What is Set: Definition and 1000 Discussions

Seth, in Judaism, Christianity, Mandaeism, Sethianism, and Islam, was the third son of Adam and Eve and brother of Cain and Abel, their only other child mentioned by name in the Hebrew Bible. According to Genesis 4:25, Seth was born after Abel's murder by Cain, and Eve believed that God had appointed him as a replacement for Abel.

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  1. Math Amateur

    MHB Units of the set of all Eisenstein Integers

    In Chapter 1: "Integral Domains", of Saban Alaca and Kenneth S. Williams' (A&W) book "Introductory Algebraic Number Theory", the set of all Eisenstein integers, \mathbb{Z} + \mathbb{Z} \omega is defined as follows:https://www.physicsforums.com/attachments/3392Then, Exercise 2 on page 23 of A&W...
  2. T

    Proving Vector Space Axioms for f(x) = ax+b, a,b Real Numbers

    Question: Show that the set of all functions of the form f(x) = ax+b, with a and b real numbers forms a vector space, but that the same set of functions with a > 2 does not. Equations: the axioms for vector spaces Attempt: I think that the axiom about the zero vector is the one I need to use...
  3. H

    ZF Set Theory and Law of the Excluded Middle

    Hello, I know that the law of the excluded middle is implied in ZFC set theory, since it is implied by the axiom of choice. Taking away the axiom of choice, does ZF set theory (with axioms as stated in the Wikipedia article http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory)...
  4. Math Amateur

    MHB Every submodule has at least one set of generators?

    In Paul Bland's book: Rings and Their Modules, we read the following text at the start of Section 2.2 Free Modules:https://www.physicsforums.com/attachments/3388In the above text we read: " ... ... Every submodule has at least one set of generators, namely the set N. ... " Now, I know that...
  5. G

    Is this the correct set up for the electric field?

    Homework Statement A spherical conductor of radius ##a## carries a charge q and also there is a jelly of constant charge ##rho## per unit volume extending from radius ##a## out to radius ##b##. I'm looking to see if I got the correct set up for the electric field of this spherical conductor for...
  6. C

    Orthonormal Set spanning the subspace (polynomials)

    Homework Statement In the linear space of all real polynomials with inner product (x, y) = integral (0 to 1)(x(t)y(t))dt, let xn(t) = tn for n = 0, 1, 2,... Prove that the functions y0(t) = 1, y1(t) = sqrt(3)(2t-1), and y2 = sqrt(5)(6t2-6t+1) form an orthonormal set spanning the same subspace...
  7. SSGD

    Variable Set Distribution - Buckingham Pi Theorum

    Background: I am trying to write a program for Buckingham Pi Groups. I need to find a way to list all the input varialbes as different sets. For example if I have 4 variables [V D p u] and I want to distribute them 3 ways I get 4 sets. Number of Sets = Binomial(Number of Variables...
  8. A

    MHB Determining if a set is a subspace.

    Hey there guys, its AngrySnorlax here again with another problem. I posted here before when I was having an issue and the responses I got were extremely helpful because there was a specific step that I just could not grasp that was explained to me and I am hoping that is the same situation here...
  9. evinda

    MHB The set of all sets does not exist.

    Hey! (Wave) Theorem (Russell's paradox is not a paradox in axiomatic set theory) The set of all sets does not exist. Proof We suppose that the set of all sets exist, let $V$. So, for each set $x$, $x \in V$. We define the type $\phi: \text{ a set does not belong to itself, so } x \notin x$...
  10. evinda

    MHB Show that it is not a set.

    Hi! (Cool) I want to show that $\cap \varnothing$ is not a set. That's what I have tried so far: We suppose that $\cap \varnothing$ is a set. Let $x \in \cap \varnothing$. Then, $\forall b \in \varnothing, x \in b$. However, $\varnothing$ does not contain any element. So, we cannot find a $b...
  11. L

    Complex Number: What's the set?

    What's the set \{ z \in \mathbb{C}| |z|^2 \geq z+ \bar{z} \}? I've set z=a+ib and found a^2 + b^2 \geq 2a \Rightarrow b^2 \geq a(2-a) I'm not sure how to interpret this geometrically ie what it looks like? I suppose it is the set of vectors whose length is bigger than twice their real part. I...
  12. evinda

    MHB Proving A is the Empty Set: Exploring Set Subsets and Implications

    Hello! (Wave) I want to show that $A \subset \varnothing \rightarrow A=\varnothing$. That's what I thought: $$A \subset \varnothing \text{ means that :}$$ $$\forall x (x \in A \rightarrow x \in \varnothing)$$ Since, there is no $x$, such that $x \in \varnothing$, there is no $x$, such that...
  13. D

    Why don't these open set axioms specify that the empty set is open?

    In all the topology textbooks I used in school, the open set axoims specified 4 conditions on a set S: (i) S is open (ii) empty set is open (iii) arbitrary union of open sets is open (iv) finite intersection of open sets is openI noticed on proofwiki, that (ii) is omitted. I was curious if...
  14. evinda

    MHB Proving Uniqueness of a Set: A Logical Approach

    Hey again! (Blush) I want to show that the set $\{ a, b \}$ is unique.That's what I have tried: We suppose that $\{a,b\}, \ \{a,b \}'$ are sets, so that each of them has as elements $a$ and $b$ and only these ,and $\{a,b \} \neq \{a,b \}'$. From the axiom of extensionality, there is, without...
  15. E

    News South Africa set to get a nuclear power plant funded by Russia

    South Africa is set to get a nuclear power plant, following the signing of a cooperation deal with Russia recently. Both sides noted that the nuclear power plant will have a production capacity of up to 9.6 GW based on Russian Technology, by 2030. Read more here...
  16. rayne1

    MHB Expressing a set as the difference between two sets.

    Let a, b, c, and d be real numbers with a < b < c < d. Express the set [a, b]U[c, d] as the difference of two sets. I know that [a,b]U[c,d] is a union and what a difference of two sets is, but I don't quite understand this question.
  17. evinda

    MHB Introduction to Set Theory: Fundamentals, Construction, and Arithmetic

    Hello! (Wave) What is the subject Set Theory about? What knowledge is required? (Thinking) That is the Course Content: Brief report on basic elements (algebra of sets, relations and functions, etc..). Construction of the set of natural numbers. Ordinal numbers and their arithmetic. The...
  18. R

    Understanding Sets & Images: A Beginner's Guide to Set Theory

    Could someone please explain how the image of a set A' ⊆ A is the set: f(A') = {b | b = f(a) for some a ∈ A'}. And how can the complement of A be a subset of A? Forgive my ignorance here, I'm a beginning student of set theory. Edit: Maybe I should rephrase my question: Could you explain what...
  19. R

    MHB Set Theory for Beginners: How is A' ⊆ A and its Complement a Subset of A?

    Could someone please explain how the image of a set A' ⊆ A is the set: f(A') = {b | b = f(a) for some a ∈ A'}. And how can the complement of A be a subset of A? Forgive my ignorance here, I'm a beginning student of set theory.
  20. B

    How many elements are in a set of unique rational numbers from 1 to 9?

    Let ##T = \{ \frac{n}{m}\in \mathbb{Q} \vert n, m \in \{ 1, 2, ..., 9 \} \}## No values can repeat (e.g. ##\frac{2}{2},\frac{3}{3},...##) How many elements does the set have. I could just go ahead and count the elements and eliminate the repeats, but I'm wondering if there is a simpler (and...
  21. A

    Equations of Motions of a Wheel Axle Set

    Hello all, I am currently studying dynamics of a wheel-axle set for my research. My problem is I could not find the same equation for the rate of the change of the momentum in the book, book is a little bit old and I could not find any errata about the book or any other references that...
  22. nuuskur

    Finding Sup A for Set {0.2, 0.22, 0.222, 0.2222,...}

    Homework Statement Find sup A if A = {0.2, 0.22, 0.222, 0.2222, ...} I'll write elements of a set with low case letters and indexes, e.g anThe Attempt at a Solution Begin by definition of supremum: \sup A = a if \forall x \in A, x \leq a and \forall b \in \mathbb R ((\forall x \in A , x \leq b)...
  23. C

    How do you calculate the power set of a set of sets?

    How are you supposed to go about putting together the power set of a set of sets such as X = {{1},{1,2}} What is the power set of X then? And what's the rule for calculating cardinality for the power set of a set that consists of elements which are sets such as the above? Because the set X...
  24. F

    MHB Proving a set is a borel set

    Let $f:R->R$ be a continuous function. Prove that set of points $f$ is differentiable at is a borel set. I need to get to this set by union/intersection of intervals but how? I guess I'm missing a theorem about differentiable points and cotinuity Thanks
  25. K

    Constructive or destructive interference after a set number of waves?

    Homework Statement Two sources, S1 and S2, send our circular waves that are in phase and of the same frequency. They have the same wavelength (0,5m) and the same amplitude. Will there be a maxima og a minima at the given points: a) S1P = 5.00 m og S2P = 6.50 m? b) S1P = 5.00 m og S2P =...
  26. N

    What equipment do I need to set up a sonoluminescence experiment?

    I am working on a high school science fair project and I wanted to study sonoluminescence. I am having trouble with the setup and I am not sure what kind of stuff I need. I've been referring to websites and papers such as: http://dave.ucsc.edu/physics195/thesis_2010/mccluney_thesis.pdf...
  27. R

    Set of all points within a distance of 1 from the box?

    Homework Statement Consider a solid box with dimensions L,W, and H. Let S be the set of all points whose distance is at most 1 from the nearest point inside or on the box. What is the volume of S? Homework Equations Not sure if there are any? The Attempt at a Solution My initial...
  28. F

    Non-parametric single set test of the mean

    Hey gang, I was wondering if there is a non-parametric version of the single set TTest? I know that often people refer to the Wilcoxon signed-rank test, but my understanding is that only tells you about the median, correct? Is there an equivalent that deals strictly with the mean? Cheers!
  29. P

    Relations, power sets and the empty/null set.

    Homework Statement Suppose R is a relation on A, and define a relation S on P (A) as follows: S = {(X, Y ) ∈ P (A) × P (A) | ∀x ∈ X∃y ∈ Y (xRy)}. For each part, give either a proof or a counterexample to justify your answer. (a) If R is reflexive, must S be reflexive? (b) If R is symmetric...
  30. L

    How many subsets are there of a set consisting of n elements?

    Hi, So I understand this problem a little, I just can't understand the ending! So saying that we have n elements, we want all the subsets consisting of r elements where r goes from 0 to n. So we want (n choose 0) + (n choose 1) + ... + (n choose n) which is the summation of n choose r for...
  31. Nathanael

    Confusion about the boundary of a simple set

    Homework Statement Determine the boundary of the following set. As usual, z=(x,y). 0<\left| z-z_0 \right|<2 2. The attempt at a solution The book's solution says "The circle \left| z-z_0 \right|=2 together with the point (0,0)" Why should the answer not be "... together with the...
  32. M

    Proving Measurability of ##A## from ##E=A \cup B## with ##|B|=0##

    Homework Statement Let ##E \subset \mathbb R^n## be a measurable set such that ##E=A \cup B## with ##|B|=0## (##B## is a null set). Show that ##A## is measurable. The Attempt at a Solution I know that given ##\epsilon##, there exists a ##\sigma##-elementary set ##H## such that ##E \subset...
  33. V

    Least amount of structure on a set to define a series on it

    I know I can define a sequence on a set ##X## as a function ##a:T\rightarrow{}X##, where ##T## is a countable totally ordered set. But what about series? Can I define a series as a function ##\omega{}:a\rightarrow{}A##, where ##A\in{}X##? Or is this too general to be a series? Do I need to...
  34. P

    Setting a family of sets equal to the empty/null set?

    Homework Statement Suppose A is a set, and for every family of sets F, if ∪F = A then A ∈ F. Prove that A has exactly one element. (Hint: For both the existence and uniqueness parts of the proof, try proof by contradiction.) Homework Equations The Attempt at a Solution Let A be...
  35. P

    Proving Existence/Uniqueness of a Set.

    Homework Statement Let U be any set. (a) Prove that for every A ∈ P (U) there is a unique B ∈ P (U) such that for every C ∈ P (U), C \ A = C ∩ B. Remark: P(U) = the Power set of U, i.e. A ∈ P (U) then A⊆U Homework Equations The Attempt at a Solution The question's form is as follows...
  36. Nathanael

    Regions; "Each point of the set is the center of a circle "

    "A set in the plane is called a region if it satisfies the following two conditions: 1. Each point of the set is the center of a circle whose entire enterior consists of points of the set. 2. Every two points of the set can be joined by a curve which consists entirely of points of the set."...
  37. R

    MHB How to find set from following condition.

    how can i find the sets from following situation. i have three numbers,{1 2 3} which will always be in this order {123}, i want to find out number of cases can be made. but 2 can come at frist position that is before 1 or at second position or at third position that is after 3. and all are...
  38. C

    Control Theory (EE): How to set up a transmittance given wn only?

    Homework Statement How do we set up a second-order plant transmittance with the only information available are: One pole is at a position where the undamped natural frequency (ωn = 0 rad/sec), and the other pole is at a position where ωn = 2 rad/sec? The question asks to build that...
  39. M

    Set inclusion in topological space

    Homework Statement . Let ##X## be a topological space and let ##A,B \subset X##. Then (1) ##A \cap \overline{B} \subset \overline{A \cap B}## when ##A## is open (2) ##\overline{A} \setminus \overline{B} \subset \overline {A \setminus B}##. The attempt at a solution. In (1), using...
  40. M

    Topology on a set ##X## (find interior, closure and boundary of sets)

    Homework Statement . Let ##X## be a nonempty set and let ##x_0 \in X##. (a) ##\{U \in \mathcal P(X) : x_0 \in U\} \cup \{\emptyset\}## is a topology on ##X##. (b) ##\{U \in \mathcal P(X) : x_0 \not \in U\} \cup \{X\}## is a topology on ##X##. Describe the interior, the closure and the...
  41. Medicol

    Conditional probability: selecting one from a set

    I have a group of dogs (3 brown male, 2 brown female, 4 white male, 4 white female, 5 black male, 4 black female) What is the probability to 1. select a female brown dog ? 2. select a female, given that is a brown dog ? 3. select a brown given that is a female dog ? Thank you. I have...
  42. K

    Prove a set is not a vector space

    Homework Statement Let b be a symetric bilinear form on V and A = \{ v\in V : b\left(v,v\right)=0\}. Prove that A is not a vector space, unless A = 0 or A = V. 2. The attempt at a solution If we suppose that A is a vector space then for every v,w\in A we must have...
  43. T

    Formula that will give the equation for the line of best fit of a data set

    Is there a formula that will give me the equation for the line of best fit of a data set, the line being a 6th degree polynomial? I know how to graph the table and add a line of best fit while showing the equation but as far as I know the equation cannot be used in the given format in excel. If...
  44. mazgan

    Set of 2 nonlinear ODE in mathamtica 9

    i just signed up here so i hope this is the right place. i need to solve a set of 2 non-linear ordinary differencial equations. i tryed using NDSolve but it doesn't really work so I am not sure what's wrong with my code. here is my code (copy paste): c = 0.1; Subscript[sys, B]...
  45. johann1301

    Is the Set N Closed for Addition in English?

    If you take two arbitrary numbers from a set N - let's say N stands for the natural numbers - and add them together, the sum will always be an element of N. In my language, there is a word for this, but i don't know what it is in english? If i translate it from norwegian, it would be something...
  46. johann1301

    Does Infinity Belong to a Set? Understanding the Concept of Infinity

    Is infinity part of any set?
  47. S

    Set theory: proofs regarding power sets

    Let X be an arbitrary set and P(X) the set of all its subsets, prove that if ∀ A,B ∈ P(X) the sets A∩B,A∪B are also ∈ P(X). I really don't know how to get started on this proof but I tried to start with something like this: ∀ m,n ∈ A,B ⇒ m,n ∈ X ⇒ Is this the right way to start on this proof...
  48. S

    Is the sample space not a set under ZFL?

    I am reading Introduction to Set Theory (Jech & Hrbacek) and in one of the exercises we're asked to prove that the complement of a set is not a set. I get that if it were a set that would imply that "a set of all sets" (the union of the set and its complement, by the axiom of pairing) exists and...
  49. T

    MHB I need the two elements with the greatest volume in a data set

    I have very large data sets: coins & amount i.e. {10 & 11, 9 & 7, 8 & 9, 4 & 5, 3 & 1} graphed would show greater volume to the left side while {1 & 0, 2 & 3, 2 & 1, 4 & 6, 9 & 10} would reflect greater volume on the right, and {1 & 4, 2 & 3, 12 & 10, 4 & 4, 3 & 2} would reflect a surge in the...
  50. L

    Weighted average of a set of slopes with different goodness of fit

    Hi there, I have a bit of a confusing question, but I'll try to be as clear as I can in asking it. I have a set of linear fits for four different sets of data. Basically, I have three sets of data, with sample sizes N1 = 5, N2 = 7, N3 = 5 respectively. I have plotted these data with...
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