Set Definition and 1000 Threads

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...

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...

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)...

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...

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...

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...

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...

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...

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 $\phi: \text{ a set does not belong to itself, so } x \notin x$...

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...

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...

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...

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...

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...

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...

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.

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...

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...

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.

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...

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...

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)...

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...

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

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 =...

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...

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...

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!

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...

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...

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...

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...

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...

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...

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...

"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."...

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...

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...

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...

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...

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...

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...

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...

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]...

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...

Is infinity part of any set?

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...

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...

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...

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...