Proofs Definition and 671 Threads

1. So firstly I would like to show that product of the matrices L21,L31,... in the derivation of the LU decmoposition is lower triangular. I have already shown that product of upper and lower triandular matices is upper or lower. Now I don't know how to show the derivation. 2. I have been given...

Hello, So I am struggling with a couple very simple proofs of properties of matrix algebra. This is the first time I have ever had real proofs in math (Linear algebra). For the first one, I have it from our text but need a little help, and I am completely lost on the second one. 1) Prove...

I find that in number theory, number theorists (and mathematicians more generally) generally prefer elementary proofs over any other kind of proof. Am I right about this? If so, why is this? Is this something to do with the content of number theory itself? Thanks!

I am no expert in formal logic, so please forgive me if this question sounds stupid. It's about a common pattern used in many mathematical proofs. For me it' "obvious" or "trivial" - but I can't prove it. For a friend of mine it's far from obvious or even wrong - but I don't get his point...

Show that for $|a| \le \frac{\pi}{2}$, $$\int_{0}^{\infty} \frac{\cos (\frac{\pi x}{2}) \cos(ax)}{1-x^{2}} \ dx = \frac{\pi}{2} \cos a$$Similarly, show that for $|a| \le \pi$, $$ \int_{0}^{\infty} \frac{\sin (\pi x) \sin(ax)}{1-x^{2}} \ dx = \frac{\pi}{2} \sin a $$

Can you think of any theorems that admit of both explanatory and non-explanatory proofs? Roughly, a proof is "explanatory" if the proof illuminates why the theorem is true.

Ok, I'm going to be taking calc III next week, so I wanted to get a head-start by doing the MIT multivariable calculus opencourseware. While most of the material was easy, these proofs are really killing me. Here are two examples: Ex.1: Using vectors and dot product show the diagonals of a...

I am trying to check whether lim h→0 (R(h)/||h||) =0 or not. I am working in ℝ2. h=h1e1+h2e2** => ||h||=(h1^2+h2^2)^1/2 I am using the definition that (R(h)/||h||)<ε * whenever 0<|h|<δ for all h. Example 1 (R(h)/||h||)=h1h2/(h1^2+h2^2)^3/2 I can see that the denominator dominates...

Homework Statement I've actually got a couple questions, I'll provide an example for each question, but I'm not really looking for an answer to the example, but an explanation of the concept. I have very little to go on from class notes. We've had some inclement weather in these parts leading...

I have been looking at the idea of 1:1 correspondence as a method of determining set size/cardinality, and have noticed that the principle allows for inductive proofs, which I think are properly constructed, that can come to conclusions which are clearly wrong under traditional set theory if...

I'm currently reading the section on relations in Velleman's "How to prove it" and I have found a statement somewhere that I want to prove, but I'm not sure whether what I have come up with is reasonable and I also have some questions on the logic used in these type of proofs. The theorem is...

Ok, I'm really confused with the reasoning behind inductive proofs. To prove some statement is true for all natural numbers, you need to assume the statement is true for some number k. But aren't you really assuming the statement is true for all natural numbers in the first place? If you can...

From Artin's Algebra: "Prove that the set ##\operatorname{Aut}(G)## of automorphisms of a group ##G## forms a group, the law of composition being composition of functions." Of course, we could go through and prove that the four group axioms in the standard definition of a group hold for...

How would I prove that if A is singular, then Av=0 has a non-zero solution?.

(I numbered my questions- it ended up being a long post!) (1) I'm also wondering if anyone has any good metaphors for difference between proof and "drills" or "techniques". Maybe learning "techniques" is sort of like getting good at scales, whereas proof is actually playing songs? I feel that...

I'm trying to really solidify my maths knowledge so that I'm completely comfortable understanding why and how certain branches of mathematics are introduced in physics and inevitably that leads me to a study of proofs. I usually skip proofs as I found them annoying, unintuitive and just...

Homework Statement Prove that if ##\lim_{x\to a}f(x)=A\neq 0## then ##\lim_{x\to a}1/f(x)=1/A## Homework Equations This is a proof, so it's just an epsilon delta proof. I know the solution. I am asking about a thought process. The Attempt at a Solution Pick a ##\delta_1## small enough such...

Let q be a natural number, show that if q is not divisible by 3, then neither is q^2 proof: if q is not divisible by 3 then q = 3k + 2 for some integer k q^2 = 4 + 3(3k^2 + 4k) = 4 + 3m for some integer m, hence q is not divisible by 3 another case, if q = 3k + 1 for some integer k, then q^2...

Please see the attached image. There are 6 figures. Please explain the proofs for sin(90 - θ) w.r.t the 2nd, 3rd and 4th images. I understand the proof w.r.t 1st image. In the 2nd image y1 and x is negative. In 3rd image y, y1, x, x1 are all negative and in the 4th image y and x1 are...

I'm a junior in high school taking calculus 1 at a local junior college, and I am getting quite bored with how very simplistic calculus seems to be in calculus 1. To me it just feels like an expansion upon algebra and trig with a few new twists, but nothing that actually requires me to use my...

I am new to tensor notation, but have known how to work with vector calculus for a while now. I understand for the most part how the Levi-Civita and Kronecker Delta symbol work with Einstein summation convention. However there are a few things I'm iffy about. For example, I have a problem where...

1) Are there any proofs that sine cosine and tangent are the same in all similar triangles? Like, that sine 30 degrees is the same no matter what the lengths are? Or is it just an axiom? 2) Is there a proof for the actual value for the sine cosine and tangent ratios (besides for common ones...

Just started this Analytical Mechanics class, so I figured this question should go here... I've been pretty stuck with a problem. I felt like I totally knew what I was doing but I've become very stumped. We're given the vector for general circular motion...

I just have a general question about limit proof using epsilon delta proofs It generally follows the algorithm from this website:http://www.milefoot.com/math/calculus/limits/DeltaEpsilonProofs03.htm focusing on the first example(example using linear functions) The first table I get, but...

does anyone have any tips on learning how to do proofs. completing a proof isn't something I am good at, at all. I know how to get answers and I can do the math but explaining it to someone step by step, using the right communication. Seems like a foreign language to me. any tips on learning...

I have trouble using isomorphism and permutation in proofs for combinatorics. I don't know when I can assume "without loss of generality". What are some guidelines to using symmetry in arguments. One problem I'm working on that uses symmetry is to "prove that any (7, 7, 4, 4, 2)-designs must be...

I've recently come to the conclusion that I might have made some mistakes along the way. I'm going into my senior year of EE and something just doesn't feel right about my abilities. Over the last couple semesters, I've fallen into the "plug and chug" mode of solving problems. I have some issues...

I don't need unit circle proof or proof using trignometric equality. I know those. I am attaching a image. Please read it. It is from the book Plane Trigonometry by S L Loney. See what it says. I know that angle MOP = angle M'P'O and both triangles formed are congruent, OM' = MP and M'P" = OM...

Hi all. This is my first post on here. I come here to read what others ask/answer so this was the first place I thought of when this dilemma came up for me. I am a Math Major and I am for sure taking a proofs course this fall. I have been allowed to take Analysis concurrently with the proofs...

Well, the title pretty much sums it up. I was reading up about combinatorial proofs and was wondering if anyone could offer an example of one or explain how they are done.

Hello guys, this is my first post on this forum. I want to learn advanced/pure mathematics basically just because I find it really interesting and challenging and I have started to learn about proofs. I'm currently reading Velleman's book and I have reached the part in which you actually...

Hi, I have my exam tomorrow and have been doing the practice questions. However we don't get the answers to these questions so I am lost as to whether I am doing them right, also I am stuck at a few points, particularly with the definition questions. \[ 1.)Prove\ lim_{x \to...

Homework Statement First Question: Be A and B square matrix. Show that if A and B are invertible matrix, then: (A + B)^(-1) = A^(-1) * [I + B*A^(-1)]^(-1) The Attempt at a Solution First Queston: (A+B)^(-1) = [A^(-1) + A^(-1)*B*A^(-1)]^(-1) (I distributed the a^(-1) outside the []). Then...

So yet we have the Standard Model which tries to explain and unify the 4 fundamental forces or atleast 3 for now since the gravity is not quite well understood in particle physics.So people search for a theory of everything which unifies all forces and yet we didn't found any good theory as...

Hi there again guys! I didnt really know what to call this thread, because my problem isn't actually to do with how to manipulate the elements of the matrix itself, but rather how to deal with the actual symbol for the matrix in equations. I'll start off with a fundamental thing, even...

I am dealing with sets of problems that go as such: "How many n-cent postages can be formed from x and y cent stamps" For instance, I am doing a problem where x and y are 4 and 11 respectively. I don't understand how to determine a base case. I know that I must proof P(k + 1) for all P(i)...

I have a problem. Here is the task. Need to prove all these claims: I was able to prove only one and half of them. b) Let A\subsetB and x \in A°. It mean that there is ε>0 that Oε(x)\subsetA. And it mean that Oε(x)\subsetA\subsetB. And that mean that x\inB, what I need to prove. c)...

There are a few theorems in my DE book whose proofs I've been trying to find, without much luck: 1) Given an nth order linear homogenous ODE with continuous coefficients and nonzero leading coefficient, any IVP of this ODE has a unique solution over some interval I centered about the IVP...

Hello, I am a senior in high school wondering if I should major in mathematics. I am developing a strong interest in the subject and am currently enjoying and doing well in my AB AP Calculus course. The problem, however, is that I have read in many places (such as on these fantastic forums) that...

Homework Statement Let G\subset L(\mathbb{R}^n;\mathbb{R}^n) be the subset of invertible linear transformations. a) For H\in L(\mathbb{R}^n;\mathbb{R}^n), prove that if ||H||<1, then the partial sum L_n=\sum_{k=0}^{n}H^k converges to a limit L and ||L||\leq\frac{1}{1-||H||}. b) If A\in...

In 1970, Solovay proved that, although (1) under the assumptions of ZF & "there exists a real-valued measurable cardinal", one could construct a measure μ (specifically, a countably additive extension of Lebesgue measure) such that all sets of real numbers were measurable...

Let $G\subset L(\mathbb{R}^n;\mathbb{R}^n)$ be the subset of invertible linear transformations. a) For $H\in L(\mathbb{R}^n;\mathbb{R}^n)$, prove that if $||H||<1$, then the partial sum $L_n=\sum_{k=0}^{n}H^k$ converges to a limit $L$ and $||L||\leq\frac{1}{1-||H||}$. b) If $A\in...

Homework Statement Write down careful proofs of the following statements: a) sqrt(6)- sqrt(2) > 1 b) If n is an integer such that n^2 is even, then n is even. c) If n= m^3- m for some integer m, then n is a multiple of 6 The Attempt at a Solution I will rely on P - > Q and not...

I have gotten to this point with a and b but do i am totally lost with c. Any help would be much appreciated Consider any three arbitrary sets A, B and C. (a) Show that if A ∩ B = A∩ C and A ∪ B = A ∪ C, then B = C. (b) Show that if A − B = B − A, then A = B. (c) Show that if A∩B = A∩C = B ∩C...

My 7th grader has enough mathematical background in algebra, geometry and trigonometry to start learning how to write out proofs. Are their good books that teach this step by step? I can certainly teach him myself with examples, but I figured there must be a systematic way to teach this...

I have a few questions for my homework assignments for solving limits, but in order to do those questions I have to use a few standard limits that we haven't been taught, which means I'll have to prove them. I know these can be done using L'Hopital's rule, but we haven't covered that yet so I...

Homework Statement I have to prove two of the curvature formulas. The first one is (V X A) / l V3l The other one is a(t) * N(t) / l V(t) I2 Homework Equations I have a hint from my professor, but it is all confusing. I need a youtube video or something to get started on these...

I googled for lens formula's proof on the internet and found one here http://www.tutorvista.com/content/physics/physics-ii/light-refraction/convex-lens-formula.php The problem is that during the course of derivation, it is assumed that image distance is -u following the sign convention...

Hello, I am working through some proofs from the following document: Function Definitions Under Calculation of Big - Oh, some theorems are provided that classify the growth rates of functions in relation to one depending on what the limit is as the input approaches infinity. One proof is...

Homework Statement Let ##\theta : G \mapsto H## be a group homomorphism. A) Show that ##\theta## is injective ##\iff## ##\text{Ker}\theta = \left\{e\right\}## B) If ##\theta## is injective, show that ##G \cong I am \theta ≤ H##. The Attempt at a Solution A)The right implication is...