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. CMATT

    Choose the element in each of the sets you would expect to have the highest IE2

    Homework Statement Choose the element in each of the sets you would expect to have the highest IE2. a. K b. Be c. Mg d. Ca e. Al Homework Equations The correct answer is K The Attempt at a Solution I do not understand why it is K ...I kind of guessed by using my Ionization Energy diagram...
  2. Battlemage!

    B How many possible relations between two sets?

    Say you have set A with n elements and set B with m elements. If I recall, there are a total of 2nm relations between them. But my question is, does this count redundancies? What I mean is, if in the relation A~B = B~A. I don't want to count identical relations twice. Thanks!
  3. F

    MHB Re: Union and Intersection of Sets

    Re: Union and Intersection of Sets Hi, Please I need a help regarding Union of sets can anybody solve this A={1,2,3} and B={{1,2},3} then what is A Union B and A Intersect B Thanks
  4. N

    MHB Proving No Set Contains All Sets Without Russell's Paradox

    Greetings: I am attempting to prove that no set contains all sets without Russell's paradox. What I have thus far is this: Let S be an arbitrary set and suppose S contains S. If X is in S for some X not=S, then S - S cannot be empty. But this is a contradiction; hence if S contains S, then...
  5. zrek

    I Venn diagram for the reals and transfinite numbers as sets

    My statement: The first transfinite ordinal, omega is the first number that cannot be expressed by any natural number, therefore it is not included in the set of natural numbers. The set of natural numbers is a subset of real numbers, every natural number can be taken out of it, but still true...
  6. Josh Terrill

    B Linear regression with two data sets?

    I want to try to predict the USA summer highs using a linear regression. I know I can probably take data from the last 10 summers and plug that in, and use that to predict, but I'd like to use two data sources. 1 data source from the historical highs from past summers in the USA, and the 2nd...
  7. Math Amateur

    MHB Categories of Pointed Sets - Aluffi, Example 3.8

    I am reading Paolo Aluffi's book: Algebra: Chapter 0 ... ... I am currently focussed on Section I.3 Categories ... ... and am trying to understand Example 3.8 which is introduced as a concrete instance of the coslice categories referred to in Example 3.7 ... Examples 3.7 and 3.8 read as...
  8. Math Amateur

    I Categories of Pointed Sets - Aluffi, Example 3.8

    I am reading Paolo Aluffi's book: Algebra: Chapter 0 ... ... I am currently focussed on Section I.3 Categories ... ... and am trying to understand Example 3.8 which is introduced as a concrete instance of the coslice categories referred to in Example 3.7 ... Examples 3.7 and 3.8 read as...
  9. Q

    Combinatorics -- Counting Sets of Binary Strings

    Homework Statement Give combinatorial proofs of the identities below. Use the following structure for each proof. First, define an appropriate set S. Next, show that the left side of the equation counts the number of elements in S. Then show that, from another perspective, the right side of the...
  10. Superposed_Cat

    Schema theorem for non binary sets?

    Hey all, the schema theorem shows that in all probability a genetic algorithm will converge to a solution. much like the second law of thermodynamics for optimization. Although, it is taught with the genes being $$ \in (0,1, *), * \in (0,1) $$ is there a proof for non binary genes? example...
  11. Alpharup

    B Can open sets be described in-terms of closed sets?

    Let A be an open set and A=(a,b). Can A be described, as closed set as "or every x>0, all the elements of closed set [a+x,b-x] are elements of A"?
  12. J

    Proving Set Equality: A Simple and Effective Method

    Homework Statement Attached is the problem Homework EquationsThe Attempt at a Solution So I have to show that each side is a subset of the other side Assume x∈ A ∪ (∩Bi) so x∈A or x∈∩Bi case 1 x∈ ∩ Bi so x∈ (B1∩B2∩B3...∩Bn) which implies x∈B1 and x∈B2 ... and x∈Bn so x∈B1∪A and x∈B2∪A...
  13. RJLiberator

    Abstract Algebra: Bijection, Isomorphism, Symmetric Sets

    Homework Statement Suppose X is a set with n elements. Prove that Bij(X) ≅ S_n. Homework Equations S_n = Symmetric set ≅ = isomorphism Definition: Let G and G2 be groups. G and G2 are called Isomorphic if there exists a bijection ϑ:G->G2 such that for all x,y∈G, ϑ(xy) = ϑ(x)ϑ(y) where the...
  14. TyroneTheDino

    Arbitrary Union of Sets Question

    Homework Statement For each ##n \in \mathbb{N}##, let ##A_{n}=\left\{n\right\}##. What are ##\bigcup_{n\in\mathbb{N}}A_{n}## and ##\bigcap_{n\in\mathbb{N}}A_{n}##. Homework Equations The Attempt at a Solution I know that this involves natural numbers some how, I am just confused on a...
  15. S

    I Finding the Number of Sets in Two Groups: A Simplified Problem

    Came to know about the following problem from a friend which can be simplified to the following: A1, A2, ...Am and B1, B2,...Bn are two groups of sets each group spanning the sample space. Now there are p elements in each of Ai and each element is in exactly p1 of the sets of the A group. Again...
  16. M

    I Partitions of Euclidean space, cubic lattice, convex sets

    If the Euclidean plane is partitioned into convex sets each of area A in such a way that each contains exactly one vertex of a unit square lattice and this vertex is in its interior, is it true that A must be at least 1/2? If not what is the greatest lower bound for A? The analogous greatest...
  17. G

    I Physical implications from Vitali sets or Banach-Tarski?

    Hi. Can we infer something about physics from stuff like Vitali sets or the Banach-Tarski paradox? Maybe if we assume the energy in a given space volume to be well defined and finite, that there must be fundamental particles that can't be split, or that there must be a Planck length and energy...
  18. G

    MHB Proving Span of $\mathbb{R}^2$ Using Sets of Vectors

    I'm given the example that the space $\mathbb{R}^2$ is spanned by each of the following set of vectors: \left\{i, j\right\}, \left\{i, j, i+j\right\}, and \left\{0, i, -i, -j, i+j\right\}. However, it's not obvious to me how. Let $i = (s, t)$ and $j= (u, v)$ then $\left\{i, j\right\}$ means...
  19. M

    Intuitive explanation of lim sup of sequence of sets

    Hi, I can derive a few properties of the limit inferior and limit superior of a sequence of sets but I have trouble in understanding what they actually mean. However, my understand of lim inf and lim sup of a sequence isn't all that bad. Is there a way to understand them intuitively (something...
  20. Math Amateur

    Tangent Spaces of Parametrized Sets - McInerney, Defn 3.3.5

    I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... I am currently focussed on Chapter 3: Advanced Calculus ... and in particular I am studying Section 3.3 Geometric Sets and Subspaces of T_p ( \mathbb{R}^n ) ... I need help with a...
  21. Math Amateur

    Geometric Sets and Tangent Subspaces - McInnerney, Example 3

    I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... I am currently focussed on Chapter 3: Advanced Calculus ... and in particular I am studying Section 3.3 Geometric Sets and Subspaces of T_p ( \mathbb{R}^n ) ... I need help with a...
  22. N

    MHB Can Algebra of Sets Determine Solutions for Set Equations Like $AX=C$?

    How would I solve this using the algebra of sets?
  23. Y

    Proofs involving Negations and Conditionals

    Suppose that A\B is disjoint from C and x∈ A . Prove that if x ∈ C then x ∈ B . So I know that A\B∩C = ∅ which means A\B and C don't share any elements. But I don't necessarily understand how to prove this. I heard I could use a contrapositive to solve it, but how do I set it up. Which is P...
  24. P

    Why am I getting complex radii for level sets of this function?

    First of all sorry for my english skills. 1. Homework Statement Im trying to get the set levels of this function: f(x,y)=(x-y)/(1+x^2+y^2)=z Homework Equations circle-> (x-xo)^2+(y-yo)^2=r^2 The Attempt at a Solution (Leaving this here just to give a graph...
  25. Kilo Vectors

    Properties of sets under operations help

    Hi So I am learning about sets and I wanted to know if these definitions was correct, specifically the properties of sets under operations, and I had a question. please help. The closure property: A set has closure under an operation if the result of combining ANY TWO elements under that...
  26. benorin

    Need some kind of convergence theorem for integrals taken over sequences of sets

    I think this be Analysis, I Need some kind of convergence theorem for integrals taken over sequences of sets, know one? Example, a double integral taken over sets such that x^(2n)+y^(2n)<=1 with some integrand. I'd be interested in when the limit of the integral over the sequence of sets is...
  27. D

    Find the sets of real solutions

    [b[1. Homework Statement [/b] ##|4^{3x}-2^{4x+2}*3^{x+1}+20*12^x*3^x| \ge 8*6^x(8^{x-1}+6^x)## The sets containing the real solutions for some numbers ##a, b, c, d,## such that ##-\infty < a < b < c < d < +\infty## is of the form ##(-\infty, a] \cup [b, c] \cup [d, +\infty)##. Prove it by...
  28. T

    Looking for a good source for problem sets

    Homework Statement i'm feeling that i didn't quite catch the whole concept of inertial forces very well , and I'm looking for an additional source for mechanics problems . so far i have been learning and solving problems for "an introduction to mechanics" by danniel kleppner , which is btw is...
  29. T

    MHB How can we prove $P(A \cap B) = P(A) \cap P(B)$?

    $P(A \cap B) = P(A) \cap P(B)$ How can we prove this to be true?
  30. R

    The union of any collection of closed sets is closed?

    I don't see how this is the case. Let ao and bo be members of [A,B] with ao<bo. Let {ai} be a strictly decreasing sequence, with each ai>A and {bi} be a strictly increasing sequencing with each bi<B. Let the limits of the two sequences be A and B, respectively. Then define Ii = [ai,bi]. It seems...
  31. O

    MHB How does ${X}_{w}\subset {X^*}_{w}$ occur in modular metric space?

    Let $d$ be a metric on $X$. Fix ${x}_{0}\in X$. Let ${d}_{\lambda}\left(x,y\right)=\frac{1}\lambda{}\left| x-y \right|$ and The two sets ${X}_{w}={X}_{w}\left({x}_{0}\right)=\left\{x\in X:{d}_{\lambda}\left(x,{x}_{0}\right)\to0\left( as \lambda\to\infty\right) \right \}$ and...
  32. C

    Solving Sets of Matrices for Proving Equivalence Relation

    Homework Statement If there are two sets of matrices ##S = \begin{Bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} | a, b, c, d \in \mathbb{C} \end{Bmatrix} ## and ##M = \begin{Bmatrix} \begin{bmatrix} a & b \\ -\overline{b} & \overline{a} \end{bmatrix} | a, b \in \mathbb{C} \wedge |a|...
  33. G

    Solving Equations of Sets in P(E)

    Homework Statement Let ##A,B \in {\cal P}(E)##. Solve in ##{\cal P}(E)## the following equations: ##X\cup A = B## ##X\cap A = B## ##X - A = B## Homework EquationsThe Attempt at a Solution We have ##A\cup B = (A\cup X)\cup A = A\cup X = B##. So ##A\subset B## and the solution cannot be less...
  34. B

    I Why can sets contain duplicate elements?

    If S={a,b,c}, what does it mean that S=T when T={a,a,a,a,a,a,b,c}? The mapping that confirms the definition of equality assumes that the duplicate symbols in a set are representative of the same entity or idea. If S={1/2} and T={ .5 , 2/4 , ,25/,5 , 4/8 } are these sets equal? At what point...
  35. W

    Help understanding a set and its distribution

    Homework Statement given set C = {(x,y)|x,y are integers, x^2 + |y| <= 2} Suppose they are uniformly distributed and we pick a point completely at random, thus p(x,y)= 1/11 Homework Equations Listing it all out, R(X) = {-1,-2,0,1,2} = R(y) The Attempt at a Solution My problem is that when I...
  36. P

    Is ε closed under countable intersections?

    Homework Statement Let ε = { (-∞,a] : a∈ℝ } be the collection of all intervals of the form (-∞,a] = {x∈ℝ : x≤a} for some a∈ℝ. Is ε closed under countable unions? Homework Equations Potentially De Morgan's laws? The Attempt at a Solution Hi everyone, Thanks in advance for looking at my...
  37. M

    Proof with sets and elements. Am I going about this right?

    Homework Statement Give an element-wise proof for the following: If A⊆B and B⊆C', then A ∩ C = ∅ Homework Equations A is a subset of B (written A ⊆ B) if every element in the set A is also an element in the set B. Formally, this means that fore every x, if x ∈ A, then x ∈ B. A ∩ B = { x ∈ U ...
  38. L

    Prove that if A and B are sets, then (A - B) U B = A U B

    Homework Statement Prove that if A and B are sets, then (A - B) U B = A U B I think I might be missing a few steps here. Homework EquationsThe Attempt at a Solution (A - B) U B = 1. (A ^ ~B) U B = 2. (A ^ ~B) U (A ^ B) = 3. A U B
  39. Math Amateur

    MHB Affine Algebraic Sets - Properties of the map I - Dummit and Foote, page 661

    I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need someone to help...
  40. J

    Mathematica: Animate multiple sets of XY data

    I have nine sets of data with x,y coords that are the position of a particle. I can ListPlot the particle positions on a single plot, but, I want to animate this. ListPlot[{mydata1, mydata2, mydata3, mydata4, mydata5, mydata6, mydata7, mydata8, mydata9}, PlotRange -> {{-1, 20}, {-1, 20}}] I...
  41. S

    Sets & Hyperplanes Homework: Convexity, Separability & More

    Homework Statement Consider the sets ##A = \left\{(x_1,x_2) \in\mathbb{R}^2: x_1+x_2 \leq 1\right\}## which is a straight line going through ##(0,1)## and ##(1,0)## and ##B = \left\{(x_1,x_2) \in\mathbb{R}^2: (x_1-3)^2+(x_2-3)^2 \leq 1 \right\}## which is a circle of radius ##1## centred at...
  42. yango_17

    Checking if sets are subspaces of ##\mathbb{R}^{3}##

    Homework Statement Is the set ##W## a subspace of ##\mathbb{R}^{3}##? ##W=\left \{ \begin{bmatrix} x\\ y\\ z \end{bmatrix}:x\leq y\leq z \right \}## Homework EquationsThe Attempt at a Solution I believe the set is indeed a subspace of ##\mathbb{R}^{3}##, since it looks like it will satisfy...
  43. shanepitts

    Are Finite Families of Closed Sets Closed?

    Homework Statement Let {Ei: 1≤i≤n} be a finite family of closed sets. Then ∪i=1n Ei is closed. Homework Equations Noting that (Ei)c is open The Attempt at a Solution Honestly, I have no idea where to start. I tried to demonstrate that Eai≥Ei if a is a constant greater than zero. Then...
  44. evinda

    MHB Understanding Open/Closed Sets in $\mathbb{R}^n$

    Hello! (Wave) The following definition is given: A set $U \subset \mathbb{R}^n$ is called open if for each $x \in U$ there is $B_d(x, \epsilon) := \{ y \in \mathbb{R}^n: d(x,y)< \epsilon\}$ -> open ball with center $x$ and radius $\epsilon$. Could you explain me why the following set is open...
  45. Math Amateur

    MHB Morphisms (or polynomial maps or regular maps) of algebraic sets - Dummit and Foote

    I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need someone to help...
  46. Math Amateur

    MHB Is that correct so far ... ?Yes, that is correct. Good job!

    I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need help to get...
  47. Math Amateur

    MHB Elementary Algebraic Geometry: Dummit & Foote Ch.15, Ex.24 Coordinate Ring

    I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need help to get...
  48. Math Amateur

    MHB Elementary Algebraic Geometry: Exercise 23, Sect 15.1 Dummit & Foote

    I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need help to get...
  49. S

    Uniqueness of familiar sets

    The formal way to define many mathematical objects is careful not to assert the uniqueness of the object as part of the definition. For example, formally, we might define what it means for a number to have "an" additive inverse and then we prove additive inverses are unique as a theorem...
  50. Math Amateur

    MHB Affine Algebraic Sets in A^2 - Dummit and Foote, page 660, Example 3

    I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need someone to help me...
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