 - Example 3 - pag

    I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, Example 3 on page 660 reads as follows: (see attachment) ----------------------------------------------------------------------------------------------...
  46. Math Amateur

    MHB Affine Algebraic Sets - D&F Chapter 15, Section 15.1 - Properties of the map I

    I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, the set \mathcal{I} (A) is defined in the following text on page 660: (see attachment)...
  47. Math Amateur

    MHB Affine Algebraic Sets - D&F Chapter 15, Section 15.1 - Example 3 - page 660

    I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, Example 3 on page 660 reads as follows: (see attachment)...
  48. Math Amateur

    MHB 'Curly' Z and I - Affine algebraic sets

    I am reading Dummit and Foote on affine algebraic sets and wish to create posts referring to such objects. The notation for a subset Z(S) of affine space is a "curly" Z - see attachment - bottom of page 658. Also the notion for the unique largest ideal whose locus determines a particular...
  49. Math Amateur

    MHB Affine Algebraic Sets - D&F Chapter 15, Section 15.1

    I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, Example 2 on page 660 reads as follows: (see attachment)...
  50. P

    Prove or disprove the following statement using sets frontier points

    if A is a subset of B and the frontier of B is a subset of A then A=B. I am pretty sure that this is true as I drew I diagram and I think this helped. A frontier point has a sequence in the set and a sequence in the compliment that both converge to the same limit. However I'm not really...
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