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. D

    Equivalence between power sets

    Homework Statement Part a: Show that X \subseteq Y and X \subseteq Z if and only if X\subseteq Y \cap Z, for sets X,Y,Z. I have done this. Part b: Use the equivalence from part a to establish the identity P(A) \cap P(B)= P(A \cap B), where P is the power set. Homework Equations...
  2. P

    Cross correlation in two data sets of the same sample population

    I have the following problem: Suppose there is a survey of persons with two properties each (a and b). a can take a relatively small number of values (say the political inclination: Conservative, Liberal, Socialist, or Other) the other one b can take a large number of values (say: the country...
  3. A

    What is the Cardinality of Sets X, Y, and A, B, C?

    Hey guys, this is my first post, (Hi) was just wondering if i could get your help. I'm studying for my repeats and you guys can save me. If X = {1,2,3,4}, Y = {2,4,6} what is the cardinality of the following sets? (i) A = {x|x mod 2 = 0 and 0 <=x<=20} (ii) B = X * X * Y (iii) C = {(x,y)|x ≠ y...
  4. C

    What is the connection between sequence of sets and sequence of functions?

    Hi context: i am trying to understand convergence of sequence of random variables. random variable are just measurable functions but I still can't get my head around the connection between sequence of functions and sequence of sets. To start suppose A_n \subset \Omega i don't even...
  5. T

    Difference between Power Sets and Sample Space

    Hey All, In my probability theory class we have just started learning about how a probability space is defined by a sample space (which contains all possible events), events and a measure. We briefly went over the idea of the Power Set, which seems to be the set of all subsets in your...
  6. K

    How can I optimize a function on a non-linear set with unknown points?

    Hey all, I'm doing some research on computing optimal controls in quantum mechanics, and need a numerical algorithm that I can try to adapt to my problem. I'm hoping that if I describe the problem, someone out there can point me in a good direction. Consider a function f: \mathbb R^n \to...
  7. radou

    How Many Continuous and Open Mappings Exist Between Discrete Topological Spaces?

    Another problem whose answer I'd like to check. Thanks in advance. Let X = {1, 2, 3, 4} and Y = {1, 2, 3}. Let P(X) and P(Y) be the power sets of X and Y, respectively. i) How many continuous mappings are there from the discrete topological spaces (X, P(X)) to (Y, P(Y))? Well, I figured that...
  8. T

    Spanning Sets in R3: Determine Vectors

    Homework Statement Determine whether the given vectors span R3: v1=(3,1,4) v2=(2,-3,5) v3=(5,-2,9) v4=(1,4,-1) Homework Equations I need to show that an arbitrary point in R3 can be written as: (b1,b2,b3)=k1(3,1,4)+k2(2,-3,5)+k3(5,-2,9)+k4(1,4,-1) The Attempt at a Solution...
  9. O

    How is it that naïve sets can be used in logic consistently before ZFC?

    I've been following the first few chapters of Yuri Manin's "A Course in Mathematical Logic for Mathematicians," and as an undergraduate who has only had basic logic and naïve set theory, the way he explained a few of the topics rubbed me the wrong way - specifically, the definitions of the...
  10. J

    Smallest Set Containing All n-point Sets

    I've been thinking about this. Suppose you have an n-point set P in Rm which has the property that for any two points x, y in P, ||x - y|| < 2. If we fix n, what can we say about the smallest set S in Rm that contains P, allowing for both translations and orthogonal transformations of S? If...
  11. A

    Understanding Index Sets & Union of Sets

    Homework Statement 1.Given a set T we say that T serves as an index set for family F={Aa} of sets if for every a in T there exists a set Aa in family F. 2. By the union of the sets Aa, where a is in T, we mean the set {x l x\inAa for at least one a in T}. We shall denote it by...
  12. F

    Extending De Morgan to infinite number of sets

    I'm using my summer vacation to try to improve my understanding of real analysis on my own but it seems it's not as easy when not having a teacher at hand so I've a small question. Homework Statement The problem concerns extending a De Morgan relation to more than two sets and then, if...
  13. Borek

    When does the Sun sets down at 71°11'

    Bear with me, I am not even sure if I know how to properly word the question. Few days ago there was summer solstice - that means Sun was visible all day north of arctic circle. My question is - when will the Sun set down at 71°11'? Now, Sun is not a point, so perhaps better question is -...
  14. M

    Compact sets in Hausdorff space are closed

    First of all I just want to rant why is the Latex preview feature such a complete failure in Firefox? Actually it is really bad and buggy in IE too... So I am reading into Foundations of geometry by Abraham and Marsden and there is a basic topology proof that's giving me some trouble. They...
  15. B

    Understanding Open Sets in Metric Spaces: A Brief Exploration

    I'm reading Analysis on Manifolds by Munkres and in the section Review of Topology Munkres states the following theorem without proof: Let X be a metric space; let Y be a subspace. A subset A of Y is open in Y if and only if it has the form A = U ∩ Y where U is open in X. All he has defined is...
  16. B

    Proving the Open Sets Theorem for Metric Spaces

    I'm reading Analysis on Manifolds by Munkres and in the section Review of Topology Munkres states the following theorem without proof: Let X be a metric space; let Y be a subspace. A subset A of Y is open in Y if and only if it has the form A = U ∩ Y where U is open in X. All he has...
  17. K

    Sets: A\B can represent A union B?

    As a event A\B stands for "A occurs but B does not." Show that the operations of union, intersection and complement can all be expressed using only this operation.A \backslash B = A \cap \bar{B} So far I have resorted to making a truth table with a bunch of A\B combinations that look at A\B...
  18. F

    Proof regarding the algebra of sets.

    I am currently trying to prove the following: An equation in X with righthand member \oslash can be reduced to one of the form (A \cap X) \cup (B \cap ~X) = \oslash. (Where A, B, and X are sets of some universal set U, and ~X is the complement of the set X). The only problem is that I'm...
  19. M

    Is the Union of Open Sets Also Open in Y?

    1. Suppose open sets V_{\alpha} where V_{\alpha} \subset Y \: \forall \alpha , is it true that the union of all the V_{\alpha} will belong in Y? (i.e. \bigcup_{\alpha} V_{\alpha} \subset Y) Thanks! M
  20. B

    Intersection of Connected Sets: True or False?

    I have been asked if the following is true or false the intersection of two connected sets is connected ? I would have thought that if their intersection was empty they wouldn't be connected. If they were disjoint or course it would be empty. any ideas?
  21. S

    Open and closed sets of metric space

    Homework Statement I am using Rosenlicht's Intro to Analysis to self-study. 1.) I learn that the complements of an open ball is a closed ball. And... 2.) Some subsets of metric space are neither open nor closed. Homework Equations Is something amiss here? I do not understand how...
  22. C

    Finding topologies of sets in complex space

    Homework Statement Consider the following subsets of \mathbb{C}, whose descriptions are given in polar coordinates. (Take r \geq 0 in this question.) \begin{align*} X_1 =& \{ (r,\theta) | r = 1 \} \\ X_2 =& \{ (r,\theta) | r < 1 \} \\ X_3 =& \{ (r,\theta) | 0 < \theta < \pi, r > 0 \}...
  23. P

    Proof on boundedness of sets in n space

    Homework Statement Let S be a bounded set in n -space. Fix a d>0. Then it is possible to choose a finite set of points {pi...pm} in S such that every point p in S is within a distance d of at least one of the points p1, p2,...pm. Homework Equations None really. The Attempt at a Solution...
  24. R

    Sets & Notations: What Do S* and S* Mean?

    I have seen different notations on different books but I couldn't find anywhere what S* and S* mean, being S a set. Anyone can help?
  25. Y

    Isomorphism as an Equivalence Relation on Sets: A Proof

    So it says here "Let S be a set of sets. Show that isomorphism is an equivalence relation on S." So in order to approach this proof, can I just use the Reflexive, Symmetrical, and Transitive properties that is basically the definition of equivalence relations? eg. suppose x, y, z are sets...
  26. C

    Question about cadinality of sets

    i was confused about what the cardinality of the set {1,2,2,3,3,1,1,1} is? is it 8 or is it 3?
  27. J

    Compactness of A in R2 with Standard Topology: Tychonoff's Theorem Applied

    1. The question is to show whether A is compact in R2 with the standard topology. A = [0,1]x{0} U {1/n, n\in Z+} x [0,1] 3. If I group the [0,1] together, I get [0,1] x {0,1/n, n \in Z+ }, and [0,1] is compact in R because of Heine Borel and {0}U{1/n} is compact since you can show that every...
  28. G

    When is a collection of sets too large to be a set?

    Is there any easy way to say when a collection of sets is too big to be a set? For example, why is the collection of all groups, vector spaces, etc. not a set anymore? How do I determine that a given collection is still a set?
  29. S

    Calculating all possible relations of 2 sets?

    A={1,3,5} B={4,6,8,10} The set AXB that we have been using had 4096 subsets. Why? Can you find a general procedure for calculating the number of possible relations where there are k ordered pairs available? I don't know how to calculate how many relations there are? The only information I...
  30. D

    Cardinality of sets: prove equality

    A=\mathbb{R}-0 (0,1)\subseteq \mathbb{R}-0 Assume A is countable. Since A is countable, then A\sim\mathbb{N}. Which follows that (0,1)\sim\mathbb{N}. However, (0,1) is uncountable so by contradiction, Card(A)=c Correct?
  31. jaketodd

    Equivalent Trajectories in Relativity: Observer Effects

    In relativity, can two sets of trajectories, carried out at different times, be considered equivalent if they only differ by when they change directions as they traverse their sets of trajectories? They traverse the same trajectories. The only difference is the rate at which they traverse the...
  32. C

    Open sets and closed sets in product topology

    Homework Statement Let (X_a, \tau_a), a \in A be topological spaces, and let \displaystyle X = \prod_{a \in A} X_a. Homework Equations 1. Prove that the projection maps p_a : X \to X_a are open maps. 2. Let S_a \subseteq X_a and let \displaystyle S = \prod_{a \in A} S_a \subseteq...
  33. O

    Abstract Alg- Group theory and isomorphic sets.

    Homework Statement I am suppose to determine if the following list of groups are isomorphic and if they are define an isomorphic function for them. a. [5Z, +],[12Z, +] where nZ = {nz | z\inZ} b. [Z6, +6]], [S6, \circ] c. [Z2, +2]], [S2, \circ] Homework Equations +6 means x +6] y = the...
  34. R

    Linear independence of orthogonal and orthonormal sets?

    (Note: this isn't a homework question, I'm reviewing and I think the textbook is wrong.) I'm working through the Gram-Schmidt process in my textbook, and at the end of every chapter it starts the problem set with a series of true or false questions. One statement is: -Every orthogonal set...
  35. Mentallic

    The size of sets and infinities

    The set of integers is smaller than the set of real numbers. This I understand through logic. I've also heard that the set of irrationals is larger than the set of rationals. How is this so? And so I'm guessing the set of reals is larger than both the set of rationals and irrationals since...
  36. C

    Cardinalities of Sets: Prove |(0, 1)| = |(0, 2)| and |(0, 1)| = |(a, b)|

    How to prove the open intervals (0,1) and (0,2) have the same cardinalities? |(0, 1)| = |(0, 2)| Let a, b be real numbers, where a<b. Prove that |(0, 1)| = |(a, b)| ----------------------------------- |(0,1)| = |R| = c by Theorem ----------------------------------- I know that we...
  37. C

    Singleton sets closed in T_1 and Hausdorff spaces

    If (X,\tau) is either a T_1 space or Hausdorff space then for any x \in X the singleton set \{ x \} is closed. Why is this the case? I can't see the reason from the definitions of the spaces. Definition: Let (X,\tau) be a topological space and let x,y \in X be any two distinct points, if...
  38. G

    Comparison of 4 sets of data: a measure of similarity

    hi, i measured the emission spectrum of an LED with a monochromator connected to a PMT tube. the spectrum was measured at four different gain levels on the PMT. i want to check the PMT's linearity at the different gain level, so i want to compare the four data sets and check their...
  39. R

    Solve Math Sets Homework: (C-\negA)\cup(A\capB)\cup(C\capB)=C

    Homework Statement (C-\negA)\cup(A\capB)\cup(C\capB) = C Homework Equations No idea what to be put here. The Attempt at a Solution (C\capA)\cup(A\capB)\cup(C\capB) = C {[(C\capA)\cupA]\cap[(C\capA)\cupB]}\cup(C\capB)=C A\cap[(C\capA)\cupB]\cup(C\capB)=C got stuck after this...
  40. C

    Funtion continuity and open sets

    Homework Statement Suppose that f : (X,d_X) \to (Y,d_Y). If f is continuous, must it map open sets to open sets? If f does map open sets to open sets must f be continuous? Homework Equations The Attempt at a Solution The answer to the first question is yes. The answer to the...
  41. G

    Infinite Sets: Are They All Equal?

    This is something I was thinking about. Note that I am not a mathematician, so don't get mad if the answer to this is really obvious. Ok, so define x as the set of all real positive integers. Clearly, this is an infinite set. Now define y as the set of all real positive even integers (or...
  42. A

    Question about open sets in [0,1]

    Isn't [0,1) open in [0,1]? I know it's not open in \mathbb R, but I sincerely hope it's open in [0,1].
  43. M

    Understanding Well-Founded Sets

    I am struggling to properly understand the concept of a well-founded set. Is this well founded, d = {{x},{x,y},{x,y,z}} because there exists an element of d i.e. {x} = e such that d n e = 0 ?
  44. R

    Adding a Vector to a Spanning Set: Will it Still Span the Vector Space?

    Homework Statement Let {x1,x2,...,xk} be a spanning set for a vector space V. a) if we add another vector, xk+1 to the set, will we still have a spanning set ? ExplainHomework Equations The Attempt at a Solution I think 'yes', but I am not sure if my explanation is correct. It doesn't matter...
  45. G

    Can a Basis B Exist in V Such That A is Contained in B and B in C?

    Homework Statement Let V be a vector space, let p ≤ m, and let b1, . . . , bm be vectors in V such that A = {b1, . . . , bp} is a linearly independent set, while C = {b1, . . . , bm} is a spanning set for V . Prove that there exists a basis B for V such that A ⊆ B ⊆ C. Homework...
  46. T

    Explore the Wonders of Polar Sets!

    polar set... Excuse me for my poor Latex ability. I type my question in WORD. The follow is the URL of the problem... http://img714.imageshack.us/img714/8185/matht.jpg
  47. Fredrik

    Is every subset of a totally bounded set also totally bounded?

    Not really homework, but a textbook-style question... Homework Statement Is every subset of a totally bounded set (of a metric space) totally bounded? Homework Equations F is said to be totally bounded if, for every \epsilon>0, there's a finite subset F_0\subset F such that...
  48. O

    Proving Sum-Free Sets: A Presentation

    I'm doing a presentation on using probability to prove various results, and one of them is that given any set of natural numbers B, it contains a set A that is sum-free, i.e. no two elements in A sum to another element in A, such that |A| \geq \frac{|B|}{3}. I looked around and found a...
  49. S

    Characteristic Sets A & B: Special Name & Importance

    I was wondering if sets A and B, with the following property: \mbox {Either } A\bigcap B=\emptyset \mbox{ or } A\bigcap B=A. have a special name. The name per se is not that important, however, what i am asking is whether these are well known/studied sets, or if they are of any special...
  50. Y

    Half Skulls and Forelimb Sets

    this is a lab question. though it's a lab question but we didn't learn anything in the lab (we only dissected a rat and that wasn't pertained to lab questions), and the teacher didn't provide us with the resources, I have spent numerous hours browsing google and had no luck (only gathered bits...
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