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. J

    Proving the triangle inequality property of the distance between sets

    Proving the "triangle inequality" property of the distance between sets Here's the problem and how far I've gotten on it: If you are unfamiliar with that notation, S(A, B) = (A \ B) U (B \ A), which is the symmetric difference. And D(A, B) = m^*(S(A, B)), which is the outer measure of...
  2. C

    Nested Open Sets: Example & Intersection

    Homework Statement Give an example of an infinite collection of nested open sets. o_1 \supseteq o_2 \supseteq o_3 \supseteq o_4 ... Whose intersection \bigcap_{n=1}^{ \infty} O_n is closed and non empty. Homework Equations A set O \subseteq \mathbb{R} is open if for all points...
  3. S

    Proof: function of int. of family of sets and int. of function of family of sets

    Homework Statement The question is "Prove f(\bigcap_{\alpha \in \Omega} A_\alpha) \subseteq \bigcap_{\alpha \in \Omega} f(A_\alpha) where f:X \rightarrow Y and \{A_\alpha : \alpha \in \Omega\} is a collection of subsets of X. Also, prove the statement's equality when f is an injective...
  4. U

    Closure of relations betweens sets

    Hi all! I am searching for an algorithm (most likely already present in the literature) that could solve the following problem: Instance: Properties of sets of elements and relations between sets of elements Question: Find the closure of the properties and relations Possible properties...
  5. C

    A perpetual machine model that sets me thinking

    Now I'm not a PMI (perpetual machine inventor). In fact I'm quite convinced that there is no such thing as that. But a while ago, I saw the schematics of a perpetual machine that is hard to debate. Well this is how the machine worked. The inventor argued that if you have two magnets as...
  6. N

    Determining open and connected sets.

    Homework Statement Are the following regions in the plane (1) open (2) connected and (3) domains? a. the real numbers; b. the first quadrant including its boundary; c. the first quadrant excluding its boundary; d. the complement of the unit circle; f. C \ Z = {z ∈ C : z \notinZ}...
  7. J

    What is the significance of JD and how is it calculated?

    Hi, When you have to calculate the rising or setting time of a celestial body, you have to handle with hour angle and sidereal time. Sidereal time for the rising is given by T = alpha - H and by T = alpha + H for the setting (alpha = right ascension). Why - H in one hand, and + H on the...
  8. L

    Closed set as infinite intersection of open sets

    This is not a homework problem, just something I was thinking about. In a general metric space, is it true that every closed set can be expressed as the intersection of an infinite collection of open sets? I don't really know where to begin. Since the finite intersection of open sets is open...
  9. E

    Cartesian product of open sets is a open set

    Homework Statement This is not really coursework. Instead, this is some sort of curiosity and proposition formulation rush. Then the initial questions are that if this is a valid result that is worth to be proven. Let X,Y be metric spaces and X\times Y with another metric the product metric...
  10. N

    Whats an infinite intersection of open sets

    whats an infinite intersection of open sets? how is it different from finite intersection of open sets and why is it a closed set in the case of ∞ intersection but open in case of finite. To quote kingwinner, is it being defined as a limit? it really does look look like a limit in the case...
  11. T

    Help with part of my Linear Algebras project - affine sets and mappings

    ℝHomework Statement I'm a 2nd year undergraduate student, so I suppose many users here won't find this too difficult, but I've had some issues with the following questions and, of course, any help would be very much appreciated: (i) Prove f: V→W is affine (where V and W are real vector space)...
  12. 6

    Exploring Closed Sets in Metric Spaces through Infinite Intersections

    Homework Statement Find (X,d) a metric space, and a countable collection of open sets U\subsetX for i \in Z^{+} for which \bigcap^{∞}_{i=1} U_i is not open Homework Equations A set is U subset of X is closed w.r.t X if its complement X\U ={ x\inX, x\notinU} The Attempt at a Solution Well...
  13. D

    Zakon Vol 1, Ch2, Sec-6, Prob-19 : Cardinality of union of 2 sets

    Homework Statement Show by induction that if the fi nite sets A and B have m and n elements, respectively, then (i) A X B has mn elements; (ii) A has 2m subsets; (iii) If further A \cap B = \varphi, then A \cup B has m+ n elements. NOTE : I am only interested in the (iii) section of...
  14. K

    Julia Sets: Periodic and Non-Periodic Points Explained

    I'm a little bewildered when reading about these Julia sets. From the definition a Julia set is the closure of all repelling periodic points of a complex map f. However I read that a Julia set always contain periodic and non-periodic points. Wasn't the definition including only periodic points...
  15. T

    Proof about the decomposition of the reals into two sets.

    Homework Statement Let S and T be nonempty sets of real numbers such that every real number is in S or T and if s \in S and t \in T, then s < t. Prove that there is a unique real number β such that every real number less than β is in S and every real number greater than β is in T. The Attempt...
  16. Z

    How can I get a function relation with these two sets?

    Homework Statement I have these two sets: Pairwise, (1, 1) (2, 4) (3, 9) (4, 16). Clearly this is just squared. How can I get a function relation with like: (1, 1) (2, 3) (3, 9) (4, 10) or like (1, 1) (2, 5) (3, 12) (4, 22) Homework Equations The Attempt at a...
  17. C

    Limits in infinite unions of sets

    Suppose I define sets D_n = \lbrace x \in [0,1] | x has an n-digit long binary expansion \rbrace . Now consider \bigcup_{n \in \mathbb{N}} D_n. This is just the set of Dyadic rationals and therefore countable for sure. Now for the question: is this equal to \bigcup_{n = 0}^{\infty} D_n...
  18. T

    Real Analysis: countably infinite subsets of infinite sets proof

    Homework Statement Prove that every infinite subset contains a countably infinite subset. Homework Equations The Attempt at a Solution Right now, I'm working on a proof by cases. Let S be an infinite subset. Case 1: If S is countably infinite, because the set S is a subset...
  19. S

    Discrete math, sets, power sets.

    Homework Statement If s (0,1), find |P(S)|, |P(P(S))|, |P(P(P(S)))| Homework Equations The Attempt at a Solution |P(S)| = {(0), (1), (0,1), ∅} = 4 |P(P(S))| = {...} = 16. |P (P(P(S)))| = {...} = 16 ^4 ...but how? as my lecturer explained it, it come from pascals...
  20. C

    Probability of 2 equivalent random selections from integer sets

    What is the probability that a number selected from 0-9 will be the same number as one randomly selected from 0-4? Relevant equations: $$P(A \cap B) = P(A)*P(B|A)$$ I used the equation above, using A as the event that the number selected from 0-9 will be between 0 and 4, and B as the event that...
  21. B

    Clopen Sets: Closure = Interior?

    For a subset which is both closed and open (clopen) does its closure equal its interior?
  22. K

    Showing two countably infinite sets have a 1-1 correspondence

    Homework Statement Suppose A and B are both countably infinite sets. Prove there is a 1-1 correspondence between A and B. Homework Equations The Attempt at a Solution Since A is countably infinite, there exists a mapping f such that f maps ℕto A that is 1-1 and onto...
  23. T

    Real Analysis: one to one correspondence between two countably infinite sets

    Homework Statement Suppose that A and B are both countably infinite sets. Prove that there is a one to one correspondence between A and B. Homework Equations The Attempt at a Solution By definition of countably infinite, there is a one to one correspondence between Z+ and A and...
  24. B

    Abstract math prove involwing sets

    Homework Statement Let Ts denote the set of points in the x; y plane lying on the square whose vertices are (-s; s), (s; s), (s;-s), (-s;-s), but not interior to the square. For example, T1 consists of the vertices (-1; 1), (1; 1), (1;-1), (-1;-1) and the four line segments joining them...
  25. M

    [Cardinality] Prove there is no bijection between two sets

    Homework Statement prove there is no continuous bijection from the unit circle (the boundary; x^2+y^2=1) to R Homework Equations The Attempt at a Solution is this possible to show by cardinality? since if two sets have different cardinality, then there is no bijection between...
  26. T

    Understand Affine Subsets & Mappings: Research Project for Undergrads

    This research project is to help me (I'm an undergraduate) get my head around this topic. It is concerned with affine subsets of a vector space and the mappings between them. As an application, the construction of certain fractal sets in the plane is considered. It would be considered pretty...
  27. Z

    Sequences, sets and cluster points

    Hello all, I am having trouble with a homework problem. The problem is as such: Let a = {zn = (xn,yn) be a subset of ℝ2 and zn be a sequence in ℝ2 such that xn ≠ xm and yn ≠ ym for n≠m. Let Ax and Ay be the projections onto the x and y-axis (i.e. Ax = {xn} and Ay = {yn}. Assume that the...
  28. 1

    Unions and intersections of collections of sets

    My proof class just took a turn for the worst for me - I don't understand this. First, the notation is extremely confusing to me, I need help to make sure I'm getting this. If An is some set for some natural number n such as [-n, n]. Then (script A) the collection is the set of all An...
  29. Useful nucleus

    Are all open sets compact in the discrete topology?

    A subset K of a metric space X is said to be compact if every open cover of K contains a finite subcover. Does not this imply that every open set is compact. Because let F is open, then F= F \bigcup ∅. Since F and ∅ are open , we obtained a finite subcover of F. Am I missing something here?
  30. I

    Let A, B and C be sets. Prove that

    *Sorry wrong section* Let A, B and C be sets. Prove that if A\subseteqB\cupC and A\capB=∅, then A\subseteqC. My attempted solution: Assume A\subseteqB\cupC and A\capB=∅. Then \veex (x\inA\rightarrowx\inB\cupx\inc). I'm not sure where to start and how to prove this. Any help would be greatly...
  31. K

    Showing two certain sets have no elements in common

    Homework Statement Let x and y be irrational numbers such that x-y is also irrational. Let A={x+r|r is in Q} and B={y+r|r is in Q} Prove that the sets A and B have no elements in common. Homework Equations The Attempt at a Solution Since x and y are in A and B, then...
  32. K

    Finding X and Y with respect to third variable if having two sets of X and Y

    My problem At temperature 5 degrees the Y=1.00228594 for X=435 and Y=1.000986038 for X=449 and Y=0.999760292 for X=463 At temperature 7 degrees the Y=1.002094781 for X=435 and Y=1.00079709 for X=449 and Y=0.999573015 for X =463 For a new temperature of 9.6 degrees how to find the Y...
  33. I

    MHB Finding a set which is not equinumerous with series of sets

    Hi Let \( A_1=\mathbb{Z^+} \) and \( \forall n\in \mathbb{Z^+}\) let \( A_{n+1}=\mathcal{P}(A_n) \) I have to come up with an infinite set which is not equinumerous with \( A_n \) for any \( n\in \mathbb{Z^+} \). Clearly \( \mathbb{R}\) will not fit the bill since \( \mathbb{R}\;\sim\; A_2...
  34. I

    MHB Can Power Sets of Equinumerous Sets Remain Equinumerous?

    Hi Here is the problem. Let A be a set with at least two elements. Also suppose. \[ A\times A \sim A \] Then prove that \[ \mathcal{P}(A)\times \mathcal{P}(A)\sim \mathcal{P}(A) \] Let a and b be the two elements of this set. Then I want to exploit the result that \[ \mathcal{P}(A)\;\sim...
  35. S

    MHB Questions about sets and subsets

    Hi, the question goes as follows: Given two subsets X and Y of a universal set U, prove that: (refer to picture) I'm having particular trouble on D and E, if anyone can clear this up or provide some clarification for me that would be appreciated! I know a venn diagram might not be possible...
  36. N

    Generating sets based on a recursive language definition

    I've searched the internet and the forums for any help on this, but I can't seem to find a topic that details what the successive sets will contain. Here is an example question: (I have many HW problems like this, I just don't know where to start) Let L be the language over {a,b} generated by...
  37. T

    Linearly Independence and Sets of Functions

    Homework Statement The Attempt at a Solution I don't think I'm really understanding this problem. Let me tell you what I know: A set is linearly independent if a_1 A_1 +...+a_n A_n = \vec0 for a_1,...,a_n \in R forces a_1 = ...=a_n = 0. If f,g,h take any of the x_i \in S, then one of the...
  38. Deveno

    What is the role of forcing in understanding uncountable sets in set theory?

    I have recently become suspicious of the real numbers. For nearly 3 decades I accepted their axiomatic existence as a complete, ordered archimedian field. The Dedekind-cut, and Cauchy sequence, and "infinite decimal" constructions all made sense to me. And then I started reading about models...
  39. T

    Proving sets with structural induction

    Consider the set S defined recursively as follows: • 3 ∈ S, • if x,y ∈ S,then x−y∈S, • if x∈S, then 2x ∈ S, • S contains no other element. Use Structural Induction to write a detailed, carefully structured proof that ∀ x ∈ S, ∃ n ∈ Z, x = 3n. What I've got is since 3 is in the set...
  40. T

    Can Structural Induction Prove All Elements of Set T Are Powers of 2?

    Consider the set T defined recursively as follows: • 2∈T, • if x∈T and x>1,then x/2 ∈T, • if x∈T and x>1,then x^2 ∈T, • T contains no other element. Use Structural Induction to write a detailed, carefully structured proof that ∀ x ∈ T, ∃ n ∈ N, x = 2n. I'm not sure how to...
  41. H

    Find the bit string for the following sets.

    Homework Statement The a universal set: u = {1,2,3,4,5,6,7,8,9,10} 1) Find the bit string for b = {4,3,3,5,2,3,3,} 2) Find the bit string for the union of two sets. Homework Equations 1)Would I first begin this problem by realizing that set b is the same as {2,3,4,5}? The...
  42. K

    What is the Proof of Equality for Sets of Random Variables?

    Hey there. I'm asked to prove: If X1,...Xn are random variables defined on a set Ω and B1,...,Bn C R1 then prove that (X1,...,Xn)^-1 (B1x...xBn) = X1^-1B1 n X2^-1B2 n ... n Xn^-1Bn so I think I can explain the proof, but just not write it out. This is my attempt if omega is on the left hand...
  43. S

    A geometric property of a map from points to sets?

    I'm interested in the proper way to give a mathematical definition of a certain geometric property exhibited by certain maps from points to sets. Consider mappings from a n-dimensional space of real numbers P into subsets of an m-dimensional space S of real numbers. For a practical...
  44. S

    Multiple sets of linearly independent vectors

    hallo I am trying to calculate the probability to obtain 2 sets of linearly independent vectors from a set of binary vectors of length k. For example: k = 4, and therefore I have 2^k = 16 vectors to select from. I want to randomly select 7 vectors (no repetition). What is the...
  45. A

    Sets closed under complex exponentiation

    The rational (and also algebraic) elements of ℂ are closed under addition, multiplication, and rational exponentiation (the algebraic numbers, that is), but not under complex exponentiation. For instance, (-1)^i=e^{-\pi}, with is not rational, and in fact it is even transcendental. Is there any...
  46. F

    Are A and B subsets of R \ {0}?

    & means belong to and # not equal to : $ subsets of A={(t-1,1/t): t&R, t # 0} B= {(x,y) &R^2:y=1/(x+1), x#-1} i started by say A$B let x= t-1 and y=1/t so we have y= 1/(t-1)+ = Y=1/t hence A$B to prove B$A is where i am stuck- as I think I have got my first part wrong anyway and I...
  47. K

    'order of operations' in sets?

    Hello all. Currently working on simplifying some Boolean expressions, one of the questions is: ( A int B U C) int B I do not know how to go about simplifying the first term because there are not any parentheses within it and I have both the intersection and union symbols. Is there an...
  48. J

    Recursion Theorem and c.e. sets

    Let \varphi_e denote the p.c. function computed by the Turing Machine with code number e. Given M=\{x : \neg(y<x)[\varphi_y=\varphi_x]\} prove that M is infinite and contains no infinite c.e. subset. I have already proved that M is infinite. A necessary and sufficient condition to prove that M...
  49. A

    Closed Sets in \mathbb{C}: Showing Unclosedness by Example

    Homework Statement Show by example that an infinite union of closed sets in \mathbb{C} need not be closed. The Attempt at a Solution In \mathbb{R} I know that an infinite union of the closed sets A_{n}=[1/n,1-1/n] is open. Not sure if it works in \mathbb{C} as well.
  50. A

    Bio mechanic force/moment problem sets

    Homework Statement Hi everyone, I have 2 questions that I think are relatively easy but are frustrating in that I just can't seem to get an answer 1st -- If a lateral-to-medial load is applied to the foot, a counteracting moment is produced at the knee joint in a lateral-medial plane, which...
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