Proof verification Definition and 21 Threads

The proof in the solutions was done much differently then mine (much simpler), so I would like feedback on whether my proof is valid or not. Assume that F(L) and F(K) are not parallel, then they necessarily intersect. Let X' be the point of intersection. Then X' lies on both F(L) and F(K). Let...

Relevant Rules: N5: -(a+b) = - a - b N4: a = -(-a) N2: a + (-a) = 0 and -a + a = 0 I tried just manipulating -(a - b) with the rules to get the answer: -(a - b) = -(a + (-b)) With N5: = - a + (-(-b)) With N4: = - a + b Commutativity: b - a The provided solution in the book used N2 to prove it...

a) Proof: Consider the system ## \dot{x}=x+x^3-y^2 ## and ## \dot{y}=x^2-x^4+y^5 ##. By theorem, Bendixson's negative criterion states that there are no closed paths in a simply connected region of the phase plane on which ## \frac{\partial X}{\partial x}+\frac{\partial Y}{\partial y} ## is of...

Say we have as special lattice ## \Lambda^{\perp}(A) = \left\{z \in \mathbf{Z^m} : Az = 0 \in \mathbf{Z_q^n}\right\}##. We define ##U \in \mathbf{Z^{m \times m}}## as an invertible matrix then I want to proof the following fact: $$ \Lambda^{\perp}(AU) = U^{-1} \Lambda^{\perp}(A) $$ My idea: Let...

I assumed three points for a triangle P1 = (a, c), P2 = (c, d), P3 = (b, e) and of course: a, b, c, d, e∈Z Using the distance formula between each of the points and setting them equal: \sqrt { (b - a)^2 + (e - d)^2 } = \sqrt { (c - a)^2 + (d - d)^2 } = \sqrt { (b - c)^2 + (e - d)^2 }(e+d)2 =...

I have already seen proofs of this problem, but none of them match the one I did, therefore I would be glad if someone could indicate where is the mistake here. Thanks in advance.**My proof:** Take a limit point x of U that is not in U, but is in K (in other words x \in K \cap(\overline{U}-U))...

Theorem: Show that the sequence ## a_n = (-1)^n ## for all ## n \in \mathbb{N}, ## does not converge. My Proof: Suppose that there exists a limit ##L## such that ## a_n \rightarrow L ##. Specifically, for ## \epsilon = 1 ## there exists ## n_0 ## s.t. for all ## n > n_0## then ##|(-1)^n-L|<1##...

#### If ##b \leq x_n \leq c## for all but a finite number of n, show that ##b \leq \operatorname{lim inf}_{n \to \infty} x_n## and ##\operatorname{lim sup}_{n \to \infty} x_n \leq c_n## (Buck, Advanced Calculus, Section 1.6, Exercise 24) Let ##\beta =\operatorname{lim inf}_{n \to \infty} x_n##...

$2^{n+2} < (n+1)!$ for all n $\geq 6$ Step 1: For n = 6, $256 < 5040$. We assume $2^{k+2} < (k+1)!$ Induction step: $2 * 2^{k+2} < 2*(k+1)!$ By noting $2*(k+1)! < (k+2)!$ Then $2^{k+3} < (k+2)!$

Homework Statement Theorem: If ##X## is a topological space, each path component of ##X## lies in a component of ##X##. If ##X## is locally path connected, then the components and the path components of ##X## are the same. I need help locating errors in my proof. Please help. Homework...

Homework Statement Let us have ##n \geq 3## points in a square whose side length is ##1##. Prove that there exists a graph with these points such that ##G## is connected, and $$\sum_{\{v_i,v_j\} \in E(G)}{|v_i - v_j|} \leq 10\sqrt{n}$$ Prove also the ##10## in the inequality can't be replaced...

Homework Statement Find ##z## in ##z^{1+i}=4##. Is my solution correct Homework Equations ##\log(z_1 z_2)=\log(z_1)+\log(z_2)## such that ##z_1, z_2\in \{z\in\Bbb{C} : (z=x+iy) \land (x\in\Bbb{R}) \land -\infty \lt y \lt +\infty\}## ##re^{i\theta}=r(\cos\theta + i\sin\theta)## The Attempt at a...

Homework Statement Let ##f:\Bbb{R} \to \Bbb{R}## be a function such that ##f## has a local minimum for all ##x \in \Bbb{R}## (This means that for each ##x \in \Bbb{R}## there is an ##\epsilon \gt 0## where if ##\vert x-t\vert \lt \epsilon## then ##f(x) \leq f(t)##.). Then the image of ##f## is...

Homework Statement In a finite-dimensional vector space, the length of every linearly independent list of vectors is less than or equal to the length of every spanning list. It's quite long :nb), hope you guys read through it. Thanks! :smile: Homework Equations N/A The Attempt at a Solution...

Homework Statement Let ##x,y\in X## such that ##X## is a metric space. Let ##d(x,y)=0## if and only if ##x=y## and ##d(x,y)=1## if and only if ##x\neq y## Homework Equations N/A The Attempt at a Solution I have already seen various approaches in proving this. Although, I just want to know if...

Homework Statement Show that ##f(x)=\frac{1}{x^2}## is not uniformly continuous at ##(0,\infty)##. Homework Equations N/A The Attempt at a Solution Given ##\epsilon=1##. We want to show that we can compute for ##x## and ##y## such that ##\vert x-y\vert\lt\delta## and at the same time ##\vert...

Proposition(Zorn's Lemma): Let ##X\neq\emptyset## be of partial order with the property that ##\forall Y\subseteq X## such that ##Y## is of total-order then ##Y## has an upperbound, then ##X## contains a maximal element. Proof: Case 1: ##B\neq\emptyset## such that ##B##=##\{####b\in X##: ##b##...

Homework Statement Let ##x\in\Bbb{R}## such that ##x\neq 0##. Then ##x=LIM_{n\rightarrow\infty}a_n## for some Cauchy sequence ##(a_n)_{n=1}^{\infty}## which is bounded away from zero. 2. Relevant definitions and propositions: 3. The attempt at a proof: Proof:(by construction) Let...

Homework Statement Let ##A,B## be subgroups of a finite abelian group ##G## Show that ##\langle g_1A \rangle \times \langle g_2A \rangle \cong \langle g_1,g_2 \rangle## where ##g_1,g_2 \in B## and ##A \cap B = \{e_G\}## where ##g_1 A, g_2 A \in G/A## (which makes sense since ##G## is abelian...

1. Statement: Prove that the EF and the MF are in phase far away from an oscillating electric dipole Homework Equations : The oscillatory motion equations for charges (q(t) = q_0sin(ωt) etc.)[/B] The Attempt at a Solution : Attached PDF file[/B]

I have a problem that I would like help on. I'm preparing for an exam, and I have provided my work below. **Problem statement:** Let $G$ act on $X$, and suppose $x,y\in X$ are in the same orbit for this action. How are the stabilizer subgroups $G_x$ and $G_y$ related? **My attempt:** $G_x =...