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. TheMathNoob

    How Does Induction Prove the Probability Limit of Nested Sets?

    Homework Statement Let A1,A 2,...be an infinite sequence of events such that A1⊂A2⊂... Prove that Pr(∪Ai)=limn→infPr(An) ∪ is also an iterator that starts from i=1 to infinity. How can you put those iterators?[/B] Homework Equations I decided to use induction The...
  2. TheMathNoob

    How Many Families Subscribe to Exactly One Newspaper?

    Homework Statement In a certain city, three newspapers A, B, and C are published. Suppose that 60 percent of the families in the city subscribe to newspaper A, 40 percent of the families subscribe to newspaper B, and 30 percent subscribe to newspaper C.Suppose also that 20 percent of the...
  3. H

    Solving Set Equality Proof Homework

    Homework Statement Let ##A, B, C## be sets with ##A \subseteq B##. Show ##(A-B)\cup C=(A\cup C)-(B\cup C)## Homework Equations None. The Attempt at a Solution So, generally, one shows two sets to be equal by showing that each is a proper subset of the other. I started with the LHS. Thus...
  4. G

    MHB  Prove |X| = |Y| When X\Y and Y\X are Equipotent Sets

    Prove that if X\Y and Y\X are equipotent sets then |X| = |Y|. The problem is that I've no clue where to start... (Futile) attempt: There is bijection $f: X\backslash Y \to Y\backslash X$. For every $r_1 \in X\backslash Y$ there exists $r_2$ s.t. $r_2 \in Y\backslash X$. That's $r_1 \in X$ and...
  5. W

    Probability, Set Theory, Venn Diagrams

    Homework Statement Let A and B be two events such that P(A) = 0.4, P(B) = 0.7, P(A∪B) = 0.9 Find P((A^c) - B) 2. Homework Equations I can't think of any relevant equations except maybe the Inclusion Exclusion property. P(A∪B) = P(A) + P(B) - P(A∩B) This leads us to another thing P(A∩B^c)...
  6. J

    Can the Infinite Sum of Natural Numbers Really Equal -1/12?

    I was some youtube videos and i got sucked into this channel called "numberphile". They were talking about infinite sets. In particular the set that is the sum of all natural numbers. Through some creative algebra they demonstrate the proof. Somehow the set that is equal to the sum of all...
  7. T

    Are set theory functions sets too?

    I read somewhere that mathematical functions can be implemented as sets by making a set of ordered tuples <a,b> where a is a member of A and b is a member of B. That should create a function that goes from the domain A to the range B. But set theory has functions too, could they be sets too...
  8. M

    How can I wire together two sets of computer sound systems?

    I want to purchase the Logitech z623 and add to it two creative inspire M2600 speakers (only the speakers, not the sub-woofer). The reason for doing this is because the Logitech system sounds really nice, it has an amazing bass, but mids and highs are not the best and I wish they was as clear as...
  9. R

    In search of Julia Sets and ChaosPro Training

    Hello everyone. I am looking for some free software to produce a Julia Set that would allow me to enter the equation. I would prefer it to be downloadable software, but if a web applications all there is that will do. It's preferable that a color scheme can be chosen, or at least used. Also, I...
  10. L

    Divergence theorem on non compact sets of R3

    So my question here is: the divergence theorem literally states that Let \Omega be a compact subset of \mathbb{R}^3 with a piecewise smooth boundary surface S. Let \vec{F}: D \mapsto \mathbb{R}^3 a continously differentiable vector field defined on a neighborhood D of \Omega. Then...
  11. M

    MHB Solving Recursive Sets with Turing Machines

    Hey! :o I have to show that a set is recursive if and only if the set and its complement is recursively enumerable. I have done the following: $\Rightarrow$ Let $A$ the recursive set, so there is a Turing machine $M$ that decides the set $A$. We construct a TM $M'$ that semi-decides the set...
  12. O

    Schools Problem sets for Young and Freedman University Physics (Y&F)

    In the last year, I took a few Mooc online and I felt like my Physics was a bit rusty. So, I found myself a copy of Young and Freedman University Physics (13th edition) to do some self-study. The thing is, there is so many problems to do, I think that I’m on chapter 5 since the beginning of May...
  13. I

    Godel's Theorem, What's it really saying?

    Hi, So I was just going through my copy of The Emperor's New Mind, and I'm having a little difficulty accepting Godel's theorem , at least the way Penrose has presented it. If I'm not wrong, the theorem asserts that there exist certain mathematical statements within a formal axiomatic system...
  14. W

    "Minimal Cover" in Finite Collection of Sets?

    Hi All, Say we have a finite collection ## S_1,...,S_n ## of sets , which are not all pairwise disjoint , and we want to find the minimal collection of the ## S_j ## whose union is ## \cup S_j ## . Is there any theorem, result to this effect? I would imagine that making the ## S_j##...
  15. B

    Set theory, intersection of two sets

    Homework Statement We have the set D which consists of x, where x is a prime number. We also have the set F, which consists of x, belongs to the natural numbers (positive numbers 1, 2, 3, 4, 5..) that is congruent with 1 (modulo 8). What numbers are in the intersection of these two sets...
  16. phoenixthoth

    A method for proving something about all sets in ZFC

    I would appreciate any and all feedback regarding this document currently housed in Google docs. Basically, I generalize induction among natural numbers to an extreme in an environment regarding what I call grammatical systems. Then an induction principle is derived from that which holds in...
  17. C

    Solving the Relation: ##n((AXB) \cap (BXA)) = n(A \cap B)^2##

    Homework Statement If I am given ##n(A)## and ##n(B)## for two sets A and B, and also provided with ##n(A\cap B)^2##. We are supposed to find ##n((AXB) \cap (BXA))##. Homework Equations My teacher said that the formula for ##n((AXB) \cap (BXA)) = n(A \cap B)^2##. I am not sure how do you get to...
  18. Demystifier

    Application of sets with higher cardinality

    Sets with cardinality ##2^{\aleph_0}##, that is, with cardinality of the set of real numbers, obviously have many applications in other branches of mathematics outside of pure set theory. For example, real any complex analysis is completely based on such sets. How about higher cardinality? Is...
  19. nomadreid

    Recursive sets and recursive numbers: relationship?

    Given the two standard definitions (1) A computable set is a set for which there is an algorithm which terminates after a finite amount of time and correctly decides whether or not a given number belongs to the set. (2) A computable number is a number which can be approximated to any degree of...
  20. nomadreid

    Recursive sets as delta^0_1 in arithmetic hierarchy.

    This is an elementary question that I may blush about later, but for now: given that a recursively enumerable set is a set modeling a Σ01 sentence, and a recursive set is a recursively enumerable set S whose complement ℕ\S is also recursively enumerable. Fine. But then, letting x→ = the...
  21. S

    How to Formally Describe Various Sets in Mathematics?

    Homework Statement Write formal descriptions of the following sets. (a) The set containing the numbers 1, 10, and 100 (b) The set containing all integers that are greater than 5 (c) The set containing all natural numbers that are less than 5 (d) The set containing the string aba (e) The set...
  22. sushichan

    Finding solution for three sets of planes

    Homework Statement (I did not copy the problem statement, but basically solve the system of equations if there is solution and give a geometrical interpretation) P1: 2x - y + 6z = 7 P2: 3x + 4y + 3z = -8 P3: x - 2y - 4z = 9 Homework Equations Scalar triple product: n1⋅(n2 × n3) The Attempt...
  23. Mastermind01

    Number Theory Book/Books on elementary number theory

    Hello all, I probably should have posted this in a math forum but I don't know of any. Can anyone recommend a book/books on elementary number theory with exercises? My math background is not very strong with very little knowledge of set theory so it should be understood by me. We're covering...
  24. S

    Let [ ] be a countable number of finite sets. Prove [ ]

    Homework Statement Problem: Let A_1 , A_2 , . . . be a countable number of finite sets. Prove that the union S = ⋃_i A_i is countable. Solution: Included in the TheProblemAndSolution.jpg file. Homework Equations Set-theoretic algebra. The Attempt at a Solution Unless I missed something, it...
  25. B

    MHB Describing the relationship between two sets A and B (probability)

    Hi I am new here! hopefully someone is kind enough to reply fast and help. so the question I am stuck is: Describe the relationship between two sets A and B ( A and B are non-empty) if: a. Pr(A|B)=Pr(A) b. Pr(A/B)=0 c. Pr(A/B)=Pr(A)/Pr(B) (Sorry guys can't get the fraction signs working! so...
  26. J

    Equivalent Metrics From Clopen Sets

    Homework Statement Prove that if ##(X,d)## is a metric space and ##C## and ##X \setminus C## are nonempty clopen sets, then there is an equivalent metric ##\rho## on ##X## such that ##\forall a \in C, \quad \forall b \in X \setminus C, \quad \rho(a,b) \geq 1##. I know the term "clopen" is not a...
  27. jdawg

    Solving Spanning Sets in R^2: Need Help with Problem Tomorrow!

    Hi! Can someone please help me with this problem? I need to understand how to do it for my test tomorrow! I know this a spanning set for R^2, but the way I saw this problem solved was kind of lazy and not very helpful. S={(-1,4),(4,-1),(1,1)} I tried testing to see if it had just the trivial...
  28. xwolfhunter

    Question about empty sets in set theory

    So I'm reading Naive Set Theory by Paul Halmos. He asks: His response is that no ##x## fails to meet the requirements, thus, all ##x##es do. He reasons that if it is not true for a given ##x## that ##x \in X~ \mathrm{for ~ every} ~X~ \mathrm{in} ~ \phi##, then there must exist an ##X## in...
  29. B

    What is the Zero Vector in a Vector Space with Unconventional Operations?

    Homework Statement Determine if they given set is a vector space using the indicated operations. Homework EquationsThe Attempt at a Solution Set {x: x E R} with operations x(+)y=xy and c(.)x=xc The (.) is the circle dot multiplication sign, and the (+) is the circle plus addition sign. I...
  30. Calpalned

    Level curves, level surfaces, level sets

    Homework Statement I know that the equation ##z = f(x,y)## gives a surface while ##w = f(x, y, z) ## gives an object that has the same surface shape on top as ##z = f(x,y)## but also includes everything below it. If these statements are correct, what is the level surface of a function of three...
  31. J

    Cylinder sets, some clarification

    Hey, In a course were we are treating phase transitions from a mathematically exact point of view, Cylinder sets were introduced. I'll first outline the context some more. So we are considering systems on a lattice with a finite state space for each lattice point, for simplicity. E.g. an Ising...
  32. K

    Prove Set of all onto mappings from A->A is closed

    Homework Statement Prove that set of all onto mappings of A->A is closed under composition of mappings: Homework Equations Definition of onto and closure on sets. The Attempt at a Solution Say, ##f## and ##g## are onto mappings from A to A. Now, say I have a set S(A) = {all onto mappings of A...
  33. M

    How math is builded up from sets

    Hi there! I would like to understand the basic concepts of how mathematics is builded up from sets of elements to anything else (i.e: an equation) For example here is a formula: 2-1 So are there different sets of elements like A=(2) B=(-) C=(1) ? And maybe there is an operation with sets...
  34. E

    Are the following Sets: Open, Closed, Compact, Connected

    Homework Statement Ok I created this question to check my thinking. Are the following Sets: Open, Closed, Compact, Connected Note: Apologies for bad notation. S: [0,1)∪(1,2] V: [0,1)∩(1,2] Homework Equations S: [0,1)∪(1,2] V: [0,1)∩(1,2] The Attempt at a Solution S: [0,1)∪(1,2] Closed -...
  35. E

    Closed/Open Sets and Natural Numbers

    Homework Statement I am trying to understand why ℕ the set of natural numbers is considered a Closed Set. 2. Relevant definition A Set S in Rm is closed iff its complement, Sc = Rm - S is open. The Attempt at a Solution I believe I understand why it is not an Open Set: Given that it...
  36. Math Amateur

    MHB Compact Sets - Simple question about their nature .... ....

    Just a simple question regarding the nature of a compact set X in a metric space S: Does X necessarily have to be infinite? That is, are compact sets necessarily infinite? Peter***EDIT*** Although I am most unsure about this it appears to me that a finite set can be compact since the set A...
  37. Math Amateur

    MHB Continuity and Compact Sets - Bolzano's Theorem

    I am reading Tom Apostol's book: Mathematical Analysis (Second Edition). I am currently studying Chapter 4: Limits and Continuity. I am having trouble in fully understanding the proof of Bolzano's Theorem (Apostol Theorem 4.32). Bolzano's Theorem and its proof reads as follows...
  38. N

    Simple question about sets in statistics

    Let's say you have 100 tickets of type A, and 100 tickets of type B in a box. Let's also say the probability to draw ticket A, for whatever reason, is twice that to draw ticket B. Is this problem, for all intents and purposes, mathematically equivalent to having 200 type A tickets and 100 type...
  39. evinda

    MHB Difference of elements of Sets

    Hello! (Wave) Consider two sets $D=\{ d_1, d_2, \dots, d_n\}$ and $E=\{ e_1, e_2, \dots, e_m \}$ and consider an other variable $K \geq 0$. Show that we can answer in time $O((n+m) \lg (n+m))$ the following question: Is there is a pair of numbers $a,b$ where $a \in D, b \in E$ such that $|a-b|...
  40. Math Amateur

    MHB Perfect Sets in R^k are uncountable - Issue/problem 2

    I am reading Walter Rudin's book, Principles of Mathematical Analysis. Currently I am studying Chapter 2:"Basic Topology". I have a second issue/problem with the proof of Theorem 2.43 concerning the uncountability of perfect sets in R^k. Rudin, Theorem 2.43 reads as follows:In the above proof...
  41. Math Amateur

    MHB Perfect Sets in R^k are uncountable

    I am reading Walter Rudin's book, Principles of Mathematical Analysis. Currently I am studying Chapter 2:"Basic Topology". I am concerned that I do not fully understand the proof of Theorem 2.43 concerning the uncountability of perfect sets in R^k. Rudin, Theorem 2.43 reads as follows: In...
  42. S

    Is the Subtraction of Power Sets Possible?

    I have two quick questions: With P being the power set, P(~A) = P(U) - P(A) and P(A-B) = P(A) - P(B) I'm told if it's true to prove it, and if false to give a counterexample. To be they're both false, since the null set is part of any power set, the subtraction of two power sets would get...
  43. evinda

    MHB The subsets of finite sets are finite sets.

    Hello! (Wave) A set is called finite if it is equinumerous with a natural number $n \in \omega$. I want to show that the subsets of finite sets are finite sets. That's what I have tried so far: Let $A$ be a finite set. Then $A \sim n$, for a natural number $n \in \omega$. That means that...
  44. B

    MHB Linear Dependence in \mathbb{R}^4?

    Question: If \textbf{v}_1,...,\textbf{v}_4 are in \mathbb{R}^4 and \{\textbf{v}_1, \textbf{v}_2, \textbf{v}_3\} is linearly dependent, is \{\textbf{v}_1, \textbf{v}_2, \textbf{v}_3, \textbf{v}_4\} also linearly dependent? My Solution: http://s29.postimg.org/4wvwjlkqd/Linearly_Independent_Sets.png
  45. evinda

    MHB Union of Disjoint Sets with at Most k Sets - Algorithm

    Hello! (Smirk)Consider an implementation of disjoint sets with union, where there can be at most $k$ disjoint sets. The implementation uses a hash table $A[0.. \text{ max }-1]$ at which there are stored keys based on the method ordered double hashing. Let $h1$ and $h2$ be the primary and the...
  46. Albert1

    MHB Proving $A=B$ from Sets $A,B,C$

    Three sets $A,B,C$ given: (1)$A\bigcup C=B\bigcup C$ and (2)$A\bigcap C=B \bigcap C$ Prove: $A=B$
  47. evinda

    MHB Proving the Same Cardinality of Sets A & B

    Hello! (Smile) If we want to show that the sets: $$A=\{ 3X^2| X \in \mathbb{Z}_p \}\ \ \text{ and } \ \ B=\{ 7-5Y^2| Y \in \mathbb{Z}_p\}$$ have the same cardinality, could we take the bijective function $f$ such that $f(x)=\frac{7-5x}{3}$ ? Or am I wrong? (Thinking)
  48. B

    Understanding Sets and Intervals: Proving Complements and Open/Closed Status

    Homework Statement Hello, I'm not sure if it's the right place to post this exercise, but I'm learning it in a calculus course. I need to prove that: a) The complement of an open set is a closed. b) An open interval is a open set, a closed interval is a closed set. Homework Equations I have...
  49. evinda

    MHB We cannot define an order over all the sets

    Hi! (Mmm) This: $$\subseteq_{\mathcal{P}A}=\{ <X,Y> \in (\mathcal{P}A)^2: X \subset Y\}$$ is a partial order of the power sets $\mathcal{P}A$. But, we have to take attention to the following fact: We cannot define an order over all the sets because if $R=\{ <X,Y>: X \subset Y \}$ is a set...
  50. N

    Proving Properties of Open Sets in $\mathbb{R}^d$

    Homework Statement Let ##O## be a proper open subset of ##\mathbb{R}^d## (i.e.## O## is open, nonempty, and is not equal to ##\mathbb{R}^d##). For each ##n\in \mathbb{N}## let ##O_n=\big\{x\in O : d(x,O^c)>1/n\big\}## Prove that: (a) ##O_n## is open and ##O_n\subset O## for all ##n\in...
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