Set Definition and 1000 Threads

Homework Statement Hi there, the issue I'm having right now is finding the right way to calculate uncertainties when trigonometric terms come in play. The question goes like this, we're given 2 sets of 5 angles in degrees, the incident angle (10.0, 20.0, 30.0, 40.0, 50.0) and the other is the...

I understand the definition of continuity on a manifold based on open sets. I was questionning myself about what is the corresponding physical meaning of an open set of a manifold (M, Power-set-of-M, Atlas). Is it a simple (maybe simplest) assumption in order to define mathematically continuity...

Homework Statement So we have been doing complex numbers for about 2 weeks and there is this one equation I just can't solve. It's about showing the set of solutions in graphical form (on "coordinate" system with the imaginary and the real axis). So here is the equation: Homework Equations...

Homework Statement A person has come to a world full of pets. He notices right away that dogs are the most prevalent of all pets. But that's not all. He notices these 8 ratios: Dog:Cat = 5:1 Dog:Rodent = 1:5 Dog:Lizard = 1:3 Dog:Snake = 2:1 Dog:Turtle = 1:3 Dog:Alligator = 3:1 Dog:Crocodile =...

Hello. A whole decade passed since I graduated mathematics and shifted to other profession, so my knowledge is very rusty. There is an important problem for a scientific work that I need help for. Let's say factor t is being calculated from factors x, y and z, all some parameters from living...

Hey! :o We have the function $\displaystyle{y=f(x_1, x_2)=x_1\cdot x_2^2}$ and the set $M_1=\{x\in [0, \infty)^2 \mid f(x_1, x_2)>1\}$. I want to check if the set is convex. Let $x=(x_1, x_2) , y=(y_1, y_2)\in M_1$, then $x_1\cdot x_2^2>1$ and $y_1\cdot y_2^2>1$. We want to show that...

In bio we are learning about genetics; DNA replication, transcription and translation. The process from DNA to proteins is extremely, extremely complex (not understanding it, just the mechanics of it). It just seems like a weird thing to come up naturally. Now, I am not a believer in inteligant...

I'd really like some help in answering the next question...anything that might help will save my life: F is defined this way: F:A→B where A,B⊂P(N) and P(N) is the power set of the naturals. Let S,R∈A such that S is a proper subset of R if and only if F(S) is a proper subset of F(R) My question...

Why does sound always move at the local speed of sound in a medium (in other words, why is there a set speed of sound for certain medium)? I understand that sound is a compression wave, but shouldn't a louder sound (i.e. one with a higher amplitude) move faster? What about a higher pitch noise...

Homework Statement A determinant with all elements of order unity may be surprisingly small. The Gilbert determinant Hij=(I+j-1)^-1, i,j=2... n is notorious for its small values. Homework EquationsThe Attempt at a Solution I just need help setting up the matrix and I can solve it myself. Thanks

Homework Statement The problem is to prove that ##A \cup (B - A) = \varnothing## Homework EquationsThe Attempt at a Solution The solution in the textbook is that ##A \cup (B-A) = \{x~ |~ x \in A \land (x \in B \land x \not\in A) \} = \{x~ |~ x \in A \land x \not\in A \land x \in B \} = \{x~ |...

How do I find the set of all eigenvalues of $F^2$, if the problem gives me that $F: X \mapsto X$ is the derivative mapping defined in the space $X$ of all infinitely differentiable real functions? I found this solution from an online source which is not known for being reliable. Therefore I am...

Hey! :o A set $R$ with two operations $+$ und $\cdot$ is a ring, if the following properties are satisfied: $(R, +)$ is a commutative/abelian group Associativity : For all $a,b, c \in R$ it holds that $(a \cdot b)\cdot c = a \cdot (b \cdot c)$. Distributive property : For all $a,b, c...

Homework Statement Liquid nitrogen is in a dewar connected to a vacuum pump. Initial pressure in a dewar is 1atm and saturated with gaseous nitrogen. If the vacuum pump started, it removes gas in it and the pressure in a dewar will be reduced under the saturation pressure of the liquid...

I'm attempting to reclaim lost knowledge... hopefully this works. I would like to take a data set I have x,y values and project the trend beyond the (10) values I currently have. For example, I have a graph with (10) values for x and y, but would like to graph the trend created by values 1-10 to...

Homework Statement Determine if the following set of equations has unique solution of the form ##g(z)=(x,y)## in the n-hood of the origin. $$\begin{cases} xyz+\sin(xyz)=0 \\ x+y+z=0 \end{cases}$$ Homework Equations I assume I am supposed to use the implicit function theorem...

Homework Statement Let Zn denote the set of integers {0,1,2,...n-1}. Let * be a binary operation on Zn such that a*b is the remainder of ab divided by n. Show that (Zn,*) is a semigroup for any n . Homework EquationsThe Attempt at a Solution [/B]For showing the algebraic system to be a...

I've started Book of Proof, the first chapter of which is an intro to sets. Q.1 Is there any particular way to approach these kinds of problems, other than using intuition / trial & error? I tend to have some difficulty in working out the best way to express the general term of a sequence, for...

Continiung from the title, it does not necessarilly follow that A will be a subset of C. I knew this for a long time, but I am unable to understand why. Elias zakons notes gave an example' here is the example: This may be illustrated by the following examples. Let a “nation” be defined as a...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ... I need some help with Theorem 1.7.4 ... and in particular with...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ... I need some help with Theorem 1.7.4 ... and in particular with the...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ...At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ... The...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis" ... ... At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ... The...

I'm an astrophysics major in college and I absolutely plan to continue on to graduate school if it gives me the best opportunities. I've grown up watching popular scientists (or science "communicators" if you will) and reading their books. Bill Nye, Carl Sagan, Neil deGrasse Tyson, Richard...

Hello I am trying to prove that $\mathbb{Q}$ is not a finite set. I proceed with path of proof by contradiction. Suppose that $\mathbb{Q}$ is a finite set. Then $\exists\; n \in \mathbb{N}$ such that $I_n \sim \mathbb{Q}$, where \[ I_n = \{i \in \mathbb{Z^{+}} |\; i \leq n\} \] This means that...

Homework Statement T/F: If a finite set of vectors spans a vector space, then some subset of the vectors is a basis. Homework EquationsThe Attempt at a Solution It seems that the answer is true, due to the "Spanning Set Theorem," which says that we are allowed to remove vectors in a spanning...

Screenshot below -Thanks

Homework Statement I am trying to show that ##X = \{z : |z| \le 1 \} \cup \{z : |z-2|<1\} \subset \mathbb{C}## is a connected set. Homework Equations Definition of connectedness that I am working with: A metric space ##(X,d)## is connected if the only subsets of ##X## that are both open and...

Hello all. I have a question about linear dependency. Suppose we have a set ##S## of functions defined on ##\mathbb{R}##. ##S = \{e^x, x^2\}##. It seems very intuitive that this set is linear independent. But, we did something in class I'm unsure about. Proof: Let ##\alpha, \beta \in...

Homework Statement Can anyone please help me solve these questions? (1) Prove that (A-B) - (B-C) = A-B (2)Simplify (A-( A N B)) N (B-(ANB)) (3) Simplify ( ( A N ( B U C)) N ( A-B)) N ( B U C') (4)Use element property and algebraic argument to derive the property (A-B) U (B-C) = (A U B) - (B N...

The empty set has a cardinality of 0, yet it is found in all sets and as we know any sets within a set are counted as a single element. But it never counted as an element in the cardinality of a set? For example: The set {1,2,3} contains 8 distinct subsets including the empty set, as taken by...

Homework Statement ##C \subseteq A \cap B \implies A \cap B \cap C = C## Homework Equations How do I get rid of the "belongs to" term on the right hand side? I know I need to prove either the left hand or the right hand side of the "or" term is correct, I'm just not sure how to get there. The...

Homework Statement A uniform solid cylinder of wood rolls without slipping down an incline of angle theta. the coefficient of static friction is μs and kinetic friction is μk. Find the acceleration of the center of mass of the cylinder. Homework Equations mgsin(theta)-μk = ma The Attempt at a...

Given an indexed collection of sets A_x the disjoint union of these sets can be thought of as the ordinary union of the sets \{ x \} \times A_x for all x. That is, it is the set of all pairs (x, a) where a \in A_x. (Correct me at this point if my understanding of disjoint union is wrong.)...

I'm having trouble in expressing the shaded region in set notation (Thinking) Many Thanks :)

Homework Statement Prove that ##A \cup (A \cap B) = A## Homework Equations In the previous exercise, we proved: Let A, B be sets. Then, the following statements are equivalent: 1) ##A \subseteq B## 2) ##A \cup B = B## 3) ##A \cap B = A## The Attempt at a Solution The proof of ##A \cup (A...

Homework Statement Show that any vector in a vector space V can be written as a linear combination of a basis set for that same space V. Homework Equations http://linear.ups.edu/html/section-VS.html We are suppose to use the 10 rules in the above link, plus the fact that if you have a lineraly...

I need help on this problem: If $X$ and $Y$ are convex sets, show that $X-Y = Z = \{x-y \mid x \in X, y \in Y\}$ is also convex. Here are the steps I have gone so far: Let $p \in Z$ such that $p = x_1 - y_1$, and let $q \in Z$ such that $q = x_2 - y_2$. Assume that $r$ lays in the segment...

Homework Statement Show that the complement of a non-measurable set is also non-measurable. The Attempt at a Solution Let A be a set that is non measurable. Let B be the complement of A. Let's assume for contradiction that B has a measure. Now from the axioms of measure theory they have...

Homework Statement This problem concerns the surface determined by the graph of the equation ##x^2 + xy -xz = 2## a) Find a function ##F(x,y,z)## of three variables so that this surface may be considered to be a level set of F. b) Find a function ##f(x,y)## of two variables so that this...

I know what a proper subset is, but I never understood why every set has the empty set as its subset? I mean, is the reasoning something primitive like this: if I have x objects, the number of unordered sets of elements I can make are 2^x, including the case where I throw out x objects and...

Hi PF! I have some data (y-value data and x-value data) which I can plot to make a curve (see attachment, the top curve). However, when trying to compute the slope of this data I have an issue (see attachment, the bottom "curve"). The formula I am using for slope is ##(y_{i+1}-y_{i-1})/(2...

(Sweating) Any Ideas on how to begin? An incomplete Venn diagram containing the information about 100 types of books displayed at bookstall is shown in the figure. (i) There were 70 translation book types and also there were 40 scientific translation book types that were not in English...

Homework Statement Let A:={x∈ℝ2 : 1<x2+y2<2}. Is A open, closed or neither? Prove.Homework Equations triangle inequality d(x,y)≤d(x,z)+d(z,y) The Attempt at a Solution First I draw a picture with Wolfram Alpha. My intuition is that the set is open. Let (a,b)∈A arbitrarily and...

Homework Statement [/B] P(A | \overline{B}) = ? Homework Equations Multiplicative rule: \begin{equation} P(A | B) = \frac{P(A \cap B)}{P(B)} \end{equation} Additive rule: \begin{equation} P(A \cup B) = P(A) + P(B) - P(A \cap B) \end{equation} Difference: \begin{equation} A \backslash B = A...

Homework Statement In this problem we will be generating and analyzing lists of normally distributed random numbers. The distribution we are sampling has true mean 0 and standard deviation 1. If we sample this distribution N=5 times, what do we expect the mean to be? How about the standard...

Homework Statement Hi everybody! I'd like to check with you guys if I tackled that problem correctly. I might have a few theoretical questions along the way :) Prove that the difference ##A \setminus B## of a closed set ##A \subset \mathbb{R}^2## and an open set ##B \subset \mathbb{R}^2## is...

Homework Statement Let ##(X,d)## be some metric space, and let ##A## be some subset of the metric space. The interior of the set ##A##, denoted as ##int A##, is defined to be ##\bigcup_{\alpha \in I} G_\alpha##, where ##G_\alpha \subseteq A## is open in ##X## for all ##\alpha \in I##. The...

Hi everyone, Thank you in advance for taking the time to read my question and for your help. I really appreciate it. The question is set out as follows: Given the set 𝑆 = {𝑎 sin (𝑥+𝑏) | 𝑎, 𝑏 E R}. The addition of elements 𝒔1, 𝒔2 ∈ 𝑆 is defined as follows: 𝒔1 +𝒔2 =𝑎1 sin (𝑥+𝑏1)+𝑎2 sin...

characters in a specified amount. I've been working on this for 10+ hours and can't seem to get it right. It passes all the unit tests I've written, but when I use it in the context of the program I'm writing it fails. The idea is like ACTAGACC { A : 2, C : 1} ------> 4, because smallest...