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. Math Amateur

    MHB Open Sets - Unions and Intersections -

    Open Sets - Unions and Intersections - Sohrab Ex. 2.2.4 (1) I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with a part of Exercise 2.2.4 Part (1) ... ... Exercise 2.2.4 Part...
  2. G

    I Can you calculate probability with infinite sets?

    Suppose set A is defined as the even integers and set B is defined as for every even integer there are two odd integers, like so: {2,3,3,4,5,5,6,7,7 ... } Can you calculate that the probability of choosing an odd number is 66%?
  3. C

    Mathematica Chisquare fit to multiple data sets

    I am looking to perform a ##\chi^2## fit to more than one data set in mathematica, I just wondered how one would set this up? Given some non-linear ansatz function ##F## depending on some parameters ##a,b,c## to describe all sets of data, i.e ##F = F_z(x,a,b,c)## I want to do a ##\chi^2## fit...
  4. SSequence

    B Subsets of Rational Numbers and Well-Ordered Sets

    This isn't original or anything, but I was thinking about how would one go about formalizing (in a general sense) an informal wikipedia picture such as this: https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Omega-exp-omega-labeled.svg/487px-Omega-exp-omega-labeled.svg.png For example...
  5. K

    MHB Proving Set Equality: {A ∪ (B ∩ C')} ∪ A ∪ C = A

    Prove that {A UNION(B INTERSECT C')} UNION (A UNION C)=A
  6. C

    I Can sets contain coordinates of points and be used in Cartesian product?

    Hi guys, I would like to ask if a set can contain coordinates of points, for example A={[1,3];[4,5];[4,7]} and if we can do Cartesian product of such sets, for example A={[1,3];[4,5]}, B={[7,8];[4,2]} A×B={[1,3][7,8];[1,3][4,2];[4,5][7,8];[4,5][4,2]} (is it correct to write it like that?). I am...
  7. AJ007

    I 2 basis sets, for which one is DFT calc faster?

    Hello fellow physicist, I'm new on this subject, I hope that you can help me clear some stuff out. If I have 2 basis sets; A containing 100 basis functions and B containing 50 basis functions, for which one of them will a DFT calculation be faster? Also, is a DFT calculation using basis sets...
  8. V

    How many sets of 6 can be made from a set of 12

    Homework Statement [/B] A bag has a total of 12 balls, 3 of which are black. If you select 6 balls from the bag. a) How many sets of 6 can be made? b) How many of the sets of 6 contain 3 black balls? c) What is the probability of selecting a set of 6 that contains 3 black balls? Homework...
  9. Bunny-chan

    Supremum and infimum of specific sets

    Homework Statement I'm in need of some help to be able to determine the supremum and infimum of the following sets:A = \left\{ {mn\over 1+ m+n} \mid m, n \in \mathbb N \right\}B = \left\{ {mn\over 4m^2+m+n^2} \mid m, n \in \mathbb N \right\}C = \left\{ {m\over \vert m\vert +n} \mid m \in...
  10. B

    Producing a Family of 0,1 Knapsack Sets

    Dear Physics Forum friends, I am currently stuck with the following question about the integer optimization: "Produce a family of 0,1 knapsack sets (having an increasing number n of variables) whose associated family of minimal covers grows exponentially with n." My thought is that I need to...
  11. stevendaryl

    A Relative frequency and nonmeasurable sets

    There is a conceptual puzzle that I don't understand about nonmeasurable sets. Take the unit interval [0, 1] and let S be some subset. Now, generate (using a flat distribution) a sequence of reals in the interval: r_1, r_2, r_3, ... Then we can define the relative frequency up to n as follows...
  12. P

    MHB Understanding Sets, Relations, and Functions for Struggling Students

    I'm having issues with the first four questions and have uploaded them. My attempts are shown below. 1. a) True, all elements of E are even b) False, 0 is not a multiple of 3 c) True, 8 is even and 9 is a multiple of 3 d) No idea e) False, 6 is an element of E and T f) No idea 2. a) You can...
  13. M

    Interpreting: Consider S & T Sets - Are they Convex?

    Homework Statement Homework EquationsThe Attempt at a Solution Consider S = {(1,1)} and T = {(0,0)} Clearly, S and T is convex S + T = S and S - T = S So both of them are convex. So answer is (E) But i feel that the answer is too simple...and seems that i wrongly interpreted the question...
  14. Eclair_de_XII

    Proving that the intersection of two sets is a subset of Q?

    Homework Statement "Let ##U={p+r\sqrt{2}:p,r∈ℚ}## and let ##V={a+b\sqrt{7}:a,b∈ℚ}##. Show that ##ℚ⊆U∩V##. Then show that ##U∩V⊆ℚ## and conclude that ##U∩V=ℚ##." Homework EquationsThe Attempt at a Solution I only knew how to prove the first part. That is: (1) "Suppose that ##ℚ⊄U∩V##. This...
  15. R

    A Formal axiom systems and the finite/infinite sets

    The hereditarily finite sets(a subclass of the Von Neumann universe) are an axiomatic model that corresponds to the usual axioms of set theory but with the axiom of infinity replaced by its negation(showing its independency from the other axioms of set theory). Some mathematicians (a minority)...
  16. Sollicitans

    Calculating events from phrasal expressions

    Homework Statement This excresice is supposed to help you understand the basic operations of sets, later used in probability. I am given the following phrases and have to write them in using mathematics. Given three events A, B and C, which belong to sample space S, calculate the following...
  17. F

    I Sets, Subsets, Possible Relations

    Given a set, there are subsets and possible relations between those arbitrary subsets. For a given example set, the possible relation between the subsets of the example set will narrow down to the "true" possible relations between those subsets. a) {1} Number of Subsets: ##2^1 = 2## (∅, {1})...
  18. sa1988

    Determine all of the open sets in given product topology

    Homework Statement ##X = \{1,2,3\}## , ##\sigma = \big\{\emptyset , \{1,2\}, \{1,2,3\} \big\}##, topology ##\{X, \sigma\}## ##Y = \{4,5\}## , ##\tau = \big\{\emptyset , \{4\}, \{4,5\} \big\}##, topology ##\{Y, \tau\}## ##Z = \{2,3\} \subset X## Find all the open sets in the subspace topology...
  19. T

    Sets and Quantifiers: Power Sets & Family Homework

    Homework Statement B ∈ {P (A) | A ∈ F}. where P(A) is the power set of A and F is Family Homework Equations N/A The Attempt at a Solution My interpretation: A: an element of the Family of sets. Hence, A is a set. P(A): the set of all the possible unique subsets of A. B: an element of the set...
  20. sa1988

    Functions and Sets: Understanding Notation and Inverse Functions

    Homework Statement ONLY QUESTION 2[/B] Homework EquationsThe Attempt at a Solution Not sure what's going on here. I think the issue is in my own flawed understanding of the notation used in sets generally. So the question states: f : R \rightarrow R such that f(x) = x^{2} My...
  21. sa1988

    Some basic questions on the way sets are defined

    Homework Statement Homework EquationsThe Attempt at a Solution Q.1 I'm a little confused with how subsets and elements are defined in the case of the given set "A" as it seems to be a set of sets, so I'm going to throw my answers out and would appreciate any guidance on where I'm wrong (if...
  22. TheChemist_

    I Question about Accumulation points

    So we just recently did accumulation points in my maths class for chemists. I understood everything that was taught but ever since I was trying to find a reasonable explanation if the sequence an = (-1)n has 2 accumulation points (-1,1) or if it doesn't have any at all. I mean it's clear that...
  23. RJLiberator

    Infimum and Supremum, when they Do not exist in finite sets

    Homework Statement Give an example of each, or state that the request is impossible: 1) A finite set that contains its infimum, but not its supremum. 2) A bounded subset of ℚ that contains its supremum, but not its infimum. Homework EquationsThe Attempt at a Solution I either understand this...
  24. W

    Show: Elementary row operations don't affect solution sets

    Homework Statement Show that elementary row operations don't affect solutions sets in linear systems Homework Equations - The Attempt at a Solution It's pretty easy to come up with a random linear system and perform ERO on them and showing that solutions are not affected, but is there all...
  25. M

    Proving Linear Independence of Av1, ..., Avk and Conditions for Basis in Rm

    Homework Statement [/B] 1. Suppose {v1, . . . , vk} is a linearly independent set of vectors in Rn and suppose A is an m × n matrix such that Nul A = {0}. (a) Prove that {Av1, . . . , Avk} is linearly independent. (b) Suppose that {v1, . . . , vk} is actually a basis for Rn. Under what...
  26. F

    MHB How do I solve a system of sets ?

    Let A,B and C be three elements of P(E) 1. Solve in P(E) the following equation : AUX=B 2. Let's suppose that C ⊂ A ⊂ B , solve in P(E) the following system : AUX=B and A⋂X=C I've already answered the first question , it's X = (B\A) U Y such that Y∈P(A) As for the second , I thought maybe X=C...
  27. F

    MHB Intersection of Sets A, B and C in ℤ

    Let A,B,C be three sets such that : A={x∈ ℤ / x=11k+8 , k∈ℤ} B={x∈ ℤ / x=4k , k∈ℤ} C={x∈ ℤ / x=11(4k+1) -3 , k∈ℤ } Prove A⋂B = C I started with this : Let x be an arbitrary element of A⋂B then ∃(k,k')∈ ℤ² such that x=11k+8 and x=4k' then 11k+8 = 4k' then 11(k+1)-3 = 4k' I don't know where...
  28. agargento

    Possible Subsets of Even Numbers in a Set of Size n?

    Homework Statement Given {1,2,...,n}, n is an even number. What are all the possible subsets that contain only even numbers? (Notice that ∅ is also defined as such a subset). Homework Equations 2n - all possibilities for group A with n objects The Attempt at a Solution I think the answer...
  29. agargento

    Possibilities for Set B with n Objects in Sample Space U

    Homework Statement Given sample space U with n objects. A ⊂ U, and A has k objects. A ∩ B ≠∅ What are all the possibilities for B? Homework Equations 2n - All possibilities for set B with n objects The Attempt at a Solution I don't know where even to begin... The question itself confuses me.
  30. M

    MHB Can Discontinuity and Non-Derivability Exist in Strongly Concave Functions?

    Hey! :o Could you give me an example of a strong concave function $f:[0,3]\rightarrow \mathbb{R}$ that is not continuous? (Wondering) We have that $f''(x)<0$. Since the function has not to be continuous, the derivatives are neither continuous, are they? (Wondering) Is maybe the...
  31. Ryaners

    Beginning Sets: Advice on Set Building Notation?

    I've started Book of Proof, the first chapter of which is an intro to sets. Q.1 Is there any particular way to approach these kinds of problems, other than using intuition / trial & error? I tend to have some difficulty in working out the best way to express the general term of a sequence, for...
  32. M

    Combinatorics: looking for an alternative solution

    Homework Statement Show that every subset with 6 elements of {1,2,3,4, ..., 9} contains 2 elements with sum 10. I solved this (solution below) but I want to do this easier using the pidgeon hole principle. Homework Equations Pidgeon hole principle Combinatorics The Attempt at a Solution...
  33. C

    MHB How Do You Convert Temperatures and Solve Inverse Functions?

    Temperatures can be converted from Fahrenheit to Celsius using the function f(x) = 5 /9 (x − 32). (a) Calculate f(59). (b) Find f −1 (x), and verify that f −1 (f(59)) = 59. (c) Let K be the set {x : f(x) = x}. Find all elements of K and list K
  34. M

    MHB Proof of Sets: Proving (i) and (ii)

    If $X$ is a set, then the power set $P(X)$ of a set is the set of all subsets of $X$. I need to decide whether the following statements are true or false and prove it: (i) If $Z = X \cup Y$ , then $P(Z) = P(X) \cup P(Y)$. (ii) If $Z = X \cap Y$ , then $P(Z) = P(X) \cap P(Y)$. By examples I...
  35. Nikto

    Homework sets for Intro Optics/Waves (e.g. MIT 8.03)

    Can anyone recommend published homework sets even without solutions for an intro to waves/optics course, the 3rd of a typical 3-course intro physics sequence? I had been following MIT's intro sequence (8.01, 8,02, 8.03) while taking a course that follows Halliday/Resnick, which I do not find...
  36. pellman

    I How does a disjoint union differ from a set of sets?

    Given an indexed collection of sets A_x the disjoint union of these sets can be thought of as the ordinary union of the sets \{ x \} \times A_x for all x. That is, it is the set of all pairs (x, a) where a \in A_x. (Correct me at this point if my understanding of disjoint union is wrong.)...
  37. J

    Where can I buy sets of small weights/masses?

    I'm looking to tutor someone, but I need to find small masses like 1 gram, 10 gram, 25 gram, 100 gram things for demonstrations/experiments etc. As small as possible would be nice, like no bigger than a quarter. But whenever I try to google for weights I just get weightlifting things like...
  38. C

    Countable union of Jordan sets is not always Jordan measurable

    Homework Statement Show that the countable union or countable intersection of Jordan measurable sets need not be Jordan measurable, even when bounded. The Attempt at a Solution For countable intersection, I think the rationals from 0 to 1 will work, each rational have jordan measure zero...
  39. B

    MHB Prove relationship between sets

    For any two sets A and B prove: (A∪B)^c=A^c∩B^c (A∩B)^c=A^c∪B^c
  40. L

    I Is this a correct way to describe number sets?

    Hello, I am excited to be learning about number sets again, :P. Ok, so this is how I describe them: N={1,2,3,4,5,6,7,8,9,10….∞} Z={-∞….-3,-2,-1,0,1,2,3…∞} Q=a/b ,a∈Z,b∈{Z∖0} The first three are correct. However, how do we describe irrational numbers? Is it just the difference set of...
  41. arupel

    I Intersection & union of closed and open sets?

    I am a little confused here: a) The number 2 which is at the beginng of one set is closed. The number 2 is open at the beginning of the other set. b) The number 2 is closed of the beginning of a set which goes to infinity. The other set begins at 0 and goes to infinity (2 is an...
  42. L

    MHB Proving a Set Theory Statement Regarding Families of Sets

    I was wondering if anyone could please check my work and reasoning for this problem. Thank-you! (Also, would this be considered a direct proof? How might a contradiction and IFF proof look like and compare?) Problem: Suppose F, G1 and G2 are nonempty families of sets. Prove that if F ⊆ G1 ∩ G2...
  43. Derek Hart

    Adequate proof? Spivak's Calculus ; Dense sets

    Homework Statement Let A be a dense set**. Prove that if f is continuous and f(x) = 0 for all x in A, then f(x) = 0 for all x. **A dense set is defined, in the book, as a set which contains a point in every open interval, such as the set of all irrational or all rational numbers.Homework...
  44. evinda

    MHB How Can I Show That the Symmetric Difference of Sets is Associative?

    Hello! (Wave)I want to show that $ (A \triangle B) \triangle C=A \triangle (B \triangle C) $. I have tried the following: $$ x \in (A \triangle B) \triangle C \Leftrightarrow x \in (A \triangle B)\setminus C \lor x \in C \setminus (A \triangle B) \\ \Leftrightarrow (x \in A \triangle B \wedge...
  45. P

    B Sets and functions that gain more structure with context

    So I have two sets, call it ##A## and ##B##. I also have a function ##f:A\rightarrow B##. By themselves, it does not matter (or at the very least make sense) to think of ##A## and ##B## as, say, groups (I'm not really thinking exclusively about groups, just as an example). For that matter, it...
  46. DavideGenoa

    I N-dimensional Lebesgue measure: def. with Borel sets

    Let us define, as Kolmogorov-Fomin's Элементы теории функций и функционального анализа does, the definition of outer measure of a bounded set ##A\subset \mathbb{R}^n## as $$\mu^{\ast}(A):=\inf_{A\subset \bigcup_k P_k}\sum_k m(P_k)$$where the infimum is extended to all the possible covers of...
  47. Math Amateur

    I Affine Algebraic Sets - General Question

    I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I am trying to gain a full...
  48. Math Amateur

    I Affine Algebraic Sets - Dummit and Foote, page 660, Ex. 3

    I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need someone to help me to...
  49. Math Amateur

    MHB Affine Algebraic Sets in A^2 - Dummit and Foote, page 660, Example 3

    [I apologise for repeating this post ... but I genuinely would like help ... and the post comes from September 2015 ... and so is not an impatient "bump" of an item ... I hope administrators will understand ...] =====================================================I am trying to gain an...
  50. W

    I How to compare two data sets with multiple samples

    I have two data sets A and B which correspond to two different settings of the system. Both sets contain 5 separate lists of integers; I took 5 samples, at different points in time and location, for each set to reduce the random error in the data. How would I go about comparing sets A and B? If...
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