Set Definition and 1000 Threads

The class is called Math for EE and CE. The professor teaches from his own notes and doesn't give many examples. Any help checking my work would be appreciated and/or if you could point me in the direction of more examples like these. I've looked trough Set Theory and discrete math books but...

Homework Statement Assume that y1 and y2 are solutions of y'' + p(t)y' + q(t)y = 0 on an open interval I on which p,q are continuous. Assume also that y1 and y2 have a common point of inflection t0 in I. Prove that y1,y2 cannot be a fundamental set of solutions unless p(t0) = q(t0) = 0.The...

I am working on this problem on measure theory like this: Suppose ##X## is the set of real numbers, ##\mathcal B## is the Borel ##\sigma##-algebra, and ##m## and ##n## are two measures on ##(X, \mathcal B)## such that ##m((a, b))=n((a, b))< \infty## whenever ##−\infty<a<b<\infty##. Prove that...

Homework Statement Show that the set S of all (x,y) ∈ ℝ2such that 2x2+xy+y2 is closed but not compact. Homework Equations set S of all (x,y) ∈ ℝ2such that 2x2+xy+y2 The Attempt at a Solution I set x = 0 and then y = 0 giving me [0,±√3] and [±√3,0] which means it is closed However, for it to...

My gut instinct is that putting all the resistors in series will give max resistance, but I am not sure how to give a more rigorous either mathematical or just in words reasoning why. Or maybe I'm wrong! But it seems like the fraction introduced from parallel won't help In any case, assume all...

I have the hardback 5 volume set of Spivak's A Comprehensive Introduction to Differential Geometry that is in pretty good shape. Is there any value to that set? I tried looking it up, but I don't really see many people selling whole sets, so I can't tell... Thanks.

Homework Statement Find a set of basic solutions and express the general solution as a linear combination of these basic solutions a + 2b - c + 2d + e = 0 a + 2b + 2c + e = 0 2a + 4b - 2c + 3d + e = 0 Homework Equations 3. The Attempt at a Solution [/B] i reduced it to: 1 2 0 0 -1 0 0 0 1...

Having trouble understanding the concept of transitivity. By definition: If (a,b)\in R\wedge (b,c)\in R \Rightarrow (a,c)\in R - Great. Consider the set \{a,b\}. What makes the relation \{(a,a)\} or \{(a,a),(a,b)\} transitive? How do I translate this in terms of the definition? What makes an...

Let $f:\mathbf R^n\to\mathbf R^m$ be a smooth function of constant rank $r$. Let $\mathbf a\in \mathbf R^n$ be such that $f(\mathbf a)=\mathbf 0$. Then $f^{-1}(\mathbf 0)$ is a manifold of dimension $n-r$ in $\mathbf R^n$. We imitate the proof of Lemma 1 on pg 11 in Topology From A...

A theorem on equivalence relation states that for any set S, the set of equivalence classes of S under an equivalence relation R constitutes a partition of a set. Moreover, given any partition of a set, one can define an equivalence relation on the set. What allows you to "create" a partition...

Homework Statement D1 = {(x,y) : x^2 + y^2 < 3, x+2y = 2} D2={(x,y) : x^2 + y^2 > 2} D3={(x,y) : x + 2y = 2} Homework EquationsThe Attempt at a Solution D1 is neither, D2 is open and D3 is closed, am I right or wrong?

Homework Statement R = all real numbers F(x) = { y in R : sin(y) = x} 1. Is F a mapping from R to R 2. Describe the three sets F(5), F(0), F(1). 3.Can F be represtented as a function from R to R 4. Give two different choices of X and Y (take both X and Y to be subsets of ?) so that F can...

Hello! (Wave) Suppose that $X$ contains a countable set. Let $b \notin X$. Show that $X \sim X \cup \{b\}$. Prove that in general if $B$ is at most countable with $B \cap X=\varnothing$ then $X \sim X \cup B$. Proof:We will show that $X \sim X \cup \{b\}$. There is a $\{ a_n: n \in \omega \}...

Hello! (Smile) Proposition: The set $\{0,1\}^{\omega}$ of the finite sequences with values at $\{0,1\}$ is not countable. Proof: $$\{ 0,1 \}^{\omega}=\{ (x_n)_{n \in \omega}: \forall n \in \omega \ x_n \in \{0,1\} \}$$ From the following theorem: No set is equinumerous with its power set...

Hello! (Smirk) Proposition The set $\mathbb{Z}$ of integers is countable. Proof $\mathbb{Z}$ is an infinite set since $\{ +n: n \in \omega \} \subset \mathbb{Z}$. $$+n= [\langle n, 0 \rangle]=\{ \langle k,l \rangle: k+n=l\}$$ We define the function $f: \omega^2 \to \mathbb{Z}$ with...

Hello! (Wave) I want to show that if $A$ is a finite set of finite sets then the set $\bigcup A$ is finite. The set $A$ is finite. That means that there is a natural number $n \in \omega$ such that $A \sim n$, i.e. there is a bijective function $f$ such that $f: A...

Homework Statement Show if the set is convex or not! S2 = Homework Equations I know that to show a set is convex you can either use the definition or show that the set can be obtained from known convex sets under operations that preserve convexity. Convex definition: x1*Theta + (1 -...

Homework Statement Problem: Given a regular deck of 52 cards, let A be the event {king is drawn} or simply {king} and B the event {club is drawn} or simply {club}. Describe the event A ∪ B Solution: A ∪ B = {either king or club or both (where "both" means "king of clubs")} Homework Equations...

Homework Statement Describe the set of points determined by the given condition in the complex plane: |z - 1 + i| = 1 Homework Equations |z| = sqrt(x2 + y2) z = x + iy The Attempt at a Solution Tried to put absolute values on every thing by the Triangle inequality |z| - |1| + |i| = |1|...

Let X be a real Banach Space, C be a closed convex subset of X. Define Lc = {f: f - a ∈ X* for some real number a and f(x) ≥ 0 for all x ∈ C} (X* is the dual space of X) Using a version of the Hahn - Banach Theorem to show that C = ∩ {x ∈ X: f(x) ≥ 0} with the index f ∈ Lc under the...

I am doing some self study of groups and can solve problem #3 but not Problem #4. Problem 3. Let A be a finite set, and B a subset of A. Let G be the subset of S_A consisting of all of the permutations f of A such that f(x) is in B for every x in B. Prove that G is a subgroup of S_A. Problem...

Proposition: The set $\omega \times \omega$ is equinumerous with $\omega$, i.e. the set $\omega \times \omega$ is countable. "Intuitive Proof" $$\mathbb{N}^2=\{ (n,m): n,m \in \mathbb{N} \}$$ $$1 \mapsto a_{11}$$ $$2 \mapsto a_{12}$$ $$3 \mapsto a_{31}$$ $$4 \mapsto a_{22}$$ $$5 \mapsto...

I think we have a problem with this rule. This is my understanding of the existing rule. Am I close? The rule for calculating the power distribution on a power split planetary gear system is based on the equal force rule. To calculate the power on the sun and ring of a planet set with input...

Homework Statement A rod of length ##L## and mass ##M## is constrained to move in a vertical plane. The upper end of the rod slides freely along a horizontal wire. Let ##x## be the distance of the upper end of the rod from a fixed point, and let ##\theta## be the angle between the rod and the...

Hello, I am just doing my homework and I believe that there is a fault in the problem set. Consider the set of functions defined by V= f : R → R such that f(x) = a + bx for some a, b ∈ R It is given that V is a vector space under the standard operations of pointwise addition and scalar...

Suppose you have two observables ##\xi## and ##\eta## so that ##[\xi,\eta]=0##, i know that there exists a simultaneous complete set of eigenvectors which make my two observables diagonal. Now the question is, if ##\xi## is a degenerate observable the complete set of eigenvectors still exist?

Hi! (Smile) According to my notes, a set $A$ is called transitive if the elements of its elements are elements of $A$. For example, the set of natural numbers $\omega$ is a transitive set. Also, if $n \in \omega$ then $n$ is a transitive set since $n=\{0,1,2, \dots, n-1 \}$ and if we take a...

Are these three sets equivalent? $$A=\left\{(x,y):x,y\in\Bbb{R},y\ge x^2-1\right\}$$ $$B=\left\{x,y\in\Bbb{R}:y\ge x^2-1\right\}$$ $$C=\left\{(x,y)\in\Bbb{R}^2:y\ge x^2-1\right\}$$ I am thinking that $A$ and $C$ are, but not $B$ as it might be ambigious as to which dimension it is in, i.e it...

Homework Statement The problem is to prove that there is only one empty set. Let A and B be empty sets, A is a subset of B and B is a subset of A (by the definition that the empty set is a subset of every set) So A=B (by definition) By convention, all empty sets are equal. Therefore, there...

Homework Statement Bob makes two sets: one with all the even integers between 1 and 30 inclusive, and another with all the odd integers inclusive. He called the sets Q and R. He multiplied each number from Q with each number in R. Then he added the 225 products together and called the result...

Hello! (Wave) According to my notes, when we consider the order $I_A$ for $A \neq \varnothing$ each element of $A$ is minimal and maximal. If, in addition, $A$ has at least two elements then there isn't neither the greatest nor the least element of $A$. Could you explain it to me? :confused...

Let's see if I have this straight: Observables are represented by Hermitian operators, which can be, for some appropriate base, represented in matrix form with the eigenvalues forming the diagonal. Sounds nice until I consider observables with continuous spectra. How do you get something like...

I'm introducing myself to set theory. My reference doesn't seem to address the fact that 1/1 = 2/2 = 1. If we make a correspondence between natural numbers and rational numbers using sequential fractions, should we just skip equivalent fractions so as to make it a bijection? In other words, does...

Homework Statement Shown in attachment The problem has been modified. All inputs and outputs are 5 gal/min. Pure water enters tank 1. Homework Equations System of equations The Attempt at a Solution Included on attachment.

Hello! (Wave) I want to describe an algorithm with time complexity $O(m)$ that, given a set $M$ with $m$ numbers and a positive integer $p \leq m$, returns the $p$ closest numbers to the median element of the set $M$. How could we do this? (Thinking)

HI all, I have the equation, 6x^2-2>9x for which I'm to find the solution set in interval notation. I've rewritten the inequalty as 6X^2-9x-2=0. I tried to factor, but no go. Then I used the quadratic and got 9+/- rad(129)/-18. The answers I get for x are -1.1309 and .1309. The calculator...

Hello, I was going through the following paper: http://www.emis.de/journals/HOA/AAA/Volume2011/142128.pdf In page 6, immediately after equation (3.15), its written that "functions of the form v(t) are dense in L^2". I have been looking for proofs online which verifies the above statement but...

Homework Statement Let X be a set. Then the set {Y:Y is a subset of X} prove this is a set.Where do i start? Really unsure, i know that i have to use the power set? I have written down; {0,1}^X

Hello! (Nerd) I am looking at the proof of the following sentence: For any natural numbers $m,n$ it holds that: $$n \in m \rightarrow n \subset m$$ Proof: We define the set $X=\{ n \in \omega: \forall m (m \in n \rightarrow m \subset n)$ and it suffices to show that $X$ is an inductive set...

Hi! (Smile) We want to show that the elements of the natural numbers are natural numbers, i.e. $(n \in \omega \wedge x \in n) \rightarrow x \in \omega$ Could you explain me why, in order to show this, it suffices to show that $X=\{ n \in \omega: (\forall y \in n)(y \in \omega)\}$ is an...

Hi! (Nerd) Sentence: There is a unique inductive set that is contained in each inductive set. Proof: Let $A$ be an inductive set (we know that there is such a set from the axiom of infinity) and we define: $$B=\{ X \subset A: X \text{ is an inductive set}\}$$ ($B$ is a set, because if $X...

Hello! (Wave) A set $A$ is called inductive set, if $\varnothing$ is an element of $A$ and for each $x \in A$ its next element, $x'=x \cup \{ x \}$ belongs to $A$. I want to show that if $B$ is a nonempty set of inductive sets, then $\bigcap B$ is an inductive set.That's what I have tried:$B$...

I would like to find the distance that a vehicle travels using a Kalman filter. The vehicle is a car that travels the road between two positions. The vehicle has a GPS/barometer/accelerometer device that collects position data, which I converted from a longitude and latitude to a North, East...

Homework Statement Let ##R## be an ordered relation on a set ##A## that is reflexive and anti-symmetric. If there is a chain of elements in ##R## that begins and ends with the same element, say the element ##x \in A## is it true that all the elements of ##R## sandwiched in between the ones...

Could the mass of a singularity be described or defined as an empty set, or else what is the term to describe it (in at least one sentence).

Let R be the class of all ordinals. If a subset C of R is unbounded (i.e. for any ordinal \alpha \in R, there is \beta in C with \beta greater than \alpha ), then it seems to me that C cannot be a set, only a class. Is this true, and if so, how does one prove it? My reading on the general...

Hello, I am an electrical engineering student and I was hoping some body here could help me out with a cantilever question. I want to model a vibrating cantilever with a mass at the end. I am doing this for a project where I wanted to model a Wurlitzer 200 Electric Piano. The way they produce...

A mass on the end of a spring which is hanging vertically is raised up and let go. It then oscillates between 2m and 1.5m above the floor and completes 32 cycles in one minute. The height, h metres, of the mass above the floor after t seconds can be modeled by the function h=acos(pi t / 180)...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 3.1 on Categories. At present I am working on Problem 2 in Problem Set 3.1 and I need some help in understanding the problem and its solution. Problem 2 (Problem Set 3.1) reads as...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 3.1 on Categories. At present I am working on Problem 1 in Problem Set 3.1 and I need some help in understanding the problem and its solution. Problem 1 (Problem Set 3.1) reads as follows...