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. ubergewehr273

    Question about a function of sets

    Let a function ##f:X \to X## be defined. Let A and B be sets such that ##A \subseteq X## and ##B \subseteq X##. Then which of the following are correct ? a) ##f(A \cup B) = f(A) \cup f(B)## b) ##f(A \cap B) = f(A) \cap f(B)## c) ##f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)## d) ##f^{-1}(A \cap...
  2. zeronem

    Is the Power of Two Sets Theorem Valid in Introductory Real Analysis?

    Just wanted to know if the work is sound and logical on my paper posted above. I realized I probably should have included notation for the power of the sets. This is my first attempt at theorem proving in Introductory Real Analysis. I realize now that I’m starting into a subject that...
  3. Math Amateur

    MHB The Category of Pointed Sets .... Awodey Example 1.8, Page 17 ....

    I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.6 Constructions on Categories ... I need some further help in order to fully understand some further aspects of Example 1.8, Page 17 ... ... Example 1.8, Page 17 ... reads as follows:I find the...
  4. cromata

    I Complete sets and complete spaces

    -Definition of complete space: if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in converges in M. (and from this definition we can define Hilbert Space) -Definition of Hilbert space: A Hilbert space is a vector space with an...
  5. T

    I Regarding cardinality and mapping between sets.

    why is not always true that if ##\vert A\vert\leq\vert B\vert## then there exist an injection from ##A## to ##B##?
  6. I

    [Linear Alg] Determining which sets are subspaces of R[x]

    Homework Statement [/B] Which of the following sets are subspaces of ##R[x]?## ##W_1 = {f \in \mathbf R[x] : f(0) = 0}## ##W_2 = {f \in \mathbf R[x] : 2f(0) = f(1)}## ##W_3 = {f \in \mathbf R[x] : f(t) = f(1-t) \forall t \in \mathbf R}## ##W_4 = {f \in \mathbf R[x] : f = \sum_{i=0}^n...
  7. Math Amateur

    MHB Modules Generated by Sets of Submodules .... .... Bland Problem 2, Problem Set 4.1 .... ....

    I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 4.1 Generating and Cogenerating Classes ... ... I need some help in order to make a meaningful start on Problem 2, Problem Set 4.1 ... Problem 2, Problem Set 4.1 reads as follows:( *** NOTE ...
  8. Jarvis323

    A Choices of Axiomatic and Number Systems / Sets and Alternatives

    I know that the number systems we use are typically constructed from axiomatic set theory, and overall our choices along the way seam to have been largely informed by practical consideration (e.g. to resolve ambiguities, or do away with limitations). Today I randomly started to think deeper...
  9. Math Amateur

    MHB Start on Bland Problem 1, Problem Set 4.1: Generating & Cogenerating Modules

    I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 4.2 Noetherian and Artinian Modules ... ... I need some help in order to make a meaningful start on Problem 1, Problem Set 4.1 ... Problem 1, Problem Set 4.1 reads as follows: Can someone...
  10. Math Amateur

    Modules Generated by Sets of Submodules .... ....

    Homework Statement I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 4.2 Noetherian and Artinian Modules ... ... I need some help in order to make a meaningful start on Problem 1, Problem Set 4.1 ... Problem 1, Problem Set 4.1 reads as...
  11. Simonel

    I The difference between a family of sets and an indexed family of sets

    What is the difference between family of sets and indexed family of sets ??
  12. M

    MHB Is Path-Connectivity Equivalent to Connectivity in $\mathbb{R}^2$?

    Hey! :o Can a set in $\mathbb{R}^2$ be path-connected only when it is connected, i.e. when we know that a set is not connected then it cannot be path-connected? (Wondering) We have the sets $\displaystyle{A=\{x\in \mathbb{R}^2 : 1\leq x_1^2+x_2^2\leq 4\}}$ $\displaystyle{B=\{x\in...
  13. P

    I What is the Definition and Understanding of Surjective Functions?

    In my book, the definition of surjection is given as follows: Let A and B be sets and f:A->B. The function f is said to be onto if, for each b ϵB, there is at least one a ϵ A for which f(a)=b. In other words, f is onto if R(f)=B. A function which is onto is also called a surjection or a...
  14. ertagon2

    MHB Maximal Elements in a Bounded Set

    Could someone please check these questions? Please correct them if necessary, with an explanation if you could.
  15. Eclair_de_XII

    Proof for convergent sequences, limits, and closed sets?

    Homework Statement "Let ##E \subset ℝ##. Prove that ##E## is closed if for each ##x_0##, there exists a sequence of ##x_n \in E## that converges to ##x_0##, it is true that ##x_0\in E##. In other words, prove that ##E## is closed if it contains every limit of sequences for each of its...
  16. ertagon2

    MHB Valid Arguments: Sets Explained

    is this right ?6(1). p$\implies$q, $\lnot$q$\therefore\lnot$p, Modus tollens 6(2). p$\implies$q, q$\implies$r, q$\therefore$r, Transitivity 6(3). p$\implies$q, q$\therefore$p. Converse fallacy 7. I have no idea what I'm doing please explain.
  17. A

    Young's double slit experiment with two sets of slits

    1. The problem statement, all variables and given/known da Homework Equations Interference minima is asinΘ=nλ/2 where n=1,3,5,7... [/B]The Attempt at a Solution Putting ab/(2sqrt(b^2/4+d^2))=λ/2 for first order minima And solving for d I get (b/2)*sqrt((2a/λ)^2-1) in which only the 2 factor...
  18. Math Amateur

    MHB Compact Sets and Closed Sets in R^n .... .... D&K Lemma 1.8.2 .... ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Lemma 1.8.2 ... ... Duistermaat and Kolk"s Lemma 1.8.2 and the preceding definition and notes...
  19. Math Amateur

    MHB Compact Sets in R^n .... .... D&K Theorem 1.8.4 .... ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Theorem 1.8.4 ... ... Duistermaat and Kolk"s Theorem 1.8.4 and its proof read as...
  20. HappyFlower

    A sailboat sets out from the U.S. side of Lake Erie

    Homework Statement A sailboat sets out from the U.S. side of Lake Erie for a point on the Canadian side, 90.0 km due north. The sailor, however, ends up 50.0 km due east of the starting point. (a) How far, and (b) in what direction must the sailor now sail to reach the original destination...
  21. J6204

    Determing the differences between two sets of differential eqs

    Homework Statement Given the following figure and the following variables and parameters, I have been able to come up with the set of differential equation below the image. My question is how does the system of equations 1 which I produced myself differ from the set of equations 2. Below I have...
  22. Math Amateur

    MHB Continuous Functions and Open Sets .... D&K Example 1.3.8 ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Example 1.3.8 ... Duistermaat and Kolk"s Example 1.3.8 reads as follows:In the above example we read the...
  23. V

    A Reducing computation for large power sets

    To illustrate my problem say I have the following table: Option, x , y A , 25 , 30 B , 5 , 12 C , 3 , 9 D, 12, 13 I want to create a graph of every possible combination in the set where the x values are added and the y values are added. For example say it was...
  24. RJLiberator

    Calculating Double Integrals over Two-Dimensional Sets

    Homework Statement For every two-dimensional set C contained in R^2 for which the integral exists, let ##Q(C) = \int \int_C (x^2+y^2) dxdy##. If ## C_1 = [{(x,y) : -1 ≤ x = y ≤ 1}], ## find Q(c).Homework EquationsThe Attempt at a Solution This was a tougher one for me (the other 2 on this...
  25. S

    Poisson Distribution -- Rental of a number of television sets

    Homework Statement A dealer has a stock of 6 similar television sets which he rents out to customers on a monthly basis. It is known from past experience of the dealer that the monthly demand for the television sets have a Poisson distribution with mean 3.56 (i) Find the probability that in any...
  26. P

    Can the Method of Least Squares be Used for Non-Linear Data?

    Many years ago I was given a problem involving the calibration of sets of weights. Each of two sets consisted of weights of 1.0 kg, 500.0 g, 300 g, 200 g, and 100 g. The two 1.0 kg weights were calibrated by the international agency in France and I was permitted to assume that their masses...
  27. Saqib Ali

    Calculus How do you choose problem sets in Courant's calculus texts?

    I'm going to use Courant's volume 1 and 2. I don't want to try every problem in the book, so how do I go about choosing problems to know if I understand the material?
  28. R

    Are the Sets {r, -r, 0} and {-r, r, 0} Identical?

    Homework Statement Is the set (r,-r,0) the same as the set (-r,r,0) Sorry I need to know the answer to this right now, somebody in class confused me and I can't talk to my teacher right now. My answer: yes it is the same set. Can anyone just give me a quick yes/no? Homework EquationsThe...
  29. E

    I Problem sets between universities

    Hey all, I'm new here. I was just looking at a problem set from Florida Atlantic University from 2010 (http://www.physics.fau.edu/~cbeetle/Fall10/PHZ5115/Assignments/pset3.pdf) and it looks absolutely ridiculous. I'm taking a course now as a sophomore called "Mathematical Methods in the Physical...
  30. M

    MHB Describe bases for the span of sets

    Hey! :o We have the subset $X_i$ of $\mathbb{R}^2$: $$X_1 := \{(x,y) \in \mathbb{R}^2 : x + y = 0\}; \\ X_2 := \{(x,y) \in \mathbb{R}^2 : x + y = 1\} \\ X3 := \{(x,y) \in\mathbb{R}^2 : x^2 + y^2 = 0\}; \\ X4 := \{(x,y) \in \mathbb{R}^2 : x^2- y^2 = 0\}$$ We want to check which of these sets...
  31. nightingale123

    Topology: Understanding open sets

    Homework Statement We define ##X=\mathbb{N}^2\cup\{(0,0)\}## and ##\tau## ( the family of open sets) like this ##U\in\tau\iff(0,0)\notin U\lor \exists N\ni : n\in\mathbb{N},n>N\implies(\{n\}\times\mathbb{N})\backslash U\text{ is finite}## ##a)## Show that ##\tau## satisfies that axioms for...
  32. Math Amateur

    MHB Antisymmetry & Partial Orderings - H&J Ch.2 Section 5 | Peter's Help

    I am reading "Introduction to Set Theory" (Third Edition, Revised and Expanded) by Karel Hrbacek and Thomas Jech (H&J) ... ... I am currently focused on Chapter 2: Relations, Functions and Orderings; and, in particular on Section 5: Orderings I need some help with H&J's depiction of...
  33. Math Amateur

    MHB Axioms of Set Theory .... and the Union of Two Sets ....

    I am reading "Introduction to Set Theory" (Third Edition, Revised and Expanded) by Karel Hrbacek and Thomas Jech (H&J) ... ... I am currently focused on Chapter 1: Sets and, in particular on Section 3: The Axioms where Hrbacek and Jech set up an axiomatic systems (which they do NOT call ZFC ...
  34. Math Amateur

    MHB Indexed Families of Sets .... Just and Weese .... Exercise 8 ....

    I am reading the book: "Discovering Modern Set Theory. I The Basics" (AMS) by Winfried Just and Martin Weese. I am currently focused on Chapter 1 Pairs, Relations and Functions ... and I am in particular focused on Cartesian Products and indexed families of sets ... I need some help with...
  35. G

    MHB How Do I Calculate Standard Deviation with Changed Values in a Set?

    Can someone guide me through the process of solving the following? Many thanks. "Given the set {11, 12, 13, 14, 15}, by approximately how much does the standard deviation change if the least value is decreased by 2 and the greatest value is increased by 2?" These questions are all over the...
  36. Math Amateur

    MHB Cartesian Products and Families of Sets .... Searcoid, Section 1.2 .... ....

    I am reading Micheal Searcoid's book: "Elements of Abstract Analysis" ... ... I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.2 Relations and Functions ... I need some help in fully understanding some remarks by Searcoid in his sub-section on products...
  37. Monoxdifly

    MHB International Notations for Number Sets

    Okay, so I know that the international notations for the set of natural numbers, whole numbers, rational numbers, real numbers, and complex numbers are N, Z, Q, R, and C, respectively. However, what are the international notations for the set of integers (natural numbers and zero), even numbers...
  38. B

    Intersection of sets in ##\Bbb{R}^2## and Open Maps

    Homework Statement No problem statement Homework EquationsThe Attempt at a Solution Let ##A = \{(x,y) ~|~ x \ge 0 \mbox{ or } y=0 \}## be a subspace, which can be shown closed, of ##\Bbb{R}^2##. If my calculations are right, isn't ##A \cap [(0,\infty) \times \Bbb{R} ] = (\{0\} \times...
  39. A

    Finding vectors needed to cancel out given sets of forces

    Homework Statement For this assignment, I am given sets of forces and asked to solve for the force(s) needed to cancel them out. For this case, I am asked to solve for the missing parts of vectors A, B and C. I was only given vector C's Mass and Direction to start with, but I managed to get all...
  40. A

    Solve for vectors needed to cancel out given sets of forces

    Homework Statement For this assignment, I am given sets of forces and asked to solve for the force(s) needed to cancel them out. For this case, I am asked to solve for the last vector, which is vector D. I've successfully found its x and y component, magnitude, and mass (kg). I've also checked...
  41. Math Amateur

    MHB Disconnected Sets .... Palka, Lemma 3.1 ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with some aspects of the proof of Lemma 3.1 ... Lemma 3.1 and its proof read as follows: In the above text from Palka Section 2.2...
  42. B

    Linearly Independent Sets in Abelian Groups

    Homework Statement [/B] ##X## is linearly independent if and only if every nonzero element of the subgroup ##\langle X \rangle## may be written uniquely in the form ##n_1 x_1 + ... n_k x_k## (##n_i \in \Bbb{Z} \setminus \{0\}##, and ##x_1,...,x_k \in X## are distinct). Homework Equations [/B]...
  43. Math Amateur

    MHB Theorem 1.8: Sets or Domains in the Complex Plane - Palka Ch.2

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with an aspect of Theorem 1.8 ... Theorem 1.8 (preceded by its "proof") reads as follows...
  44. R

    I Numeric impact of constraints on sets (passwords)

    Hi folks, I've got everything I need to write a paper on the effect of constraints on passwords... except the math. I actually could write it, but the math would make it much meatier. Here is what I know about math. If the answer to a math problem is obvious, it is probably wrong. If the...
  45. Drakkith

    Solution Sets of Linear Systems

    Homework Statement Q. Let A be an m by n matrix. Assume that the equation Ax=b is consistent for some b≠0 in Rn. Moreover, assume that Ax=b has a unique solution. Does the equation Ax=0 have a non-trivial solution? Is it possible to find a vector V in Rm so that the system Ax=V has infinitely...
  46. B

    Can Measurable Sets Be Written as Disjoint Union of Countable Collection?

    Homework Statement I am working through a theorem on necessary and sufficient conditions for a set to be measurable and came across the following claim used in the proof: Let ##E## be measurable and ##m^*(E) = \infty##. Then ##E## can be written as a disjoint union of a countable collection of...
  47. Math Amateur

    Open Sets - Unions and Intersections - Sohrab Ex. 2.2.4 (1)

    Homework Statement I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with a part of Exercise 2.2.4 Part (1) ... ... Exercise 2.2.4 Part (1) reads as follows: In the above...
  48. Math Amateur

    MHB Open and Closed Sets - Sohrab Exercise 2.4.4 - Part 3

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with a part of Exercise 2.2.4 Part (3) ... ... Exercise 2.2.4 Part (3) reads as follows: I am unable to make a meaningful...
  49. Math Amateur

    Open and Closed Sets - Sohrab Exercise 2.4.4 - Part 3 ....

    Homework Statement I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with a part of Exercise 2.2.4 Part (3) ... ... Exercise 2.2.4 Part (3) reads as follows: Homework...
  50. Math Amateur

    MHB Closed Sets - Sohrab Exercise 2.2.4 - Part 2

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with a part of Exercise 2.2.4 Part (2) ... ... Exercise 2.2.4 Part (2) reads as...
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