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. S

    Metric spaces and the distance between sets

    Homework Statement Okay, so we've moved on from talking about R^n to talking about general metric spaces and the differences between the two. We're given that X (a metric space) satisfies the Bolzano-Weierstrass Property and that A and B are disjoint, compact subsets of X. Dist(A,B) is defined...
  2. R

    Linear Algebra- Spanning Sets definition

    Linear Algebra- Basis Homework Statement Is {e1,e2} a basis for R3 ? Homework Equations The Attempt at a Solution I know that {e1,e2,e3} is a basis for R3 same here, Is this one a basis for R3 {(1,1,2)T,(2,2,5)T} I know that {(1,1,2)T,(2,2,5)T,(3,4,1)T} is a basis...
  3. K

    Q3) Proving the Cardinality of Infinite Sets: A Rigorous Approach

    Homework Statement Q1) Assuming that |R|=|[0,1]| is true, how can we rigorously prove that |R2|=|[0,1] x [0,1]|? How to define the bijection? [Q1 is solved, please see Q2] Q2) Prove that |[0,1] x [0,1]| ≤ |[0,1]| Proof: Represent points in [0,1] x [0,1] as infinite decimals...
  4. qspeechc

    Compact Sets, Unit Balls, Norms, Inner Products: Delightful Reads

    Hi everyone. I wasn't really sure where to put this thread so I stuck it here, which seems the closest fit. Anyway I've been thinking about this for too long: what characterises the unit ball of a norm? Let's be specific: consider a finite-dimensional vector space, which may as well be...
  5. I

    Counting infinite sequence of sets

    Let K1, K2, K3, . . . be an infnite sequence of sets, where each set Kn is countable. Prove that the union of all of these sets K = Union from n=1 to infinity, Kn is countable. I tried to start, but I don't even understand the question Need some idea on how to start
  6. H

    Linearly Independent Sets and Bases

    Homework Statement So I'm trying to find a basis for the space that is spanned by the given vectors. {(1,0,0,1) (-2,1,-1,1) (6,-1,2,-1) (5,-3,3,-4) (0,3,-1,1)} These are written as column vectors. Homework Equations None really (that I know of) The Attempt at a Solution So I...
  7. S

    Binary in Real Analysis & Sets?

    Hi, I have a few questions because I'm watching a lecture on real analysis & I'm a little bit unsure of a few things. I have them in point form for your convenience in answering. http://www.youtube.com/watch?v=lMHR6d0leKA&NR=1 1. (from 2.30 in the video - no need to watch) A & B are sets &...
  8. Somefantastik

    Complete, Equivalent, Closed sets

    If a set A and a set B are equivalent, and it is known that A is complete, can it then be said that B is also complete? What if it is known that A is closed, can it then be said that B is also closed?
  9. S

    Exploring Infinite Pairwise Coprime Sets: Examples and Possibilities

    Hi all, I am looking for examples of infinite pairwise coprime sets. A set P of integers is pairwise coprime iff, for every p and q in P with p ≠ q, we have gcd(p, q) = 1. Here gcd denotes the greatest common divisor.(Wikipedia) Also from wikipedia the following are some examples of...
  10. B

    Exploring Unbounded and Bounded Sets

    Is a bounded set synonymous to a set that goes to infinity? I feel like unless a set is (-infinity, n) or [n, infinity) it is not going to be unbounded. The other thing that I was wondering is can a set be neither open nor closed AND unbounded? Doesn't the definition of open/closed imply...
  11. D

    Sets: Is A ⊆ B? A={x ∈ ℤ | x ≡ 7 (mod 8)} B={x ∈ ℤ | x ≡ 3 (mod 4)}

    A={x ∈ ℤ | x ≡ 7 (mod 8)} B={x ∈ ℤ | x ≡ 3 (mod 4)} Is A ⊆ B? Yes Since x ∈ A, then xa = 7 + 8a = 8a + 7 = 2(4a + 3) +1. And since the ∈ B are of the form xb = 3 + 4b = 4b + 3 = 2(2b + 1) + 1, both ∈ A,B are odd. A ⊆ B since the ∈ of both sets are of 2p + 1. Q.E.D. Is this correct?
  12. K

    Closure & Closed Sets in metric space

    Definition: Let F be a subset of a metric space X. F is called closed if whenever is a sequence in F which converges to a E X, then a E F. (i.e. F contains all limits of sequences in F) The closure of F is the set of all limits of sequences in F. Claim 1: F is contained in the clousre of F...
  13. T

    Topology : 3 sets on the Real line with the wada property.

    Homework Statement Find three disjoint open sets in the real line that have the same nonempty boundary. Homework Equations Connectedness on open intervals of \mathbb{R}. The Attempt at a Solution If this is it all possible, the closest thing I could come up with to 3 disjoint...
  14. D

    Are problem sets like theoretical physics?

    Hi PF, This semester I'm finally starting to enjoy problem sets. I was wondering, how similar are the skills used to solve problem sets to the skills theoretical physicists use to solve actual problems? Thanks, DoD
  15. K

    Interior, Closure, Complement of sets

    Homework Statement Let (X,d) be a metric space and E is a subset of X. Prove that (c means complement, E bar means the closure of E) Homework Equations N/A The Attempt at a Solution Let (X,d) be a metric space and B(r,x) is the open ball of radius r about x. Definition: Let F be...
  16. D

    Sets help interpreting question

    I have this question but I don't get it at all. Here goes: Let X be {x, y, z} P(X) is the power set. For all Y,Z is an element of P(X), Y R Z where The number of elements in Y intersect Z is 1. So I worked out P(X) to be: {(null), (x), (y), (z), (x,y), (x,z), (y,z)} Then I don't...
  17. M

    What is the purpose of a Borel sigma-algebra in defining probability?

    can anyone intuitively explain me what does a borel field and a borel set mean?Why do we need a Borel field to define all our definitions in probability?
  18. P

    Exploring Numerical Equivalence of Sets: A Brief Definition

    1. Define numerical equivalence of sets 2. I'm not sure how in depth the definition needs to be, how is my current def? 3. X is numerically equivalent to Y if \existsF:X\rightarrowY that is bijective or there are two injective functions f:X\rightarrowY and g:Y\rightarrowX
  19. M

    Are Linear Transformations of Linearly Dependent Sets Also Linearly Dependent?

    If A is a 3x3 Matrix and {v1, v2, v3} is a linearly dependent set of vectors in R^3, then {Av1, Av2, Av3} is also a linearly dependent set? Is this true? Can someone please explain why or why not?? What I think: I think it is true because I read that a linear transformation preserves the...
  20. J

    Proving Equality of Sets - A Quick Question

    I just have a very quick (and simple) question: When trying to prove equalities like A \cup (B \cup C) = (A \cup B) \cup C, is it sufficient to note that both sets consist of all elements x such that x \in A, x \in B or x \in C? Or do I need to go through proving that each set is a subset of the...
  21. H

    Can someone give me an example of these sets to help me understand it better?

    Can someone provide me an example of three sets of integers A, B and C such that A\cupB=A\cupC, but B≠C. And also, A\capB=A\capC, but B≠C. Thanks a lot :)
  22. K

    Why is the empty set open in X?

    1) Fact: Let X be a metric space. Then the set X is open in X. Also, the empty set is open in X. Why?? 2) Let E={(x,y): x>0 and 0<y<1/x}. By writing E as a intersection of sets, and using the following theorem, prove that E is open. Theorem: Let X,Y be metric spaces. If f:X->Y is...
  23. M

    Small letters to represent sets

    Hi, I'd like to know if using small letters to represent sets violates rules? From what I've been taught capital letters are pretty much used to denote sets. Is this a strict rule?
  24. L

    Classifying Functions from {1,2,3} to {1,2} and Finding Right Inverses

    Homework Statement List all the functions from {1,2,3} to {1,2} representing each function as an arrow diagram. Which of these functions are (a) injective, (b) surjective, (c) bijective? For each surjective function write down a right inverse. Homework Equations The Attempt at a...
  25. G

    Vector Sets being Linearly Dependent

    I have a quick regarding a definition for linear dependence that my professor gave in class... A set of vectors {v_{1},v_{2},...v_{k}}, are considered linearly dependent provided there are scalars c_{1},c_{2},...c_{k} that are not all zero, such that c_{1}v_{1} + c_{2}v_{2} + ... c_{k}v_{k} =...
  26. A

    Understanding the Difference between \subseteq and \subset in Sets

    If A \subseteq B does that mean A = B which means B = A because if A is a proper \subset of B then A does not equal B right. I am wrong right?
  27. J

    Is the uncountably infinite union of open sets is open?

    This is not a homework problem, just a question from a discussion with my classmates about the Cantor set. The original goal is to prove Cantor set is closed. My earlier attempt is to show the complement of the Cantor set is open. Since when construct the Cantor set each time the sets removed...
  28. A

    Prove that X ⊆ X U Y for all sets X and Y

    hi guys... this is my first thread in this forum.. i hope i'll learn math more easily with this forum... mmm.. i've a some math homework that i must submit it this Thursday... :-p but, i don't know how to answer it... the questions is... 1. Prove that X ⊆ X U Y for all sets...
  29. S

    Understanding Open and Closed Sets in Topology

    I'm self studying topology and so I don't have much direction, however I found this wonderful little pdf called topology without tears. So to get to the meat of the question, given that \tau is a topology on the set X giving (\tau,X), the members of \tau are called open sets. Up to that point...
  30. King Tony

    Proving equalities with operations on sets

    Homework Statement Let A, B, C be any sets. Prove that if C\subseteq A, then (A\cap B)\cup C = A\cap (B\cup C) Homework Equations ? The Attempt at a Solution Don't even know where to begin, If someone could point me in the right direction, that would be the best.
  31. S

    Convergence of Sequences and closed sets

    Homework Statement This is the Theorem as stated in the book: Let S be a subset of a metric space E. Then S is closed if and only if, whenever p1, p2, p3,... is a sequence of points of S that is convergent in E, we have: lim(n->inf)pn is in S. Homework Equations From "introduction to...
  32. L

    Connected vs. Path Connected Sets

    In general, if S is a connected set, can I conclude that S must be path connected? Definition 1: S is connected if it is not disconnected. A set S is disconnected if it can be written as the union of two mutually separated sets, where mutually separated sets are two nonempty sets that do not...
  33. M

    Exploring Sets: a\b, f:a→b & Image of f

    Let a and b be sets. Show that the following constructions are sets stating clearly which axioms you need (a) a\b. (b) A function f:a→ b. (c) The image of f. (d) Given that a and b have ranks α and β respectively, what are the maximum possible ranks of a\b, f:a→ b and the...
  34. R

    Exploring Power Sets and Cartesian Products: Explaining the Answers

    Homework Statement (1)If C is a set with c elements , how many elements are in the power set of C ? explain your answer. (2)If A has a elements and B has b elements , how many elements are in A x B ? explain your answer. Homework Equations The Attempt at a Solution (1)The...
  35. M

    MATLAB Averaging Two Sets of Data with Different Lengths in Matlab

    Hi, I have 2 sets of data, one is 472 data points long, the other 370. They are both a function of the same variable "x", they both have the same value for "x" as the last point and the same value for "x" as the first point. I'm being asked to average the two sets of data, but obviously I...
  36. S

    Correspodance between infinite sets

    Take the set of positive integers plus 0 and the subset of all positive integers ending in, say 5. I see no reason why there can't be a one to one correspondence between the subset and the set. Am I wrong? (I have been challenged on this assertion.) EDIT: The challenge was: The subset is a...
  37. W

    Logical distinction between sets and algebraic structures

    Let's say we have a set S, and a function f : S -> S. Now let S be endowed with a binary operation, forming a group G. Is it correct to write f : G - > G? Up to now I have been operating on the assumption that yes, although G is not technically a set, there is little harm in being sloppy and...
  38. D

    Understanding Julia Sets: Simplified Explanation & Assistance | Expert Help

    I need someone to help explain to me in simple terms how Julia sets work. I understand how the equations governing the Mandelbrot set work, but am finding Julia sets to be a little more complex and difficult to understand.
  39. L

    Which sets are open, closed, and compact?

    Homework Statement Can someone check this for me? Problem: determine which, if any of the sets if open? closed? compact? R=reals; Q=rationals and Z=integers. A= [0,1) U (1,2) is NEITHER B=Z is CLOSED C=(.5,1) U (.25,.5) U (.125, 25) U... is OPEN D={r*sqrt(2) such that r is an element of...
  40. M

    Linear Independence of Sets in a Linear Space

    Let V be a linear space and u, v, w \in V. Show that if {u, v, w} is linearly independent then so is the set {u, u+v, u+v+w}
  41. Z

    Exploring the Countable Infinity of Disjoint Sets and their Cartesian Product

    Hi, I was wondering if two sets are disjoint countably infinite sets why is their Cartesian product also countably infinite? Thanks!
  42. R

    Exploring Categories Without Sets: A Comprehensive Guide to Category Theory

    Does anyone know of a book on Category Theory that purposely attempts to teach category theory without explicitly basing it upon set theory?
  43. Z

    Comparing Measures on Finite & Countably Infinite Sets

    I just started learning some basic measure theory. Could someone explain the difference between \overline{F(A \times A)} and \overline{F(A) \times F(A)} where A is a finite set. Also, how would this be different in A was an countably infinite set? Thanks!
  44. Z

    Measuring Sets: Finite vs. Countably Infinite

    I just started learning some basic measure theory. Could someone explain the difference between \overline{F(A \times A)} and \overline{F(A) \times F(A)} where A is a finite set. Also, how would this be different in A was an countably infinite set? Thanks!
  45. S

    Relation between two circles sets

    Good day all I have two lists of data A & B. The max & min for A is different from B. I want to draw circles, by specific software, representing each one of the two list of data. The max of both A&B should have the large circle and the min of both of them should have the smallest...
  46. J

    Proving Closed Rectangle A is a Closed Set

    Homework Statement Prove that a closed rectangle A \subset \mathbb{R}^n is a closed set. Homework Equations N/A The Attempt at a Solution Let A = [a_1,b_1] \times \dots \times [a_n,b_n] \subset \mathbb{R}^n, then A is closed if and only if its complement, \mathbb{R}^n - A, is...
  47. W

    Distance between two sets A and B in Rn

    Homework Statement A is a sequentially compact set, and B is a point ⃗v in Rn−A. define the distance between A and B as dist( ⃗u0 , ⃗v), where you showed that ⃗u0∈ A exists such that dist( ⃗u0 , ⃗v)≤dist(⃗u, ⃗v) for all ⃗u in A. a) Use this example to state a definition of the distance...
  48. O

    Set Theory proof on well ordered sets

    Homework Statement Without using the Axiom of Choice, show that if A is a well-ordered set and f : A -> B is a surjection to any set B then there exists an injection B -> A. Homework Equations The Attempt at a Solution I was wondering if the existence of the surjection from a well...
  49. 2

    How often do researchers from peer-review journals let others see their data sets?

    I asked this in the mathematics section, but they told me it was probably best to ask it in a different PF section. So I'm interested in getting a hold of peer-review journal articles' "original data sets". I'm curious because I want to play around with data for fun on some statistics software...
  50. A

    Metric Space, open and closed sets

    Homework Statement Let X be set donoted by the discrete metrics d(x; y) =(0 if x = y; 1 if x not equal y: (a) Show that any sub set Y of X is open in X (b) Show that any sub set Y of X is closed in y Homework Equations In a topological space, a set is closed if and only if it...
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