Sets Definition and 1000 Threads

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...

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...

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...

-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...

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

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...

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 ...

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...

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...

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...

What is the difference between family of sets and indexed family of sets ??

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...

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...

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

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...

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.

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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?

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...

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...

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...

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...

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...

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 ...

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...

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...

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...

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...

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...

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...

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...

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...

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]...

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...

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...

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...

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...

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...

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...

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...

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...