Sets Definition and 1000 Threads

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. O

    MHB Are All Countable Sets Closed?

    Hello everyone! I want to show that all countable sets are closed. I can show that finite sets are closed, and the set of all natural numbers is closed by showing its complement to be a union of open sets. Now, can I start like this: A is a countable set. Every element in A can be "mapped" to...
  2. S

    Is Musical Set Theory Adequately Defined in Contemporary Music Analysis?

    A Formalization Musical "Sets" For those of you who have taken Music Theory IV (or upper division or even graduate courses on 20th Century Music Analysis), musical "set theory" should be a familiar concept. I use quotation marks because, as those who are familiar with mathematical set theory...
  3. S

    Labelling sets in a Venn diagram

    Homework Statement Label each set in the following Venn diagram as described in the attached pdf file. Homework Equations The Attempt at a Solution Unable to find a relationship that can describe the second set B. Obviously set A is a collection of Odd numbers.
  4. B

    Bijection between sets of functions

    For two sets X and Y let X^Y be the set of functions from Y to X. Prove that there is a bijection between (X x Y)^Z and X^Z x Y^Z. Attempt: I could not get any further from that "there must be a function S with S(f)=g and S(f')=g for any g, f' in X^Z x Y^Z, and where f is in (X x Y)^Z."
  5. N

    MATLAB Analyzing Wall Thickness vs Force Data Sets

    Attached are three figures. The problem in question essentially involves extracting data from a .CSV (MS Excel) file and .mdb (MS Access) file which I have done already but not correctly. Data set #1: gives me 'Wall Thickness vs. Time' ( [3400 - 8200] s , [0,1] in.) as shown in figure3.jpg...
  6. T

    Compactness of sets in Banach spaces

    Homework Statement Working in a banach space (X,\|\cdot\|) we have a sequence of compact sets A_k\subset X. Assume that there exist r_k>0 such that \sum_{k\in\mathbb{N}}r_k<\infty and for every k\in\mathbb{N}: $$A_{k+1}\subset\{x+u|x\in A_k,u\in X,\|u\|\leq r_k\}.$$Prove that the closure of...
  7. S

    Determine minmal path sets by connectivity matrix

    The attached file illustrate the method I didnt understand this actually from lecture,and i try to search inside books but also didnt find any thing. The notes are brief so u can't get complete understanding So anyone could help me find relevant resources or such one example only to get the...
  8. R

    Homeomorphism between the open sets of the circle and the open sets of real line

    I'm trying to prove the homeomorphism between the open intervals of the real line and the open sets of the circle with the induced topology of R^2. Notice that the open sets of the circle is the intersection between the open balls in R^2 and the circle itself. Anyone can help me...
  9. S

    Probability problem(minmal paths sets and cuts sets

    The solution attempt in the attacment
  10. J

    Sets of all functions. Countable and Uncountable sets.

    I have some confidence that this is the right idea. But whenever I have the slightest shred of doubt, I turn to the experts! :-pThus, before I write up my proof in my notes, does this look somewhat coherent? The problem states: “Determine whether the set of all functions from {0,1} to Z+ is...
  11. K

    Problem Involving Counting of Elements in Three Sets

    Problem: -The Union of set A, set B and set C has 104 elements. -The Union of Set A and B has 51 elements -The Union of Set A and C has 84 elements -The Union of Set B and C has 97 elements -The Intersection of Set A and the Union of Set B and C has 17 elements. -Set C has twice as many...
  12. W

    Proof by induction: multiplication of two finite sets.

    Homework Statement prove by induction that if A and B are finite sets, A with n elements and B with m elements, then AxB has mn elements Homework Equations AxB is the Cartesian product. AxB={(a,b) such that a is an element of A and b is an element of B} The Attempt at a Solution...
  13. I

    Open sets, countable unions of open rectangles

    Homework Statement So here is a "proof" from my measre theory class that I don't really understand. Be nice with me, this is the first time I am learning to "prove" things. Show that a connected open set Ω (\mathbb{R}^d, I suppose) is a countable union of open, disjoint rectangles if and...
  14. J

    Set Theory Proof. Inductive sets.

    Claim: If A is an inductive set of postive integers, then A is Z+. I tried to prove this two different ways for the fun of it. I would like to get some feedback concerning the correctness of both. Thank you. :-p Proof: By definition, Z+ is the intersection of all inductive subsets of ℝ. Since...
  15. K

    What is the Definition of Closed Sets in Topology?

    Good day! Im currently reading the book of Steven R. Lay's "Analysis with an Introduction to Proof, 3rd ed.". According to his book, if a subset S of ℝ contains all of its boundary then it is closed. But i find this wrong since if we consider S={xεQ;0≤x≤2}, then it can be shown that S...
  16. F

    Sets and functions proofs needed

    Hello there, I am extremely new to mathematical analysis and do not have an idea how to prove the following questions. Could you please give me a hand and show me a way? Let At , t ∈ T, be a family of sets, and let X be a set. Prove the identities...
  17. A

    Proofs for Sets: Expert Help and Tips for Math Homework

    Homework Statement hopefully the writing is readable: http://i.imgur.com/VJ8vN.jpg All three if possible. Homework Equations none The Attempt at a Solution To be completely honest, I missed that whole week of lectures due to personal problems and I've had no chance to talk to an...
  18. D

    Totally ordered and Partially ordered Sets

    Hi Everyone, What are the difference between totally and partially ordered sets? Any examples would help except the fact that one holds reflexivity and another totality. Clarification of this would also be fine. Thank You
  19. D

    MHB Closed sets intersection of countable open sets

    Prove that every closed set in $\mathbb{R}$ is the intersection of a countable collection of open sets. Let $G_n$ be a countable collection of open sets. Then we would have 2 cases either $x\in\bigcap G_n$ which is a point which is closed. Or we could $(a,b)$ in all $G_n$ but how to show that...
  20. M

    Calculators TI-83/84 sets of numbers/set theory

    Here's what I want to do on the calculator. 1) Input sets of numbers, for example 3,6,9... in any notation, for example {3,6,9,infinity} or {x|x/3 >= 1 >= infinity} 2) use set theory (not now, but when I get into pre-cal/college.)
  21. T

    Indexed Sets and Their Intersections

    Homework Statement For a real number r, define A_{r}={r^{}2}, B_{r} as the closed interval [r-1,r+1], C_{r} as the interval (r,∞). For S = {1,2,4}, determine (a) \bigcup_{\alpha\in S} A{_\alpha} and \bigcap_{\alpha\in S} A{_\alpha} (b) \bigcup_{\alpha\in S} B{_\alpha} and...
  22. M

    Proof of Disjoint Sets: A Simple Induction Argument Using Probability Functions

    Hi. I need some help with a proof. The question says: Let P be a probability function. Prove that for any finite collection of sets, the sequence A1,A2,...,An of pairwise disjoint sets, P(Union from i=1 to n of Ai)=Ʃ from i=1 to n of Ai I think there must a mistake in question. My...
  23. D

    Open sets preserved in linear transformation that isn't bijective?

    Hi, I'm not sure how else to phrase this.Let's say I have a linear transformation from R3 to R2. Let's assume in both spaces, I am using the standard topology with the standard euclidean distance metric. Does this mean that open sets in R3 will be mapped to open sets in R2 under this...
  24. K

    Solve Sets & De Morgan Homework - Introduction to Analysis 5th Ed. Gaughan

    Homework Statement I'm using Introduction to Analysis 5th edition by Edward D. Gaughan. The question is: Prove (De Morgan) S\(\bigcap A_{\lambda}) = \cup(S\A) \lambda\epsilon \Lambda Where \Lambda A and S are sets (doesn't specify real or complex but assuming real) Homework...
  25. G

    Food for thought Sets logic and applied science

    I was just reading an article the other day about the debate in public schools about teaching evolution as an absolute truth as it has been taght for the past umteen years. Not saying that I'm a proponent of creationism or even that I'm not, but there are some serious flaws in teaching it as the...
  26. N

    Confusion about notation regarding compliments/universal sets

    Im not sure how to interpret the notation, specifically the one on the left, the one on the right seems like you just include everything in the universal set? what does it mean when the line goes over everything? what does it even mean when the line is over the and/or symbol
  27. L

    Number of Pairs of Subsets in X with Unique Elements: Sets and Subsets

    For a pair (A,B) of subsets of the set X=(1,2,...100), let A*B denote the set of all elements of X which belong to exactly one of A or B. what is number of pairs (A,B) of subsets of X such that A*B=(2,4,6,...100)? I let A =(1,2,3...50) and B=(51,52,...100) so there are 25 elememnts of...
  28. K

    Proof on Family of sets - Looking for Help OR Feedback

    Homework Statement Given that F is a family of sets, that \bigcup F is the union of the sets members of the family F, that A is a set, assume that (1) \hspace{1cm} \forall F (\bigcup F = A \rightarrow A \in F) then prove that (G) \hspace{1cm} \exists x (A= \left\{ x \right\} )...
  29. C

    Question about a collection of sets in the plane.

    Homework Statement Show that the collection \{ \{a\}\times(b,c) \subset \mathbb{R^2} |a,b,c \in \mathbb{R} \} of vertical intervals in the plane is a basis for a topology on \mathbb{R^2} The Attempt at a Solution My question is just really about (a)X(b,c) am I just basically...
  30. B

    Comparing Open Sets in Metric Spaces

    Homework Statement Let M be a metric space with metric d, and let d_{1} be the metric defined below. Show that the two metric spaces (M,d), (M,d_{1}) have the same open sets. Homework Equations d_1:\frac{d(x,y)}{1+d(x,y)} The Attempt at a Solution I tried to show that the neighborhoods...
  31. A

    Subsets and sets symbol explenation (Very simple question please have a look)

    Homework Statement Hi I would like you please to look my attachement ,and explain to me the meaning of the line above M and N what's the meaning of this line?It seems to me that it acts as we should take the complementary collection of numbers . Homework Equations The Attempt at a...
  32. J

    Discrete Mathematics - Void Sets being Subsets of other Void Sets

    Homework Statement Hello. Here is the question: Determine whether or not R is some sort of order relation on the given set X. X = {∅, {∅}, {{∅}} } and R ε ⊆. I can't seem to figure out why the ordered pairs given are what they are. Homework Equations None. The Attempt at...
  33. O

    Regularity and self containing sets

    Hey all, I was reading Terence Tao's text on analysis. After stating the axioms of pairing and regularity, he asks for proof of the statement that no set can be an element of itself, using the above two axioms. He has not defined any concepts like hierarchy or ranks. I can see how, A...
  34. H

    Proof about continuous function related to balls and sets

    Homework Statement Let \displaystyle f:{{\mathbb{R}}^{n}}\to \mathbb{R} a continuous function. Proove that: If \displaystyle f\left( p \right)>0 then there's a ball \displaystyle {{B}_{p}} centered at p such that \displaystyle \forall x\in {{B}_{p}} we have \displaystyle f\left( x...
  35. C

    Unique numbers in two sets of three

    Hi, sorry if this is in the wrong section. Some of the stuff in this section is way over my head anyway. I have 10 sets of 3 numbers ranging from 1 to 10. They are interesting in that each number appears three time, no number appears twice in the same set, and no two numbers appear together in...
  36. L

    Set Theory Basic Proof, showing two sets are equal

    Hello, I am trying to teach myself set theory...main problem is, as an engineer, mathematical proofs were never exactly stressed in my curriculum. (Scary, right?) The problem is stated as follows: "Prove the following, {x\inZ|for an integer y, x=6y}={x\inZ|for integers u and v, x=2u...
  37. C

    Proof about size of a union of sets.

    Lets say I have \aleph_1 numbers of sets that each have \aleph_1 number of elements and I want to show that the union of all of these sets has \aleph_1 number of elements. I start with the line segment [0,1] and shift this line segment up by all the reals from 0 to 1. So now...
  38. N

    Regarding Upper and lower integral sets.

    I am having some doubts in the definitions of the upper and lower integrals in apostol. There is a statement saying "Let S denote the set of all numbers _{a}\int ^{b} s(x) dx obtained as s runs through all step functions below f i.e. S = { _{a}\int ^{b} s(x) dx | s < f} " I did not get...
  39. F

    Linearly independent sets within repeated powers of a linear operator

    Homework Statement Suppose that T:W -> W is a linear transformation such that Tm+1 = 0 but Tm ≠ 0. Suppose that {w1, ... , wp} is basis for Tm(W) and Tm(uk) = wk, for 1 ≤ k ≤ p. Prove that {Ti(uk) : 0 ≤ i ≤ m, 1 ≤ j ≤ p} is a linearly independent set.Homework Equations The Attempt at a Solution...
  40. B

    Lebesgue Measurable but not Borel sets.

    Hi, All: I am trying to find a construction of a measurable subset that is not Borel, and ask for a ref. in this argument ( see the ***) used to show the existence of such sets: i) Every set of outer measure 0 is measurable, since: 0=m* (S)≥m*(S) , forcing equality. ii) Every...
  41. B

    Basic stats question involving borel sets

    Homework Statement http://i.imgur.com/tjpka.png (the actual problem is the third part down) Homework Equations the first two parts are the definition of borel sets,and the second part is a relevant theorem. The Attempt at a Solution so I'm new to Borel sets. And I feel like I'm...
  42. PhizKid

    Understanding Sets Defined by Specification

    Does this mean that whenever the function P(x) is true, then x is an element of X, and when P(x) is false, then x is not an element of X? I'm confused because the wording says that "...a sentence P(x) that is either true or false whenever x is any particular element of X..." which leads me to...
  43. J

    Discrete Mathematics - Operations with sets

    I apologize for the repost, but I had no replies to my previous post. I figured that I didn't put down a good enough attempt of a solution. I will try to explain what I did in more detail. I have read the rules for the forum, but if I'm still doing something wrong, please tell me. I want to...
  44. A

    Multiplying two data sets which don't have the same length or spacing

    As the post title describes, I have two data sets and want to multiply them together (and finally plot them). The problem I have is that the data are different length, so for example using Matlab: >>> data_set1 .* data_set2 But this would not give me the correct answer for a number of...
  45. QuestForInsight

    MHB Real Number Sets: Notation Explained | Additive Expressions

    Hello, everyone. I've trouble attempting to read the following. One of the things we assume for the set of real numbers is. A map $\left(\xi, \eta\right) \to \xi+\eta$ from $\mathbb{R} \times \mathbb{R}$ into $\mathbb{R}.$ Could someone read the above in plain English, please. Does it mean all...
  46. G

    How to show that these sets are nonempty

    How to show that these sets are nonempty (\mid means "divides")? Here N is an arbitrary large integer and q is some fixed integer. {R_{k,q}} = \{ k \in {\mathbb N}:(kN\mid k!) \wedge ((k - 1)N\mid k!) \wedge \cdots \wedge (N\mid k!) \wedge (k > Nq)\} {S_{k,q}} = \{ k \in {\mathbb...
  47. A

    Union and Intersection of empty class of sets

    why intersection of empty class of sets is the whole space while their union is null set? Book writes that an element will fail to be in the intersection if it fails to be in one of the sets of the class but since there is nothing in the empty class so there is nothing in the empty class that...
  48. C

    Modelling Long Sets of Data: Measuring "Harshness

    say i have 500,000 0s or 1s. say i have 50 such sets, each that i have ranked or assigned a value to - "harshness". can i then extrapolate - is that the right word - to find the perfect dataset that instantiates the property of harshness? and can i measure the harshness of other datasets...
  49. D

    Question about power sets and cartesian product

    Let A={1, 2} and B={∅}. First, I find the power set of A and the power set of B: P(A)= { ∅, {1}, {2}, {1, 2} } P(B)= { ∅, {∅} } I believe the power sets are correct. I'm still new to the concept of power sets. Anyway, my main question is regarding cartesian product of power sets. I'm...
  50. A

    Solving Coverage with Sets: Min Info Needed?

    A question about sets?? I have a number of weird shaped flat objects. I am interested in covering as much of the floor as I can. After placing the objects on the floor, the only info I have is: Choosing any two objects on the floor, the overlap between them is at a minimum possible...
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