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. Fernando Revilla

    MHB Marcus 's question at Yahoo Answers (Bijectivity on finite and infinite sets)

    Here is the question: Here is a link to the question: Abstract math question: bijectivity on finite and infinite sets? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  2. S

    Do we have two sets of co-ordinate systems when space-time is bent

    my knowledge of time-space is limited, so my question might be poorly/wrongly constructed/verbalized: Do we have two sets of co-ordinate systems when space-time is bent (by say, mass)? in one system the circle becomes, say, an ellipsoid while in other it remains a circle? in one...
  3. D

    MHB Countable Sets: Exploring the Union of Countable Sets

    What is a countable set exactly? HELP? Can someone help guide me through this problem? I'm a bit lost on how to show this... Countable union of countable sets: Let I be a countable set. Let Ai , i ∈ I be a family of sets such that each Ai is countable. We will show that U i ∈ I Ai is countable...
  4. S

    Example: intersection of compact sets which is NOT compact

    Homework Statement Let S = {(a,b) : 0 < a < b < 1 } Union {R} be a base for a topology. Find subsets M_1 and M_2 which are compact in this topology but whose intersection is not compact. Homework Equations The Attempt at a Solution I'm not even sure what it means for an element of S to be...
  5. S

    Subspace topology and Closed Sets

    Homework Statement Hi, This is my first post. I had a question regarding open/closed sets and subspace topology. Let A be a subset of a topological space X and give A the subspace topology. Prove that if a set C is closed then C= A intersect K for some closed subset K of X. Homework...
  6. I

    Riemann integral is zero for certain sets

    Homework Statement The question is: Let ##\pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}##. Prove that if ##E\subset\pi## is a closed Jordan domain, and ##f:E\rightarrow\mathbb{R}## is Riemann integrable, then ##\int_{E}f(x)dV=0##. Homework Equations n/a...
  7. I

    MHB Riemann integral is zero for certain sets

    The question is:Let $\pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}$. Prove that if $E\subset\pi$ is a closed Jordan domain, and $f:E\rightarrow\mathbb{R}$ is Riemann integrable, then $\int_{E}f(x)dV=0$.(How to relate the condition it's Riemann integrable to the value is $0$...
  8. B

    Open sets and cartesian products

    Let f be a continuous function from R to R and let A be a subset of R^2. Define A={(x,y): y<f(x)}. Can you express A as a cartesian product of two open sets? I tried RxU alpha_x where alpha_x = {y:y<f(x)}. But that didn't work, i need to change something about R.
  9. B

    Is S a closed subset of ℝ^n if it is compact?

    Theorem: Let S be a compact subset of ℝ^n. Then S is closed. Before looking at the book I wanted to come up with my own solution so here is what I've thought so far: Fix a point x in S. Let Un V_n (union of V_n's...) be an open covering of S, where V_n=B(x;n). We know that there is a...
  10. Petrus

    MHB Max and min value, multi variable (open sets)

    Calculate max and min value of the function f(x,y)=x^2+y^2-2x-4y+8 in the range defined by the x^2+y^2≤9 Progress: f_x(x,y)=2x-2 f_y(x.y)=2y-4 So I get x=1 and y=2 We got one end point that I don't know what to do with x^2+y^2≤9 If I got this right it should be a elips that x can max be 3,-3 and...
  11. B

    Troubleshooting Code for Sets: \mathbb{N}

    I searched for this but couldn't find a sol. when entering the code for sets i.e. \mathbb{N} I get this error message: ! Undefined control sequence. <recently read> \mathbb l.32 $\mathbb{N}$ The control sequence at the end of the top line of your error message was never \def'ed. If...
  12. micromass

    Foundations Theory of Sets by E. Kamke | Amazon

    Author: E. Kamke Title: Theory of Sets Amazon Link: https://www.amazon.com/dp/0486601412/?tag=pfamazon01-20
  13. L

    Uniform continuity proof on bounded sets

    Homework Statement Prove that if f is uniformly continuous on a bounded set S, then f is a bounded function on S.Homework Equations Uniform continuity: For all e>0, there exist d>0 s.t for all x,y in S |x-y| implies |f(x)-f(y)| The Attempt at a Solution Every time my book has covered a...
  14. F

    Are measurable sets open or closed?

    I'm seeing the term "measurable sets" used in the definition of some concepts. But when comparing with other concepts that rely on "closed sets", I can't seem to easily find whether measureable sets are open or closed. Does anyone have any insight into that? Thanks.
  15. S

    Couple of questions about sets

    Homework Statement I am confused with sets- just wanted some clarification. Say, I have a set A={b, {1,a},{3}, {{1,3}}, 3} What are the elements of set A? What are the subsets of set A? Are the subsets also the elements of the set A? The Attempt at a Solution I think the elements of the...
  16. M

    Determining if sets are subspaces of vector spaces

    Homework Statement Are the following sets subspaces of R3? The set of all vectors of the form (a,b,c), where 1. a + b + c = 0 2. ab = 0 3. ab = ac Homework Equations Each is its own condition. 1, 2 and 3 do not all apply simultaneously - they're each a separate question. The...
  17. D

    Why do charts map to open sets?

    My textbook says that "a chart or coordinate system consists of a subset U of a set M, along with a one-to-one map \phi :U\rightarrow\mathbf{R}^n, such that the image \phi(U) is open in \mathbf{R}^n." What's the motivation for demanding that the image of U under \phi be open?
  18. D

    Bounded sets, Limits superior and convergence

    (Hey guys and gals!) Homework Statement Given a bounded set x_n and for any y_n the following condition holds: \limsup_{n \rightarrow ∞}(x_n+y_n) = \limsup(x_n)+\limsup(y_n) Show that x_n converges. Homework Equations Definition of limsup(x_n) = L: \forall \epsilon > 0 \mid...
  19. B

    Comparing weighted means in two sets of data

    Homework Statement For simplicity, I'm leaving out extraneous details (like actual numbers). Also, apologies for my formatting; I don't know how to use Latex, but I tried to make this as readable as possible. I have a set of N measurements for τ which each have their own standard deviations...
  20. Fantini

    MHB Continuity in terms of closed sets

    Hello. I wish to prove this: $$\text{A function } f: X \to Y \text{ is continuous if and only if the inverse image of any closed set is closed.}$$ Proof: $(\implies)$ Let $V \subset Y$ be a closed se. By definition, $Y-V$ is an open set, and by the continuity of $f$ it follows that...
  21. F

    Question about infimums and closed sets

    Homework Statement So this question arose out of a question about showing that a set χ is dense in γ a B* space with norm ||.||, but I think I can safely jump to where my question arises. I think I was able to solve the problem in another way, but one approach I tried came to this crux and I...
  22. P

    Among the following sets, identify all pairs of equal sets?

    1. Among the following sets, identify all pairs of equal sets? What is the cardinality of each one of the sets? a) ∅ b) {∅} c) {{∅}} d) {∅,{∅}} e) {∅} \bigcap {{∅}} f) {{∅},∅} I would truly appreciate if you explain a bit. Thank you in advance. _____________________________________________ my...
  23. A

    Is My Understanding of Vitali Sets Accurate?

    I'm not sure if I understood Vitali Sets correctly, so I just want to write what I understood (because I don't know if it's right): We have an equivalence relation where x \sim y \iff x-y \in Q. So if we look at the interval [0,1], each irrational number will have its own equivalence...
  24. M

    Finding Limits of Functions with Multiple Sets of Variables

    I'm familiarized with finding limits of most kinds of functions. I was struck by a problem: What if the variables of the function belong to different sets of numbers? My point being, given the function: f(n,q)=\frac{n}{q} With n belonging to the set of natural numbers and q belonging to the...
  25. K

    How Does Subset Proof in Abstract Algebra Work?

    Homework Statement Question 1. Let U be a universal set, A and B two subsets of U. (1) Show that B ⊆ A ∪ (B ∩ A^c). (2) A = B if and only if there exists a subset X of U such that A ∪ X = B ∪ X and X\A^c = X\B^c. The Attempt at a Solution My attempt at a solution is as follows...
  26. jbunniii

    Countable intersection of F-sigma sets

    My question concerns F_\sigma subsets of \mathbb{R}. An F_\sigma set is one which can be expressed as a countable union of closed sets. I have several books that state that a countable intersection of F_\sigma sets need not be an F_\sigma set (indeed, such sets have their own designation...
  27. I

    Are open sets in R^n always homeomorphic to R^n?

    I know that open intervals in R are homeomorphic to R. But does this extend to any dimension of Euclidean space? (Like an open 4-ball is it homeomorphic to R^4?) My book doesn't talk about anything general like that and only gives examples from R^2.
  28. M

    Understanding generating sets for free groups.

    I was thinking about the following proposition that I think should be true, but I can't pove: Suppose that F is a group freely generated by a set U and that F is also generated by a set V with |U| = |V|. Then F is also freely generated by V. This is something that I intuitively think must...
  29. B

    Solving Complementary Sets: n(A U B)

    Homework Statement If n(A - B) = 5, n(A' - B) = 4, n(A') = 10, n(B'-A') = 12. What is n(AUB) = ? Homework Equations The Attempt at a Solution I drew a big rectangle and inside 2 intersected diagrams A and B. I drew 5 dots in the (( of diagram A. Now that A' is complementary that...
  30. B

    What is the intersection of A and the union of B and C?

    Homework Statement A = {a,b,c} ; B U C = {c,d,e,f} ; (A∩B) U (C∩A) = ? Homework Equations The Attempt at a Solution A U B U C = {a,b,c,d,e,f} A ∩ {B U C } = {c} My answer: {c}
  31. C

    Proving Dedekind Infiniteness of Countable Sets | Solution Attempt

    Homework Statement Call a set X Dedekind infinite if there is a 1-to-1 mapping of X onto its proper subset. Prove that every countable set is Dedekind infinite. The Attempt at a Solution I want to say that every countable set can be well ordered. I guess I could just pick some...
  32. P

    MHB Families of holomorphic functions and uniform convergence on compact sets

    Consider the sequence $\{f_n\}$ of complex valued functions, where $f_n=tan(nz)$, $n=1,2,3\ldots$ and $z$ is in the upper half plane $Im(z)>0$. I want to show two facts about this sequence: 1) it's uniformly locally bounded: for every $z_0=x_0+iy_0$ in the upper half plane, ther exist...
  33. R

    Prove |A-B|=|A|: An Uncountable Set Solution

    hello I am struggled with a qustion let B be a countable subset of uncountable set A. Prove |A-B|=|A| i know how to prove that A-B is uncountable but how do i show 1:1 with A? thanks ahead guys
  34. M

    Proof of existence of nonmeasurable sets

    Hi, I'm reading through a proof of the existence of a nonmeasurable set. I've copied down the proof below more or less verbatim: In particular, I am trying to understand the significance of why ##\alpha## has to be an irrational number. Would the proof not hold if we used any other...
  35. G

    How to Prove it book help with ch 1.4 operation on sets problem (symmetric diff)

    Homework Statement Use any method you wish to verify the following identity: (A \cap B) Δ C = ( A Δ C) Δ (A \ B) Homework Equations A Δ B = (A \ B) \cup (B \ A) = (A \cup B) \ (A \cap B) The Attempt at a Solution http://img17.imageshack.us/img17/48/14question14b.jpg I...
  36. H

    Volume of a convex combination of convex sets ,sort of

    Volume of a convex combination of convex sets,,,,sort of Hi all, I hope someone can tell me whether this is true or not: Let A_{i},i=\{1,...,m\} be m \times n matrices, and let H_{i}=\{x\in \mathbb{R}^{n}:A_{i}x\geq 0\},i=\{1,...,m\}. Also let a probability measure \mu be given. Define...
  37. D

    How to calculate possible combinations of sets

    Posts: 10 I am a woodworker, and am designing a two part magnetic/spring lock for my blanket chest. The first part has 3 master buttons (primary buttons A, B, C), and the second part has 11 secondary buttons (1, 2, 3, ...11). What you do first is choose 1 of the 3 master buttons that opens...
  38. R

    Does Equal Cardinality in Nested Infinite Sets Imply Equality Throughout?

    Homework Statement Prove that if A,B, and C are nonempty sets such that A \subseteq B \subseteq C and |A|=|C|, then |A|=|B| The Attempt at a Solution Assume B \subset C and A \subset B (else A=B or B=C), and there must be a bijection f:A\rightarrowC...
  39. H

    Infinite and finite countable sets

    Ok I understand the concept of infinite countability and that say the set of all rational #s is infinitely countable, but if I needed to represent the set how do I do that? S={xε rat. # : x= k , k ε a rational #}? that doesn't seem right. Also say I wanted to show a set of finite countable...
  40. N

    Is the Distributive Property Applicable in this Set Theory Problem?

    Hello all, while practicing set theory, I cam across this problem: If A and B are sets, prove that A x (B-C) = (AxB) - (BxC). This looks suspiciously like the distributive property but it's not. Is this simply a typo? Shouldn't the problem look like this: A x (B-C) = (AxB) - (AxC) Thanks...
  41. Y

    Infinite Union of Non-disjoint Sets

    Homework Statement To give some context, I'm trying to show that \mu(\bigcup^{\infty}_{k=1}A_{k})\leq \sum^{\infty}_{k=1}\mu(A_{k}) where μ is the Lebesgue measure and the A's are a countable set of Borel sets. Since the A's may not be disjoint, I'm trying to rewrite the left side of the...
  42. J

    Fintie point sets in a Hausdorff space are closed.

    This may seem like a silly question, but I'll ask it anyways. :) In the Munkres text, he proves this by showing that one-point sets are closed, which I completely understand why it follows that finite point sets are closed. He does so by showing that the arbitrary one-point set {x0} equals...
  43. V

    Complete sets and eigenvalues question

    Let's say I'm looking at the infinite square well. Typically, given some arbitrary initial (normalized) wavefunction, we can decompose it into a linear combination of components of the complete set (on the interval [-a,a] or whatever) of sin's and cos's. Then, if you measure something like the...
  44. STEMucator

    Prove if S and T are sets with outer content zero, SUT has outer content zero.

    Homework Statement Suppose that S and T are sets with outer content 0, prove that SUT also has outer content zero. Homework Equations C(S) denotes the outer content. C(S) = C(T) = 0 Also : C(S) = inf \left\{{ \sum_{k=0}^{n} A_k}\right\} where Ak is the area of one of the...
  45. Logic Cloud

    Boolean algebras, sets and logic

    I know that propositional logic and Boolean algebra's are related in the sense that disjunction, conjunction and negation behave the same as join, meet and negation. Similarly, we also have union, intersection and complement when talking about sets. It's obvious that all these notions are...
  46. T

    Countable union of countable sets, proof without AC?

    Pretty much every proof of this I've seen uses the axiom of countable choice at some part or another, and I never got why, since it's pretty cumbersome. Here's the sketch of a proof I wrote for the "fact" that a countable union of countable sets is countable: Let \ P:=\{\pi\in\mathbb{N}|\ \pi \...
  47. X

    Are Empty Sets Equal? Exploring the Concept of Equality in Mathematics

    I want to ask that, are you empty sets equal ?
  48. M

    Finding average acceleration with two sets of velocity vecotors

    Homework Statement A jet plane is flying at a constant altitude. At time t1=0 it has components of velocity vx=95m/s, vy=115m/s. At time t2=33s the components are vx=172m/s, vy=35m/s. Find average acceleration. Homework Equations avg acceleration=vfinal-vinitial/change in time The...
  49. M

    Spanning Sets in Vector Spaces

    Homework Statement True or False: If S is a spanning set for a vector space V, then every vector v in V must be uniquely expressible as a linear combination of the vectors in S. Homework Equations The Attempt at a Solution For some reason, the answer to this question is false...
  50. Also sprach Zarathustra

    MHB Transitive Sets: Prove, Show With $n$ Elements

    Hello, I need a help with the following: 1. Let $A$ be a transitive set, prove that $A\cup \{A \}$ is also transitive. 2. Show that for every natural $n$ there is a transitive set with $n$ elements.
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