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. Decimal

    A Unstable sets embedded in a chaotic attractor

    I am having a hard time understanding the discussion of chaotic sets on invariant manifolds as given in Chaos in Dynamical Systems by Edward Ott. If the invariant manifold of a particular system contains a chaotic attractor ##A##, then the transverse Lyapunov exponent ##h## will generally...
  2. R

    Manual vs. Auto Drain Air Sets: Which is Best for Filter Regulators?

    Difference between manual drain air sets (filter regulators) and auto drain air sets?
  3. M

    Drawing sets of Complex Variables

    I tried saying z = x + iy, then squared both sides so that I would get something that looked like: |z - i|^2 + |z + i|^2 + |z - i||z + i| = 3, where the first two terms are simple but the third term is what I don't know what to do with. I'm wondering if I'm using the wrong approach. For that...
  4. benorin

    I Convergence of a sequence of sets

    I need a little help with Baby Rudin material regarding the convergence of a sequence of sets please. I wish to follow up on this thread with a definition of convergence of a sequence of sets from Baby Rudin (Principles of Mathematical Analysis, 3rd ed., Rudin) pgs. 304-305: (pg. 304)...
  5. L

    B Are set notations simplifyable?

    I tried to name the shaded area of a Venn diagram using numbers to isolate the regions. And I found that there are several ways to get the same region. Can the set notations simplfy
  6. opus

    LaTeX Making "tabs" in LaTeX for problem sets

    I'm trying to recreate this document in LaTeX, but I'm not sure how they aligned "Factor" and "Solution" after 1.1.1. Any ideas?
  7. F

    Do Inclusion-Exclusion and Pigeonhole Principles Apply to Overlapping Sets?

    Is this related to pigeon principle? $$S_1=\{1,2,3,4\},$$ $$S_2=\{2,3,4,5\},$$ $$S_3=\{4,5,6,7\},$$ $$S_4=\{5,6,7,8\},$$ $$S_5=\{7,8,9,10\},$$ $$S_6=\{8,9,10,11\},$$ $$S_7=\{5,6,2,4\},$$ $$S_8=\{1,5,7,9\},$$ $$S_9=\{4,8,10,11\},$$ $$S_{10}=\{5,7,10,11\}$$ When we choose two of them, there is...
  8. Math Amateur

    MHB Relatively Open Sets .... Another Queston Regarding Stoll, Theorem 3.1.16 (a) ....

    I am reading Manfred Stoll's book: Introduction to Real Analysis. I need further help with Stoll's proof of Theorem 3.1.16 Stoll's statement of Theorem 3.1.16 and its proof reads as follows: Can someone please help me to demonstrate a formal and rigorous proof of the following:If the subset U...
  9. Math Amateur

    MHB Relatively Open Sets .... Stoll, Theorem 3.1.16 (a) ....

    I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's proof of Theorem 3.1.16 Stoll's statement of Theorem 3.1.16 and its proof reads as follows: Can someone please help me to demonstrate a formal and rigorous proof of the following:If U = X \cap O for some...
  10. heff001

    A Higher Set Theory – Cantorian Sets / Large Cardinals in the Infinite

    Zermelo-Fraenkel Axioms - the Axiom of Choice (ZFC), is conceptually incoherent. To me, they stole Cantor’s brilliant work and minimized it. Replies?
  11. Demystifier

    A Comparing infinite countable sets

    The uncountable sets [0,1] and [0,2] have the same cardinality ##2^{\aleph_0}##. Yet the second set is twice as big as the first set, in the sense of measure theory. Is there something similar for countable sets, by which we can say that the set of integers is twice as big as the set of odd...
  12. M

    MHB Sets so that the cartesian product is commutative

    Hey! :o Let $A,B$ be sets, such that $A\times B=B\times A$. I want to show that one of the following statements hold: $A=B$ $\emptyset \in \{A,B\}$ I have done the following: Let $A$ and $B$ be non-empty set. Let $a\in A$. For each $x\in B$ we have that $(a,x)\in A\times B$. Since...
  13. M

    MHB Decide if the sets are subspaces or affine subspaces

    Hey! :o We have the subsets \begin{equation*}V:=\left \{\begin{pmatrix}x_1 \\ x_2 \\ x_3\end{pmatrix}\mid x_1=0\right \}, \ \ \ W:=\left \{\begin{pmatrix}x_1 \\ x_2 \\ x_3\end{pmatrix}\mid x_2=2\right \}, \ \ \ S:=\left \{\lambda \begin{pmatrix}1 \\ 0 \\ -1\end{pmatrix}\mid \lambda \in...
  14. M

    MHB Can These Mathematical Sets Be Considered Subspaces?

    Hey! :o We have the following subsets: \begin{align*}&U_1:=\left \{\begin{pmatrix}x \\ y\end{pmatrix} \mid x^2+y^2\leq 4\right \} \subseteq \mathbb{R}^2\\ &U_2:=\left \{\begin{pmatrix}2a \\ -a\end{pmatrix} \mid a\in \mathbb{R}\right \} \subseteq \mathbb{R}^2 \\ &U_3:=\left \{\begin{pmatrix}x...
  15. J

    I Discrete Topology and Closed Sets

    I am trying to learn some topology and was looking at a problem in the back of the book asking to show that a topological space with the property that all set are closed is a discrete space which, as understand it, means that all possible subsets are in the topology and since all subsets are...
  16. B

    B Equal sets with different symbols?

    {1, 2 ,3} = {1, 2, 3, 3, III}? {1, 2 ,3} = {one, dos, three}? {Tom, Dick, Harry} = {Thomas, Richard, Harrison}? Seems to me, these are undetermined until the set's "type" or "category" definition of its members is defined so as to determine what elements are members of the set... whether...
  17. Demystifier

    I Cardinality of non-measurable sets

    The interval ##[0,1]## of real numbers has a non-zero measure. The set of all rational numbers in the interval ##[0,1]## has zero measure. But there are also sets that are somewhere in between, in the sense that their measure is neither zero nor non-zero. They are sets for which measure is not...
  18. christang_1023

    Can Algebraic Calculations Alone Determine Vector Set Constraints Accurately?

    1. I consider this problem algebraically, ##c\cdot \vec{u}+(1-c)\cdot \vec{v}=c(1,2)+(1-c)(2,1)=(c,2c)+(2-2c,1-c)=(2-c,1+c)##; since the constraint I know is ##c\geq 0##, I can conclude the expected vectors##(x,y)## must have ##x\leq2, y\geq 1##. 2. Similarly, I get...
  19. Math Amateur

    MHB Functions Continuous on Comapct Sets .... Apostol, Theorem 4.25 ....

    I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.25 ... ... Theorem 4.25 (including its proof) reads as follows: In the above proof by...
  20. MidgetDwarf

    Intro Real Analysis: Closed and Open sets Of R. Help with Problem

    For the set A: Note that if n is odd, then ## A = \{ -1 + \frac {2} {n} : \text{n is an odd integer} \} ## . If n is even, A = ## \{1 + ~ \frac {2} {n} : \text{ n is an even integer} \} ## . By a previous exercise, we know that ## \frac {1} {n} ## -> 0. Let ## A_1 ## be the sequence when n...
  21. Math Amateur

    MHB Carothers' Definitions: Neighborhoods, Open Sets, and Open Balls

    The Definition of a Neighborhood and the Definition of an Open Set ... Carothers, Chapters 3 & 4 ... I am reading N. L. Carothers' book: "Real Analysis". ... ... I am focused on Chapter 3: Metrics and Norms and Chapter 4: Open Sets and Closed Sets ... ... I need help with an aspect of...
  22. J

    Has anyone moved 40lbs using 4 sets of 10lb motors?

    Problem Statement: 40 lbs Relevant Equations: 10 times 4 I wanted to know if anyone has moved 40lbs using 4 sets of 10lbs motors instead of a motor that can move 40lbs I am testing using 4 small motors instead of 1 big motor thank you
  23. Y

    MHB Operations on Sets: Proving A⊆B⊆C & A∪B=B∩C

    Dear all, I have two small questions regarding operations on sets. (1) Prove that \[A\subseteq B\subseteq C\] if and only if \[A\cup B=B\cap C\]. (2) What can you say about sets A and B if \[A\B = B\] ? In the case of (1), I have used a Venn diagram and I understand why it is true, but...
  24. Math Amateur

    MHB Understanding Topology: Closure, Boundary & Open/Closed Sets

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am reading Chapter 6: Topology ... ... and am currently focused on Section 6.1 Topological Spaces ... I need some help in order to fully understand a statement by Browder in Section 6.1 ... ... The...
  25. L

    A Definitions of Cylinder Sets and Cylinder Set Measure

    I'm trying to learn about Abstract Wiener Spaces and Gaussian Measures in a general context. For that I'm reading the paper Abstract Wiener Spaces by Leonard Gross, which seems to be where these things were first presented. Now, I'm having a hard time to grasp the idea/motivation behind the...
  26. Math Amateur

    I Continuity and Open Sets .... Sohrab, Theorem 4.3.4 .... ....

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of R and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.3.4 ... ... Theorem 4.3.4 and its proof read as follows: In the above proof by...
  27. Math Amateur

    MHB Continuity and Open Sets .... Sohrab, Theorem 4.3.4 .... ....

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of R and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.3.4 ... ... Theorem 4.3.4 and its proof read as follows: In the above proof by...
  28. Math Amateur

    MHB Open and Closed Sets in R^n .... Duistermaat and Kolk, Lemma 1.2.11 ....

    I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with the proof of Lemma 1.2.11 ... Duistermaat and Kolk"s Lemma 1.2.11 reads as follows: Can someone please demonstrate...
  29. Math Amateur

    MHB Open Sets in R^n .... Duistermaat and Kolk, Lemma 1.2.5 ....

    I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Lemma 1.2.5 ... Duistermaat and Kolk"s Lemma 1.2.5 reads as follows:In the above proof by Duistermaat and Kolk...
  30. Math Amateur

    MHB Norm bounded Sets .... remarks by Garling in Section 11.2 Normed Spaces ....

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help in order to understand some...
  31. U

    I Missing(?) rigor in proof involving countable union of countable sets

    My question concerns the portion of the proof stating, “...we set up a correspondence between the elements of U(A_n), for n in N, and a subset of S by making the element a correspond to (m, n) if A_m is the first set in which a appears, and a is the nth element of A_m.” In particular, I am...
  32. Math Amateur

    MHB Open and Closed Sets .... Conway, Example 5.3.4 (b) .... ....

    I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 5: Metric and Euclidean Spaces ... and in particular I am focused on Section 5.3: Open and Closed Sets ... Conway's Example 5.3,4 (b) reads as follows ... ... Note that Conway defines open and closed...
  33. Math Amateur

    I Open and Closed Sets .... Conway, Example 5.3.4 (b) .... ....

    I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 5: Metric and Euclidean Spaces ... and in particular I am focused on Section 5.3: Open and Closed Sets ... Conway's Example 5.3,4 (b) reads as follows ... ... Note that Conway defines open and closed sets...
  34. J

    Finding the cardinal number for the intersection of two sets

    My Question : 1.Why are the inequalities considered? Why not simply use ##n(A\cap B) = n(A)+ n(B)-n(A\cup B)## to get ## n(A\cap B) = 39## ? 2. The way I interpret this is : If the set for people liking cheese was to be a subset of the set for people who like apples then the most number of...
  35. V

    Proving or disproving operations on sets

    Homework Statement Prove or disprove: if A⊆B∪C, then A⊆B or A⊆C. Homework EquationsThe Attempt at a Solution I am unsure of how to go about proving this. I know that A is a subset of B union C then A is a subset of B or A is a subset of C and I understand what a subset is and what a union is...
  36. Mr Davis 97

    Property of compact convex sets of width 1

    Homework Statement A strip of width w is a part of the plane bounded by two parallel lines at distance w. The width of a set ##X \subseteq \mathbb{R}^2## is the smallest width of a strip containing ##X##. Prove that a compact convex set of width ##1## contains a segment of length ##1## in every...
  37. V

    Operation on Sets Homework: B-A & C-A

    Homework Statement Let A= {1, 2, 3}, B= ℤ+, C= [1, infinity) That is C= {x∈ℝ:x≥1} What is B - A and C - A? Homework EquationsThe Attempt at a Solution I am unsure of how to go about answering this. I know that B - A means what elements are in B that aren't in A. Would that make the answer...
  38. E

    MHB How many subsets of set A satisfy given conditions?

    Consider a set $A$ and its subsets $B$ and $C$. It is known that $|A-(B\cap C)|=8$, $|B|=5$, $|C-B|=1$ and $|B\cap C|=3$ (here $-$ denotes set difference). How many subsets $X\subseteq A$ are there if $X\cap B\cap C\ne\emptyset$, $|X-(B\cup C)|\ge3$ and $|X\cap (B-C)|=2$?
  39. L

    A Same open sets + same bounded sets => same Cauchy sequences?

    Let ##d_1## and ##d_2## be two metrics on the same set ##X##. Suppose that a set is open with respect to ##d_1## if and only if it is open with respect to ##d_2##, and a set is bounded with respect to ##d_1## it and only if it is bounded with respect to ##d_2##. (In technical language, ##d_1##...
  40. Ventrella

    I Mutually disjoint sets of all integer powers?

    I identified what appears to be a partitioning of all integers > 1 into mutually disjoint sets. Each set consists of an infinite series of integers that are all the powers of what I am calling a "root" r (r is an integer that has no integer roots of its own, meaning: there is no number x^n that...
  41. S

    MHB Proper Subsets and Relations of Sets

    Q1: Write all proper subsets of S = {1, 2, 3, 4 }. Q2: Let S = {1,2,5,6 } Define a relation R on S of at least four order pairs, as (a,b)  R iff a*b is even (i.e. a multiply by b is even)...
  42. Mr Davis 97

    Infinite union of closed sets is not closed

    Homework Statement Show that it is not necessarily true that the infinite union of closed sets is closed Homework EquationsThe Attempt at a Solution From intuition, I came up with the following counter-example: ##\displaystyle \bigcup_{n=2}^{\infty} \left[ \frac{1}{n}, \frac{n}{n+1} \right] =...
  43. W

    Set Theory: Power sets of Unions

    Homework Statement I'm having issues understanding a mistake that I'm making, any assistance is appreciated! I know a counterexample but my attempt at proving the proposition is what's troubling me. Prove or disprove $$P(A \cup B) \subseteq P(A) \cup P(B) $$ Homework EquationsThe Attempt at...
  44. W

    Relations on Sets: Need help understanding a mistake

    Homework Statement Suppose ##R## and ##S## are relations on a set ##A##. If ##R## and ##S## are transitive, is ##R \cup S## transitive? Why? Homework EquationsThe Attempt at a Solution Suppose that ##a## is an arbitrarily but particularly picked element of ##R \cup S##, then $$a \in R \...
  45. Carrie233

    Why Can't All Subsets of A×B Be Expressed as Cartesian Products?

    Homework Statement Prove: If A and B each have at least two elements, then not every element of P(A×B) has the form A1 ×B1 for some A1 ∈ P(A)and B1 ∈ P(B). Homework EquationsThe Attempt at a Solution Suppose A = {1, 2}, B = {3, 4}. AXB = {(1,3), (1,4), (2,3), (2,4)} P(A) = {{1}, {2}, {1,2}...
  46. A

    MHB Inequality of Cardinality of Sets

    I am working on a proof problem and I would love to know if my proof goes through: If $A, B$ are sets and if $A \subseteq B$, prove that $|A| \le |B|$. Proof: (a) By definition of subset or equal, if $x \in A$ then $x \in B$. However the converse statement if $x \in B$ then $x \in A$ is not...
  47. M

    MHB Describing Sets: A Comprehensive Guide

    Hey! :o I want to describe in words the following sets: 1. $A:=\{(x,y)\in \mathbb{R}^2\mid x>0, y\leq 1\}$ $A$ is the set of all pointgs where the first coordinate is positiv and the second one is less or equal to $1$. It is the subarea of the plane that is under the point $(0/1)$ to...
  48. hilbert2

    A Linearly independent function sets

    It is well known that the set of exponential functions ##f:\mathbb{R}\rightarrow \mathbb{R}_+ : f(x)=e^{-kx}##, with ##k\in\mathbb{R}## is linearly independent. So is the set of sine functions ##f:\mathbb{R}\rightarrow [-1,1]: f(x) = \sin kx##, with ##k\in\mathbb{R}_+##. What about...
  49. M

    MHB Proving Countability: Positive Rational Numbers & Union of Countable Sets

    Hey! :o Show that the set of all positive rational numbers is a countable set. (Hint: Consider all points in the first quadrant of the plane of which the coordinates x and y are integers.) Show that the union of a countable number of countable sets is a countable set. I have done the...
  50. evinda

    MHB Finding Supremum and Infimum of Sets with Inequalities

    Hello! (Wave) I want to find the supremum, infimum of the following sets: $$\{ x \in \mathbb{R}: 0<x^2-1<3\}, \{1+\frac{(-1)^n}{n}: n=1,2, \dots \}$$ For the first set I have thought the following: $$ 0<x^2-1<3 \Rightarrow 1<x^2<4 \Rightarrow x^2>1 \text{ and } x^2 <4 \Rightarrow (x>1 \text{...
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