Proofs Definition and 671 Threads

Homework Statement See question 5 Homework Equations The Attempt at a Solution For part a, it is very easy. Multiply the inverse of A 2 times on both side, we can see the B=inverse of A. i.e. The required B is inverse of A, then the proof is finished. But how about part b...

Homework Statement Given a compact set A\subset\Re^{n} and a point x\in\Re^{n} define the distance from x to A as the quantity: d(x, A)=inf({\left\|x-y\right\|: y\inA}) Given two compact sets A, B \subset\Re^{n}, define the Hausdorff distance between them to be: d(A, B)=max(sup{d(x, B) ...

Hey, I know that questions about learning materials (like books) are supposed to be posted in the learning materials category but for some reason it is saying i cannot post there (it is saying i do not have privilieges/am trying to access administrative stuff?)...so i will just ask here...

"set builder" notation and proofs I'm curious about the references to "set builder" notation that I see in forum posts. Is this now a popular method of teaching elementary set theory and writing elementary proofs? I haven't looked at materials for that subject in the past 20 years. The...

Homework Statement Let n be a fixed positive integer greater than 1. If a (mod n) = a' and b (mod n) = b', prove that (a+b) (mod n) = (a'+b') (mod n) and that (ab) (mod n) = (a'b') (mod n) Homework Equations When a = qn + r a mod n = r The Attempt at a Solution (a'+b') (mod n) = (a...

I was doing a couple proofs (since I'm new to them) involving gcds and I just would like you guys to check them to see if I actually proved anything. There are 2 separate problems here. For both problems, a,b,c are in Z with a and b not both zero. PROBLEM 1 Homework Statement Prove...

Hello, I have stumbled upon a couple of proofs, but I can not seem to get an intuitive grasp on the what's and the whys in the steps of the proofs. Strictly logical I think I get it. Enough talk however. Number 1. "Let f be a continuous function on the real numbers. Then the set {x in R ...

Hello all :smile: I have started the problem set for Chapter one (basic properties of numbers) in Spivak's Calculus (self study). I think I am doing these right, but I have some questions. As a solid example, problem 1-(iv) says to prove the following: x^3 - y^3 =...

I think the FAQ on proofs would be improved if it emphasized the use of defintions. It says that theorems and axioms are used in proofs, but many many textbook type proofs hinge on "parsing" definitions correctly. As alluded to in the FAQs related to "is .999.. = 1?", many difficulties that...

ϵ-N proof Homework Statement Homework Equations The Attempt at a Solution I've tried to make the denominator smaller as is usual with ϵ-N proofs. But the sqrt(2) confuses me. Any help is much appreciated.

I can never seem to create proofs the way it is shown in every textbook I've seen. To be honest, I don't really know how to write the proofs correctly. I've seen sometimes my reasons are flawed and other times I go around aimlessly and get home after some unnecessary steps. So I would just like...

Hey all, I have a pretty solid background in what would be best described as applied or engineering math. However, I have had a very limited exposure to proofs. This fall, I will be taking a course covering linear algebra and vector calculus in an entirely proof-based manner. I'm looking...

I believe I have proven a famous open problem in mathematics, and no, it is not the Riemann Zeta hypothesis although that would be nice. Anyway, I want to know how I can submit my proof online and if anyone can give me pointers on preparing my paper. Thanks.

http://www.mathsisfun.com/geometry/" http://www.mathsisfun.com/algebra/trigonometry-index.html"

I'm starting to learn how to write proofs, and I am wondering how to become fluent in proofs. Is it necessary to do problems that are IMO/Putnam? Can anyone give me some advice? Thanks in advance.

Sometimes I find that while a proof can be carried out "by contradiction", this is a pretty sloppy way of proving the desired statement. I wonder if the "←" direction of the following proof is sound presentation of proof by contradiction. Statement. An integer is even if and only if its square...

Homework Statement Point A is on a circle whose center is O, AB is a tangent to the circle, AB = 6, D is inside of the circle, OD = 2, DB intersects the circle at C, and BC = DC = 3. Find the radius of the circle. Homework Equations Power of a point theorem (several cases found online...

I'm writing some of my proofs up in french in order to practice the language and I have a few questions for any of you french speakers out there. First, what voice is generally used 'nous' or 'on'. Second when I am telling the reader to perform this or that mathematical procedure, do I use...

Homework Statement Prove or falsify the statement (see picture) The Attempt at a Solution I've got the answer already but I want to make sure I know is what is meant by f(x)>g(x) for x in [a,b]. Does it mean f(x) lies above g(x) throughout the entire interval?

I'm currently in the middle of the Calc I-III string and I was wondering how much time I should dedicate to studying the proofs, if any time at all. I'm a physics major but I do plan on going on in math after I've taken all of the general curricula because I intend to pursue theory.

Homework Statement Let V and W be vector spaces and T: V-> W be linear. a) Prove that T is one to one if and only if T carries linearly independent subsets of V onto linearly independent subsets of W. b) Suppose that T is one to one and that S is a subset of V. Prove that S is linearly...

Firstly, I find the math syntax on this board incredibly difficult to use, so bear with me. Using any symbol makes the text appear on the next line... I don't know if it is my browser, or what, but I tried to make due. Sorry. Homework Statement Construct an "epsilon minus delta" proof for...

Homework Statement Prove the Identity (show how the derivatives are the same): arcsin ((x - 1)/(x + 1)) = 2arctan (sqr(x) - pi/2) Homework Equations d/dx (arcsin x) = 1/ sqr(1 - x2) d/dx (arctan x) = 1/ (1 + x2) All my attempts have been messy and it may be because I didn't...

I would like to start getting familiar with doing proofs, and I was wondering if someone could give me a good start. I am starting my "collection" in a sort of math notebook. Right now this is extra-curricular from my studies so I don't have time for anything complex. I would just like to...

I have problem with prooving those two identities. Any help would be much appriciated! Show that: a) \begin{Bmatrix} m+n+1\\ m \end{Bmatrix} = \sum_{k=0}^{m} k \begin{Bmatrix} n+k\\k \end{Bmatrix} b) \sum_{k=0}^{n} \begin{pmatrix} n\\k \end{pmatrix}...

I'm reading Calculus on Manifolds by Munkres and I often encounter multiple page proofs that are very technical. I can verify the argument in a reasonable amount of time, but to actually digest the proof (i.e. learn it such that I can reproduce it by memory weeks later) takes an inordinate...

Homework Statement I need help with trigonometry proofs. the question asks me to prove the following and show all my steps. Prove that: Tan(x) – ½sin(2x) = tan(x)sin2(x)Homework Equations I don't know :( The Attempt at a Solution No attempt as I don't get it.Any help at all would be...

Don't get how to write proofs! I'm a high school student who really wants to major in mathematics. I love reading proofs, but when the book [What is Mathematics by Courant] asks me to do proofs, I have absolutely no idea of where to start. Should I just give up my aspiration to major in math...

I made two attempts at proofs. I feel the second one is ok, but the first one feels lacking. I'm not sure if I could represent it in a better way. Homework Statement Prove the following statements Homework Equations a) If x is real, and x > 1, then x^n > 1 b) If x is real, and x...

Homework Statement For every ϕ in Aut(G), ϕ(Z(G))= Z(G). Homework Equations Z(G):={g in G| gh=hg for all h in G} The Attempt at a Solution I haven't made too much progress on this one. I know that if I let g be an element of Z(G) that I need to prove that For every ϕ(g) is also...

1) logba + logcb + logac = 1/logab + 1/logbc + 1/logca 2) logrp = q and logqr = p, show logqp = pq 3) if u = log9x, find in terms of u, logx81 4) log5x = 16logx5, solve for x attempt I know the change of base formula logax = logbx/logba, but I'm not sure if/how to apply it in any...

Hello. Where can I find proofs for the solution of d. equations? I can find the solutions but I cannot find the proofs in any textbook. Specifically, how can I prove the solution for: y''+ay=0 y''+ay'+by=0 Thank you.

[b]1. First one is (sin2x+sinx)/(cos2x+cosx+1)=tanx Second one is (sec∂-tan∂)²=(1-sin∂)/(1+sin∂) [b]2. Sec=1/cos tan=sin/cos cos²x+sin²x=1 [b]3. 1. I think eventually the sinx/cosx need to cancel to make tanx and the 1 could be used to create a lot of options 2. I have tried to...

What's the most Beautiful proof of a mathematical theorem you've seen?

Prove that a sequence of uniformly convergent bounded functions is uniformly bounded. Attempt at proof: So first we observe the following: ||fn||\leqMn. Each function is bounded. Also, |fn-f|\leq\epsilon for all n \geq N. First off, we observe that for finitely many fn's, we have them...

Homework Statement Show that the operation of taking the gradient of a function has the given property. Assume that u and v are differentiable functions of x and y and that a, b are constants. Homework Equations Δ = gradient vector 1) Δ(u/v) = vΔu - uΔv / v^2 2) Δu^n = nu^(n-1)Δu...

Please Help! proofs using contrapositive or contradiction Homework Statement Prove using contrapositive or contradiction: For all r,s∈R,if r and s are positive,then √r+ √s≠ √(r+s)

I'm not using the suggested formatting because my question doesn't fit it. If this belongs elsewhere, I apologize for posting it here. I am currently in Calc II and we are starting to prove certain things. I found out that I really enjoy this and want to do a few proofs a week just for fun...

I'm not quite sure if this is the correct subforum. I was wondering if anybody knew where I could find some decent real analysis notes or lectures online, specifically on the formal definition of a limit. My prof is great, I just missed the class and the textbook and notes aren't quite making...

I was accepted into a top tier Ph.D. Operations Research program. I have six months to prepare independently on my own (at home). Everybody told me real analysis is the first thing I should look at (which makes sense, because I don't have proof experience). Can you please recommend me a book...

Looking for a little advice regarding proving things in mathematical way. I am a physics major currently taking a math methods course where we are asked to prove things, basically for the time in my schooling career. Sometimes I have trouble formulating a mathematically rigorous way of...

Any resources to learn how to do proofs or view abstract math better in linear algebra? A lot of time when I read the solution to proof questions, I don't even see how that proves the statement. The way they are written seems unintuitive. This is my first abstract math course. I've never had...

Hi, Could you clarify the relationship between proofs that use ≤ and those that use <? For example, if it's already proven that "abs(b) ≤ a if and only if -a≤ b≤a" can we say this implies that "abs(b) < a if and only if -a< b<a"? It seems that since the first statement holds for all abs(b)...

URGENT Field Proofs help. I need to prove the following: 1) Prove that if x, y are elements of a field, and X x
Y = 0 then either x = 0 or y = 0 . Write a detailed solution. and mention which of the eld axioms you are using. 2) Let F be a field in which 1 + 1 = 0 . Prove that for any...

Hi I'm a math layman so please be simple and straightforward. Thanks. Are axioms/postulates in any field always so self-evident that they don't need any proofs? I could say '1+1=2' is an axiom, am I allowed to say this, or are there some requirements for an axiom to qualify as one? Could...

Homework Statement [PLAIN]http://img641.imageshack.us/img641/2494/mathg.png I've worked through it and at the 1st step I get: (1/-e+1)<x<(1/e+1) How do they have (1/e+1)<x<(1/-e+1)? Do you switch the signs of an inequality when you take the inverse of both sides?

Homework Statement a.) Given \delta_n=\frac{ne^{-{n^2}{x^2}}}{\pi} Show: x{\frac{d}{dt}\delta_n}=-\delta_n b.) For the finite interval (\pi,-\pi) expand the dirac delta function \delta(x-t) in sines and cosines, sinnx, cosnx, n=1,2,3... They are not orthogonal, they are normalized to...

Hi all, I have taken Calc III, Linear Algebra (Bretscher's book), and an ODE class, which have all been mostly computational. I plan on taking upper level math courses such as abstract algebra and analysis, and my understanding is that the latter are proof based rather than computational. Are...

A Free Book "Proofs and Types" http://www.mpi-sws.org/~dreyer/tor/papers/girard.pdf Proofs and Types Jean-Yves Girard [FONT="Franklin Gothic Medium"]Translated and with appendices by: Paul Taylor Yves Lafont Cambridge University Press New York Melbourne New Rochelle Sydney...

It looks like memorization plays a key component of recognizing/remembering when to use certain "rewriting" tricks to get the desired result in the string of deduction. I had to think for a minute about some of the equations involving absolute values...