Proof Definition and 999 Threads

∈Was wondering if anyone here could help me with an explanation as to how Axler arrived at a particular step in a proof. These are the relevant definitions listed in the book: Definition of Matrix of a Linear Map, M(T): Suppose ##T∈L(V,W)## and ##v_1,...,v_n## is a basis of V and ##w_1...

My question is how to show ##\mathcal{F} \subset \sum'##. Here is my work for the problem: Proof of hint: First we'll show ##\sum## is a monotone class. Let ##(A_n)_{n\in\mathbb{N}} \subset \sum## and ##F \in \mathcal{F}##. There are two things to verify. Suppose ##(A_n) \uparrow A =...

Ok, so here is what I have so far: Suppose ##T_1## is infinite and ##\varphi : T_1 \rightarrow T_2## is a bijection. Reasoning: I'm thinking I would then show that there is a bijection, which would be a contradiction since an infinite set couldn't possibly have a one-to-one correspondence...

I read in one book proving one nature of variation(variation of high-order derivative). It writes that "##\delta(F^{(n)}) = F^{(n)} - F_0^{(n)} = (F - F_0)^{(n)} = (\delta F)^{(n)}##". But I don't understand where this ##F_0## comes out from.

Summary:: prove that (n 0) + (n 1) + (n 2) + ... + (n n) = 2^n is true using mathematical induction. note that (n n) is a falling factorial Hello! I have trouble dealing with this problem: Mod note: Thread moved from math technical section, so is missing the homework template. Prove that (n...

equation of motion : while

Definition: Let ##G## be a graph. ##G## is a functional graph if and only if ##(x_1,y_1) \in G## and ##(x_1,y_2) \in G## implies ##y_1=y_2##. Problem statement, as written: Let ##G## be a functional graph. Prove that ##G## is injective if and only if for arbitrary graphs ##J## and ##H##, ##G...

The significant digits of numbers in sets of numerical data supposedly follows "Benford's Law", which asserts that the probability that the first digit in a given data point is ##D## is about ##\log_{10}(1+ \frac{1}{D})##. An upshot is that we expect ~30% of significant digits to be ##1##. The...

Set ##\epsilon=\frac{1}{2}##. Let ##N\in \mathbb{N}## and choose ##n=N,m=2N##. Then: ##\begin{align*} \left|s_N-s_{2N}\right|&=&\left|\sum_{l=1}^N \frac{1}{l} - \sum_{l=1}^{2N} \frac{1}{l}\right|\\...

Angular momentum can be exchanged between objects in a closed system, but total angular momentum before and after an exchange remains constant (is conserved). There is a proof about this conservation?

My only qualm is that the statement “Let G be a functional graph” never came into play in my proof, although I believe it to be otherwise consistent. Can someone take a look and let me know if I missed something, please? Or is there another reason to include that piece of information?

I typed this up in Overleaf using MathJax. I'm self-studying so I just want to make sure I'm understanding each concept. For clarification, the notation f^{-1}(x) is referring to the inverse image of the function. I think everything else is pretty straight-forward from how I've written it. Thank...

For fun, I decided to prove that two timelike never can be orthogonal. And for this, I used the Cauchy Inequality for that. Such that The timelike vectors defined as, $$g(\vec{v_1}, \vec{v_1}) = \vec{v_1} \cdot \vec{v_1} <0$$ $$g(\vec{v_2}, \vec{v_2}) = \vec{v_2} \cdot \vec{v_2} <0$$ And the...

Although I am not too sure how to answer this quesion I have tried below. I realize that an electromotive force is a supply voltage, the energy transferred per unit charge when one type of energy is converted into electrical energy. However, EMF is not actually a force. It is usually measured...

Here is my solution. I used mathjax to type it up in Overleaf. I feel like it makes sense, but I also have a feeling I might have "jumped the gun" with my logic. If it is correct, I would appreciate feedback on how to improve it. Thanks!

##a^2= \frac {sin^4θ} {cos^2θ}## ##b^2=\frac {cos^4θ} {sin^2θ}## therefore ##a^2b^2= sin^2θ. cos^2θ##...1 and ##a^2+b^2+3= \frac {sin^6θ + cos^6θ + 3 sin^2θ cos^2θ} {sin^2θ cos^θ}##...2 multiplying 1 and 2, we get, ##sin^6θ + cos^6θ + 3 sin^2θ cos^2θ## how do i proceed from...

Sir/madam, I request you to solve 2 questions ( q-3 and q-5 ) of symbolic logic ( Strenthened method of conditional proof ). These questions are taken from I.M.Copi's 'symbolic logic' ( edition -5, sec. 3.8, pg- 61 ) File is being attached. thank you yours truly Deep Kumar Trivedi

I have asked this question twice and each time, while the answers are OK, I am left dissatisfied. However, now I can state my question properly (due to the last few responses). Go to this page and scroll down to the matrix for sixth row of the proper Euler angles...

My try: ##\begin{align} \dfrac{d {r^2}}{d r} \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \tag1\\ \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \dfrac{1}{\dfrac{d r^2}{d r}}=\dfrac{p-a\cos\theta}{r} \tag2\\ \end{align}## By chain rule...

I want to make certain that my proof is correct: Since ## P^2 = P_\nu P^\nu=P^\nu P_\nu ##, then ## [P^2,P_\mu]=[P^\nu P_\nu,P_\mu]=P^\nu[P_\nu,P_\mu]+[P^\nu,P_\mu]P_\nu=[P^\nu,P_\mu]P_\nu=g^{\nu\alpha}[P_\alpha,P_\mu]P_\nu=0 ##, since ## g^{\nu\alpha} ## is just a number, I can bring it...

The moving magnet and conductor problem is an intriguing early 20th century electromagnetics scenario famously cited by Einstein in his seminal 1905 special relativity paper. In the magnet's frame, there's the vector field (v × B), the velocity of the ring conductor crossed with the B-field of...

(x)(x)>0 (D) (x+(-x))(x+(-x)) >0 (A4) x^2 + 2(-x)(x) + (-x)^2 >0 (D) x^2 - 2x^2 + (-x)^2 >0 -x^2 + (-x)^2 >0

(NOTE: I have had a few similar postings lately on this subject, but they were much broader in scope, so I am posting only for this particular case; everything else has been figured out.) If given that limx -> a f( x ) = +∞ limx -> a g( x ) = +∞ what is the epsilon-delta formulation for...

The proof for the ET I've found in some of the undergrad books for statistical physics (for example in Reif's "Statistical and Thermal Physics") assumes the form of the Hamiltonian of the system to be: $$H = bp_i^2 + E'(q_1,...,p_f)$$ where ##b## is a constant. My professor in his notes, says...

I was looking at some websites that show the proof of addition of limits for a finite output value, but I don't see one for the case of infinite output value, which has a different condition that needs to be met - i.e., | f( x ) | > M instead of | f( x ) - L | < ε...

I'm trying to come up with a proof of the operator identity typically used in the Mori projector operator formalism for Generalized Langevin Equations, e^{tL} = e^{t(1-P)L}+\int_{0}^{t}dse^{(t-s)L}PLe^{s(1-P)L}, where L is the Liouville operator and P is a projection operator that projects...

Hi All, I try to prove the following commutator operator Identity used in Harmonic Oscillator of Quantum Mechanics. In the process, I do not know how to proceed forward. I need help to complete my proof. Many Thanks.

AIUI, this is a law of proofs: lim x→a f( g( x ) ) = f( lim x→a g( x ) ) I have searched for an explanation of this proof, but have been unable to find one, although I did find a page that was for certain types of functions of f( x ), just not a proof for a function in general.

I am having difficulty located experimental or observational proof that a barycenter exists between the Earth and Moon. All I can find seems to just assume that a barycenter exists because the Moon revolves around the Earth based on the assumption that the Earth and Moon are tied as a unit with...

Hi. I'm trying to grasp what the PBR theorem is about. I'm not tackling the full version, but rather the simple example in @Demystifier's summary. While I think I understand the mathematical steps, my question is why you need two systems to prove it. Is this only technical or more fundamental...

I need to prove this using the given equations. $$\vec{N}(t) = \frac{\vec{a}_{v\perp}}{|\vec{a}_{v\perp}|}$$ Here is the entirety of my work up to this point. So far I've wanted to use what I have to find something that is perpendicular to the velocity vector and maybe show that with the dot...

I search for a vector space based proof of the following : The logic on values implies ~~v(A)=v(A) If the value of A is v(A)##\in\{0,1\}## then it is simply ##1-(1-v(A))=v(A)## But if we suppose A="the sky is red" Then as on operator acting on A, ~A is not defined since for example ~A="the...

Hi! I don't understand how to demonstrate the following exercise. Let ##F: R^{n} \rightarrow R^{n}## be a linear map which is invertible. Show that if ##A## is the matrix associated with ##F##, then ##A^{-1}## is the matrix associated with the inverse of ##F##.

See attached image

Hi, My question pertains to the question in the image attached. My current method: Part (a) of the question was to state what Stokes' theorem was, so I am assuming that this part is using Stokes' Theorem in some way, but I fail to see all the steps. I noted that \nabla \times \vec F = \nabla...

Hello all, In the attached picture there is an equation. I need to fill the general expression on the left hand side, and to prove by induction that the sum is equal to the expression in the right hand side. I am not sure how to find the general expression. Can you kindly assist ? Thank you !

How is it proved that Van Der Waals gas is a second order phase transition? The second order derivative of the pressure (P ) with respect to volume ( V ) don't have a discontinuity ( except at point V = Nb , but the pressure is not existent for V<=Nb ). So how come Van der waals gas describes...

The proof of magnetic forces do no work is given in Introduction to Electrodynamics by David J. Griffiths like this My problem is why he has replaced d\mathbf{l} with \mathbf{v}dt? This substitution implies that the charged particle was moving with \mathbf{v} only and no force acted on it...

When proving that Newton's Universal Law of Gravity will produce an elliptical orbit, the radial acceleration is confusing to me. Can someone please explain how the radial acceleration is equal to d^2/dt^2 - r(d x theta / dt)^2. Could someone please detail how that acceleration is derived...

The Completeness Proof for First-Order Predicate Logic depends on if $\Phi$ is a set of consistent $\mathcal L$-formulas, then $\Phi$ is satisfiable. How is that constructed? There are a large number of Lemmas working from Machover's text Set theory, Logic and Their Limitations but I'm having...

$2^{n+2} < (n+1)!$ 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 $2^{k+3} < (k+2)!$

Given a function F(t) $$ F(t) = \int_{-\infty}^{\infty} C(\omega)cos(\omega t) d \omega + \int_{-\infty}^{\infty} S(\omega)sin(\omega t) d \omega $$ I am looking for a proof of the following: $$ \int_{-\infty}^{\infty} F^{2}(t) dt= 2\pi\int_{-\infty}^{\infty} (C^{2}(\omega) + S^{2}(\omega)) d...

Point D divides side AC, of triangle ABC, so that |AD|: |DC| = 1:2. Prove that vectors \vec{BD} = 2/3 \vec{BA} + 1/3 \vec{BC}.

My question is simple. Can one prove any theorem in mathematics by having only a pen and a paper, or a super-computer for that matter? Since math is essentially all about theorems, and we usually take them as true. I guess someone went in and proved them at some point in our history. But some...

Brahmagupta's theorem: A cyclic quadrilateral is orthodiagonal (diagonals are perpendicular) if and only if the perpendicular to a side from the point of intersection of the diagonals bisects the opposite side. But I don't understand the first step of the proof for the necessary condition...

Can someone please help me to prove that the function f(x) = Arcsin x is continuous on the interval [-1, 1] ... Peter

Book shows a proof where a conclusion is reached of: ##\neg r##. The next step says ##\neg r \lor \neg s## using the rule of disjunctive amplification. The rule of disjunctive amplification as I know it is ##p \implies p \lor q##. I don't see how from this you can also say ##\neg p \implies...

I found this video showing an elementary proof of the FTA.

I assumed that this would be a straightforward proof, as I could just make the substitution l=j and m=l, but upon doing this, I end up with: δjj δkl - δjl δkj = δkl - δlk Clearly I did not take the right approach in this proof and have no clue as to how to proceed.

In axioms containg S one invariably finds: Sx = Sy -----> x = y The converse, which characterizes S as a function: x = y ------> Sx = Sy Is never shown. Neither is it shown as an Axiom of FOL or formal Theory of Arithmetic. From the basic axioms and rules of FOL, how does one go about...