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. M

    Closed separated sets in disjoint open sets

    Hi, I was reading over a solution after working on a problem and got confused about some parts: http://nweb.math.berkeley.edu/sites/default/files/pages/f10solutions.pdf (first problem) First, how do we know that there are disjoint open sets U and V for each of the separated sets? (does...
  2. J

    Calculating match between two data sets

    Calculating "match" between two data sets Hey guys, I'm developing a program for comparing the effects of various terms in a Monte Carlo experiment. Right now I have it so you can visually see the effect of "switching" terms on and off and need a way of quantifying how much two lines "match"...
  3. J

    For sets A,B, if I can that show |A|=|B| and A is a subset of B, then A=B?

    [Note: If this is posted in the wrong forum, I'm very sorry. It is directly related to a textbook question.] This may be a silly question. I know that I can prove two sets to be equal by showing that they are subsets of each other. But, what if I have that two sets have the same...
  4. X

    How to Express Sets with Specific Cardinality Restrictions?

    In the expression of sets: B={X \in A:|X|<3} the expression is saying that B is a set that contains at most 3 sets X that belongs to A, right? How do we say, B is a set that contains elements of X that belongs to A, and all X elements contains at most 3 x elements (the cardinality of X is at...
  5. L

    How to randomize sets in math equations ?

    for example i have this : F={1,2,3,4,5} so F=1,2,3,4,5 but how to randomize the set ? i want to say F=5,3,4,2,1 or 2,3,1,4,5 or ... do i have to say like this? : F=(1)/(2)/(3)/(4)/(5)
  6. C

    Points of concurrency and sets of parallel lines

    My question is mainly concerned with discovering the allowable set of "configurations" of the given problem: We have a two-dimensional board composed of three sets (of infinite size) of parallel lines \P_1, \P_2, \P_3, where the lines in \P_2 form a 60 degree angle with lines in \P_3 and \P_1...
  7. K

    How to correctly average over multiple data sets

    Homework Statement In a particular Magneto-Optic Kerr Effect (MOKE) experiment I have taken data for 20 hysteresis loops in which Kerr Voltage is measured as a function of Applied Field. I wish to obtain an average curve. The problem is this; Even though the settings for each loop were...
  8. Useful nucleus

    What is the definition of the empty intersection?

    I read that an empty collection of sets, denote it by λ, is a little problematic when one considers \bigcup_{A\inλ}A and \cap_{A\inλ}A. I can see that the union should be ∅. However, for the intersection it was argued that if one considers a set X to be the universe of the discussion then the...
  9. T

    Power sets and cardinalities (proof)

    Homework Statement Let A be a set. Show that there is no surjective function phi: A --> P(A), where P(A) is the power set of A. What does this say about the cardinalities of A and P(A)? Homework Equations Assume that phi is a surjection of A onto P(A) and consider the set U= {a in A : a...
  10. A

    [Topology] Find the open sets in the subspace topology

    Homework Statement Suppose that (X,\tau) is the co-finite topological space on X. I : Suppose A is a finite subset of X, show that (A,\tau) is discrete topological space on A. II : Suppose A is an infinite subset of X, show that (A,\tau) inherits co-finite topology from (X,\tau). The...
  11. K

    Well Ordered Sets: Disjoint Union w/ R, S, A x B

    Homework Statement Show that for two well ordered sets, (A, R) and (B, S), the disjoint union of A and B will be well ordered by the relation R \cup S \cup A \times B . The Attempt at a Solution ... I honesly don't know how to start at this one..
  12. K

    Can Every Ordinal Number Be Proven to Be a Transitive Set in Set Theory?

    I continue with a questions regarding proofs in set theory.. :) Halmos just writes that every ordinal number is a transitive set but doesn't prove it. Is there any simple proof of this?
  13. Chris L T521

    MHB Lebesgue Integrable Functions on Measurable Sets

    Hello everyone! Welcome to the inaugural POTW for Graduate Students. My purpose for setting this up is to get some of our more advanced members to participate in our POTWs (I didn't want them to feel like they were left out or anything like that (Smile)). As with the POTWs for the...
  14. K

    What is the Set Union formula for A ∪ B = A' ∪ B?

    (a u b)'=a' u b'
  15. 9

    Open Sets in Topological Spaces: Understanding U=intcl U

    Regular open sets,,,, If U is an open set in a topological space (X,τ),is it true that U=〖int〗_X 〖cl〗_X U?Justify.
  16. R

    Epislon & Delta for Open / Not Open Sets

    Hello All, I am finding the hardest time in understanding how to work δ & ε Open Set Problems? Can someone please explain this approach to me? Thanks in Advance
  17. R

    MHB Infinite elements in the universal sets

    is it possible to make a venn diagram wherein the elements are infinite elements? ex. V = { is the set of all odd numbers) W = { 5, 15, 25, 45, 55,...} thanks a lot
  18. R

    MHB How Do Sets W, Y, and Z Intersect?

    W = { x| 0< x < 3} Y = { x| x > 2 } Z = { x | 0 <= x < = 4} then the problems: 1. (WUY) intersects Z = 2. (W intersects Y) intersects Z = do my propose answers below correct sir/mam? 1. 0 < x < = 4 2. 2< x < 3 hope you can help me on this im using the line number ... but all i see in the...
  19. C

    Continuous - how can I combine these open sets

    continuous -- how can I combine these open sets Homework Statement let ##X,Y## be compact spaces if ##f \in C(X \times Y)## and ## \epsilon > 0## then ## \exists g_1,\dots , g_n \in C(X) ## and ## h_1, \dots , h_n \in C(Y) ## such that ##|f(x,y)- \Sigma _{k=1}^n g_k(x)h_k(y)| < \epsilon...
  20. Math Amateur

    Topologising RP2 using open sets in R3

    I am reading Martin Crossley's book - Essential Topology - basically to get an understanding of Topology and then to build a knowledge of Algebraic Topology! (That is the aim, anyway!) On page 27, Example 3.33 (see attachment) Crossley is explaining the toplogising of \mathbb{R} P^2 where...
  21. J

    Liminf and limsup of sequence of sets

    Homework Statement I am attempting to learn some measure theory and am starting with liminf and limsup of sequences of sets. I found an example that is as follows: A_n={0/n, 1/n, ... , n^2/n} and I am trying to find the limsup and liminf. Homework Equations liminf \subset limsup...
  22. E

    Help with a bijection proof involving sets

    Homework Statement I was given a pdf document containing questions that require me to prove set rules. However, the third question (the one that starts at the bottom of the first page and runs into the second page) is giving me problems. I might be able to prove it if he wants a proof by...
  23. E

    Proving Open and Closed Sets for Sequence Spaces

    I have 3 questions concerning trying to prove open and closed sets for specific sequence spaces, they are all kind of similar and somewhat related. I thought i would put them all in one thread instead of having 3 threads. 1) Given y=(y_{n}) \in H^{∞}, N \inN and ε>0, show that the set...
  24. S

    Associativity and commutativity of sets

    Hi all. In chapter 9 of Halmos's book titled Naive set theory, he talks about families of sets. He then talks about the associativity of sets as follows "The algebraic laws satisfied by the operation of union for pairs can be generalized to arbitrary unions. Suppose, for instance, that {Ij}...
  25. M

    Sets in Paint Doc: True or False? | Intersection of Infinite Sets

    Question is in paint doc. Determine if the statement is true or false. My solution: I have two solutions Sol 1: FalseIf A1 contains A2 and A2 contains A3 then the number of elements of A3 contained in A1 is less than the number of elements in A2 contained in A1. In other words the...
  26. C

    Proving Equality of Image and Eigenspace for Eigenvalue 1

    Homework Statement It's given or I've already shown in previous parts of the question: A \in M_{nxn}(F)\\ A^{2}=I_{n}\\ F = \mathbb{Q}, \mathbb{R} or \mathbb{C}\\ ker(L_{I_{n}+A})=E_{-1}(A) Eigenvalues of A must be \pm1 Show im(L_{I_{n}+A})=E_{1}(A) where E is the eigenspace for the eigenvalue...
  27. I

    Is the product of dense sets a dense set in a metric space?

    Just a quick question. If Q is a dense set of a metric space X, and P is a dense set of a metric space Y, then is Q x P a dense set of X x Y? I am fairly sure this is the case. If this is true, then I want to use this statement to show that the open sets of the product of finite number of...
  28. N

    Combining two sets of regular sequence

    This question is in regards to higher dimensional algebraic geometry. The actual problem is very complicated so here is my question which is substantially simplified. Suppose {f_1,... f_k} is a set of quadratic polynomials and {g_1,...,g_l} is a set of linear polynomials in a polynomial ring...
  29. C

    Question about Dense sets in R.

    Homework Statement Decide wheter the following sets are dense in ℝ, nowwhere dense in ℝ , or somewhere in between. a) A= \mathbb{Q} \bigcap [0,5] b) B= \{ \frac{1}{n} : n \in \mathbb{N} d) the cantor set. The Attempt at a Solution a) so we have the rationals intersected with...
  30. S

    Convergence of sequence of measurable sets

    Given a totally finite measure μ defined on a \sigma-field X, define the (pseudo)metric d(A,B)=μ(A-B)+μ(B-A), (the symmetric difference metric), it can be shown this is a valid pseudo-metric and therefore the metric space (X',d) is well defined if equivalent classes of sets [A_\alpha] where...
  31. R

    Elements in sets that are common

    let A={z|z^6=√3 + i} B=(z|Im(z)>0} and C={z|Re(z)>0} find A∩B∩C the part previous to this qn asks me to find the roots of z^6 and I've already down that. but i have no idea how to proceed with this, so do i draw my unit ciorcle with the hexagon and then follow to see what regions satisfies with...
  32. M

    Find two open sets A and B, such that A is subset of B, A is not equal

    Find two open sets A and B, such that A is subset of B, A is not equal to B, and m(A)=m(B) Can I use these two sets? A=(0,2) B=(0,1) U (1,2) thanks
  33. B

    Can We Prove that an Open Ball is an Open Set Using Rectangles?

    Could someone please show that an open ball is open where the definition of "open" is: A set is open if for each x in U there is an open rectangle A such that x in A is contained in U. Where an open rectangle is (a_1,b_1)×…×(a_n,b_n). I also realize that one can use rectangles or balls, but I...
  34. B

    Understanding Indexed Sets: Exploring the Concept and Notation

    I barely started out learning on my own about proofs from this book called A transition to advanced math 2nd edition by chartrand. I am having trouble understanding what an indexed set is and the notation. Is there any online resources I can use to help me understand this concept?
  35. C

    Are All Countable Sets Compact? Proof or Counterexample Required.

    Homework Statement Decide whether the following propositions are true or false. If the claim is valid supply a short proof, and if the claim is false provide a counterexample. a) An arbitrary intersection of compact sets is compact. b)A countable set is always compact. The Attempt at a...
  36. K

    Measure Zero Sets: Proving \sigma(E) Has Measure Zero

    Homework Statement Let \sigma (E)=\{(x,y):x-y\in E\} for any E\subseteq\mathbb{R}. If E has measure zero, then \sigma (E) has measure zero. The Attempt at a Solution I'm trying to show that if \sigma (E) is not of measure zero, then there exists a point in E such that \sigma (\{e\}) that has...
  37. D

    Question about open sets in (-infinite,5]

    The stupid question of the day. If S is the real interval (-infinite, 5], and I can find a metric d so that (S,d) is a metric space, then, is, for example, (4, 5] an open set in (S,d) ? I say this because, the way I'm reading the definition of an open ball, the open ball B(5,1) is the...
  38. B

    Abstract math, sets and logic proof

    Homework Statement If A is a set that contains a finite number of elements, we say A is a finite set. If A is a finite set, we write |A| to denote the number of elements in the set A. We also write |B| < ∞ to indicate that B is a finite set. Denote the sets X and Y by X = {T : T is a proper...
  39. Femme_physics

    Successive Approximation ADC: What Sets It Apart?

    Wiki says: Isn't this exactly what every A/D converter does? For a graph of Vin to digital output it basically approximates the nearest digital value to the continuous signal -> So I don't see the difference between them.
  40. M

    Why are physics problem sets so depressing?

    When you spend hours and days tediously plugging away at the mathematics of a problem, you lose sight of the actual physics of the problem (in addition to losing sight of what you found interesting about physics in the first place). The problem statements are always innocuous, but as soon as...
  41. A

    Level Sets and Degenerate Critical Points

    How would one show that if there is a number c for which g'(c)=0, then every point on the level set {(x,y)|H(x,y)=c} is a degenerate critical point of f? I know that the question may seem vague, but this is the question as it was given to me by my professor. It is something to think about...
  42. estro

    Proving S+T is Open Set: Step-by-Step

    "Adding" 2 open sets Homework Statement I'm trying to prove that If both S and T are open sets then S+T is open set as well.Homework Equations S+T=\{s+t \| s \in S, t \in T\}The Attempt at a Solution S+T is open if every point x_0 \in S+T is inner point. Let x_0 be a point in S+T, so there...
  43. K

    Showing infinite sets are countable using a proper subset

    Homework Statement Show if a set is infinite, then it can be put in a 1-1 correspondence with one of its proper subsets. Homework Equations This was included with the problem, but I am sure most already know this. A is a proper subset of B if A is a subset of B and A≠B The...
  44. H

    Sum of sets with positive measure contains interval

    The original problem is as follows: IF E,F are measurable subset of R and m(E),m(F)>0 then the set E+F contains interval. After several hours of thought, I finally arrived at conclusion that If I can show that m((E+c) \bigcap F) is nonzero for some c in R, then done. But such a...
  45. H

    Sum of two closed sets are measurable

    I tried very long time to show that For closed subset A,B of R^d, A+B is measurable. A little bit of hint says that it's better to show that A+B is F-simga set... It seems also difficult for me as well... Could you give some ideas for problems?
  46. B

    Prove Finite Orthogonal Set is Linearly Independent

    Folks, I am looking at my notes. Wondering where the highlighted comes from. Prove that a finite orthogonal set is lineaarly independent let u=(x_1,x_2,x_n) bee an orthogonal set set of vectors in an ips. To show u is linearly independent suppose Ʃ ##\alpha_i x_i=0## for i=1 to n...
  47. J

    Proof of a theorem about spanning sets

    Homework Statement Let W be a subspace of R^n and let A be a subset of R^n. Then A spans W if and only if <A> = W (<A> is the set of all vectors in R^n that are dependent on A). Prove it. Homework Equations The Attempt at a Solution Ok the book goes likes this; First, it proves...
  48. J

    Can the Empty Set Span the Zero Subspace? Insights on Spanning Sets

    I have some questions about spanning sets 1. Why does empty set spans the zero subspace? 2. Why is this true: Since any vector u in A is dependent on A, A⊆<A>? (<A> is the set of all vecotrs in R^n that are dependent on A)
  49. 6

    Finite and Countable union of countable sets

    Homework Statement Show the following sets are countable; i) A finite union of countable sets. ii) A countable union of countable sets. Homework Equations A set X, is countable if there exists a bijection f: X → Z The Attempt at a Solution Part i) Well I suppose you could start by considering...
  50. J

    Does a Non-Empty Dense Set Have Isolation Points?

    Hello. I was wondering whether a non-empty dense set has any isolation points. From my understanding, when a set is dense you can always find a third point between two points that is arbitrarily close to them so any ball you "create" around a point will contain another point hence a non-empty...
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