Proofs Definition and 671 Threads

I don't do well by just reading a proof and internalizing it. I need problems to solve and would LOVE to internalize epsilon delta proofs by practicing 100s of them. It's how I got decent at integrals. It's how anybody gets good at math and music and in general your craft right? I Don't know a...

Hello there. I have been reading G.H. Hardy's book "A Course of Pure Mathematics". It is a fantastic introduction to Analysis. I have no problems with the book so far, however, it does assume some knowledge in number theory. I just want to make sure that the following proofs for properties of...

Hello I'm really confused with this and would appreciate any help. Homework Statement a) Show that the work done on a gas during a quasistatic adiabatic compression is given by: W = \frac{P_f V_f - P_i V_i}{\gamma - 1} b) Show that the work done by a gas during a quasistatic...

These are potential proofs for the discrete math exam on Tuesday. I haven't been able to find proofs online. I have senioritis, and I'm graduating in a few weeks. Is a proof by contraposition the best way to prove this? If you assume h is not a function or g is not a function, then that would...

Hello, I am a college freshman currently taking Real Analysis. Calculus was fairly mechanical, and dare I say it trivial, the concepts were easy to grasp and it required little memorisation. As I have began to study more abstract areas of mathematics, I have found my speed and confidence have...

First post on this forum, that IMO is amazing! I was reading the introduction of the book “A gentle introduction to the art of Mathematics” and I was wondering about what the authors wrote on whom the book is for. In particular he stated that the book is in particular for people who can...

Homework Statement Prove that the functions: (u+v)'(x0) and αu and u*v are derivable. Homework Equations in other words prove that : (u+v)'(x_{0})=u'(x_{0})+v'(x_{0}) (\alpha u)'(x_{0})=\alpha u'(x_{0}) (u\cdot v)'(x_{0})=u'(x_{0})\cdot v(x_{0})+u(x_{0})\cdot v'(x_{0}) The...

I'm a physics major a bit of inclination to mathematics. The semester just ended, and I didn't particularly have a bad one. It's just I had a really mediocre grade after the semester, I'm a bit disappointed since while I'm busy reading through the proofs it seems it didn't really do me much good...

Hello, all :) I was just wondering a few things: 1) what is the difference between pure mathematics and applied mathematics, and which classes do you need to take in order to get your Ph.D. in either subject? 2) I know this is a really large branch off, but I was wondering how do you start...

We used to have a bunch of problems and proofs that were in a pdf could be downloaded by anyone. Since we aren't able to upload pdf files of a certain size, I provided a link to google docs. If there is an error, typo, or something is just drastic wrong let me know. Undgraduate Final Review...

Hey guys, I'm in a proof class right now. We've covered direct proofs and moved on, but I'm still curious about them. Is there any important theorem that has even been derived using a direct proof (assume p to show q) or are they mainly just used to introduce proofs? In class, we only ever...

I am a freshman in High school, however I've been working quite a lot in the field of number theory for quite some time. However, I've been beginning to feel slightly bad that I haven't actually proven anything. It's not like I want to make a brand new theorem, no; but I would like to start to...

There's something I can not understand about proofs in combinatorics. Whenever I solve a counting problem, there's a non-negligible amount of uncertainty about the solution which I really don't feel when I solve problems in other fields, say in analysis or abstract algebra. It happens too often...

I'm reading a math book and found a couple of proofs I can't do. 1. Given x \in R^n, a \in R, \sum\limits_{i=1}^n{x_i}=na, prove that \sum\limits_{i \in A}\prod\limits_{j = 1}^k {x_{i_j}} \leq \binom{k}{n}a^k where A = \{i \in \{1, 2, ... n\}^k : i_1 < i_2 < ... < i_k\} which essentially...

Homework Statement Prove the following: (i) ##|x|-|y| \le |x-y|## and (ii) ##|(|x|-|y|)| \le |x-y|\qquad## (Why does this immediately follow from (i) ?) Homework Equations ##|z| = \sqrt{z^2}## The Attempt at a Solution (i) ##(|x|-|y|)^2 = |x|^2 - 2|x||y| + |y|^2 = x^2 - 2|x||y| + y^2...

Homework Statement Let C be the set of all points (x,y) in the plane satisfying x≥0, y≥0, -x-2y≤-8. a. Show that C is nonempty and unbounded. b. Prove that the LP problem: Max M=2x+3y subject to the constraint that (x,y) lie in C has no feasible, optimal solution. c. Show that the LP...

Homework Statement I am wondering if the general approach to these proofs involving absolute values and inequalities is to do them case-wise? Is that the typical approach (unless pf course you see some 'trick')? For example, I have: Prove that if |x-xo| < ε/2 and Prove that if |y-yo| <...

Hey! I tried to make the title as descriptive as possible, but ran out of characters. I am trying to prove that.. Homework Statement "There exists x \in (1, \infty) such that for all y \in (0,1), xy\geq1. \exists x \in (1, \infty) s.t. \forall y \in (0,1), xy\geq1. Homework...

Homework Statement Let M be a symmetric matrix. The eigenvalues of M are real and further M can be diagonalized using an orthogonal matrix S; that is M can be written as M = S^-1*D*S where D is a diagonal matrix. (a) Prove that the diagonal elements of D are the eigenvalues of M...

After 1-2 years of writing formal math proofs in undergraduate school, I now speak and write much more eloquently than I used to. Now, before uttering or writing a statement, I take a quick pause to ask myself whether it's logically valid; it's unambiguous; it's relevant and sequentially...

Math proofs vs physics "proofs" I'm a senior level physics major interested in taking a 400-level class in the math department for which I do not meet a prereq for (Graph Theory requires Intro to Abstract Math). I emailed the professor, and he stressed to me that a very important part of the...

Homework Statement The proofs: show (A')^-1 = (A^-1)' and (AB)^-1 = B^-1A^-1Homework Equations The Attempt at a Solution for the first one: (A^-1*A) = I (A^-1*A)' = I' = I A'(A^-1)' = I but I am not sure how this proves that a transpose inverse = a inverse transpose... the second i have...

I've been looking at this proof thinking that if I read it over and over that what I am reading that seems so obvious that something else will actually pop out that I am not realizing, but what I realized the proofs that I am reading seem meaningless and pointless. I added a paint doc with...

I'm having trouble with these two proofs. cos(n∏+θ)=(-1)^n cos θ ln|sec x|= -ln|cos x| I know for the first one that I have to incorporate log somehow but that's about all I got from it.

I am a high school senior who is planning to major in math in college. I am currently in a break until the second semester of calculus at a local college starts at the end of January. I took the first half as an AP class at my school last year. I have been going back and reviewing topics from...

Where would be the best place to find every trigonometric identity, from sin[2] + cos[2] = 1, to the matrix identities (and Euler's equation would be helpful, also) Also the location of mathematical analysis symbols would be helpful, also. Thank you very much in advance :)

Homework Statement I'm working on some set theory stuff to prepare for Topology next semester. I'm actually working out of a Topology book from Dover Publications. I could really use some direction/correction. 1. If S ⊂ T, then T - (T - S) = S. 2. If S is any set, then ∅ ⊂ S. The...

Homework Statement This isn't a homework question so I apologize if I'm in the wrong section, but I'm wondering if proofs are 'easy' or 'intuitive' to you. I recently took a linear algebra course in which I was sometimes able to get through the proofs without any trouble but was completely...

Homework Statement Let a be a fixed positive rational number. Choose(and fix) a naural number M > a. a) For any n\inN with n\geqM, show that (a^n)/(n!)\leq((a/M)^(n-M))*(a^M)/(M!) b)Use the previous prblem to show that, given e > 0, there exists an N\inN such that for all n\geqN, (a^n)/(n!)...

Mathmatical proofs help please! [b]1. Must two uncountable sets have the same cardinality? a countable union of countable sets is countable. Is a finite set necessarily countable? If the union of A and B is infinite, then A or B must be inifinte [b]2. Just use definitions of...

If lim x_n=x n to infinity and lim y_n=y n to infinity prove rigorously lim n to infinity (x_n/5+10y_n)=x/5+10y. My attempt let ε>0. Must find n_0 \in \mathbb{N} such that ||(x_n/5+10y_n)-(x/5+10y)||<ε for all n>n_0 ||(x_n/5+10y_n)-(x/5+10y)||=||(x_n/5-x/5)||+||10y_n-10y|| \le...

Suppose that f, g : \mathbb{R} \rightarrow \mathbb{R} are surjective (ie onto functions with domain \mathbb{R} and allowable output values \mathbb{R}). Prove that f \circ g is also surjective (ie, prove f \circ g is also onto). First of all, I have absolutely no math theory experience, so I...

Suppose there's a difficult proof on one of my homework problems in an undergrad course, and suppose I find on the internet a clever, elegant proof whose basic framework I use to construct a slightly modified proof, perhaps with some added explanation (for example, add a "because" or "since"...

Homework Statement (a) Let a and b be integers with gcd(a,b)=d, and assume that ma+nb=d for integers m and n. Show that the solutions in Z to xa+yb=d are exactly x=m+k(b/d), y=n-k(a/d) where k∈Z. (b) Let a and b be integers with gcd(a,b)=d. Show that the equation xa+yb=c...

In all the problems I have done so far, the limit was already given. So the goal is to utilize the theorem to see whether the limit really holds. But what's the point? If we already know how to find the limit, why must we go through a process of ingenuity algebra to tell ourselves, "okay it...

1. Given: Let f: X → Y be a function. Then we have an associated function f-1: P(Y) → P(X), where f-1 (B)⊂X is the inverse image of B⊂Y. Question: Show that f^(-1) is one-to-one if and only if f is onto. [Notes: ⊂ represents subspace, I just couldn’t find a way to put the line under the...

Homework Statement (a) we define the improper integral (over the entire plane R2) I=\int\int_{R^2}e^{-(x^2+y^2)}dA=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dy dx=\lim_{a\rightarrow\infty}\int\int_{D_{a}} e^{-(x^2+y^2)} dA where Da is the disk with radius a and center the...

Homework Statement No matter what positive real number x we choose, there exists some positive real number y such that yz2 > xz + 10 for every positive integer z. Translate the above statement to predicate logic and prove it using a direct approach. Homework Equations I don't...

Homework Statement My intro to Proofs class uses How to Prove It, 2nd edition by Velleman. I would like a couple other references on introduction to proofs. What do you recommend? I don't mind spending hours agonizing over proofs, but I'd like to be able to check my work with answers...

Homework Statement The following is all the information needed: Homework Equations There are, of course, all the basic rules of logic and set identities to be considered. The Attempt at a Solution Not really sure how to attempt this one, to be honest. I know that (A ⊆ B) can...

Homework Statement Let A =[A11 A12; A*12 A22] be Hermitian Positive-definite. Use Cholesky factorizations A11 = L1L*1 A22 = L2L*2 A22-A*12 A-111 A12 = L3L*3 to show the following: ||A22-A*12 A-111 A12||2≤||A||2 Homework Equations The Attempt at a Solution Using the submultiplicative and...

For proofs such as the derivative of cos or sin.. should I study them both analytically and geometrically? By analytically I mean to derive them by algebraic means. Or should I also study the geometrical "intuition" behind it? I love proofs but aren't completely fond of the geometrical...

I've just started Spivak's Calculus and I'm having a few questions concerning the validity of certain of my proofs since some of mine are not the same as the ones in the answer book. Homework Statement Here is one of the proof: I need to prove that (ab)^{-1} = (a)^{-1}(b)^{-1}...

Homework Statement Thanks to everyone who has helped me so far - I'm very grateful. (1) Prove that the multiplicative inverse in any field is unique (2) Prove the cancellation law | ab = ac => b=c (3) Prove (-1)a = -a Homework Equations The field axioms...

ive been trying to understand a few of the identities my professor gave me and i can get a few of them down such as \nabla(\vec{A}\vec{B})=\vec{B}\nabla\vec{A} - \vec{A}\nabla\vec{B} and i can break it down through cartesian and product rules but when i try to do \nabla X (\vec{A}ψ) =...

Homework Statement Dear Mentors, Please guide me in solving the circled questions on the 2 attachments. Homework Equations Thank you The Attempt at a Solution

Dear Mentors, Please guide me in solving the circled questions. Thank you

I was very discouraged when I couldn't do a couple proofs myself in calculus such as the squeeze theorem. My textbook has very little steps into some of the proofs and assumes that the student should infer most of the information. Not being able to follow the proofs made me feel that I hated...

1. prove that if 0<a<b, then a<\sqrt{}ab<a+b/2<b 2. \sqrt{}ab\leq(a+b)/2 holds for all a,b \geq 0 [b]3. Where do I begin? I have no clue! Thank you to anyone who can help!

Homework Statement Q. Prove by induction that... (please see attachment). Homework Equations The Attempt at a Solution The end result should be divisible by 6, but hasn't worked out for me. Can someone help me spot where I've gone wrong? Thank you.