Proofs Definition and 699 Threads

This is a famous proof that utilizes a common notion. Theorem. Limits are unique. let n>N_1 such that blah blah blah is less than epsilon over 2, let n>N_2 such that blah blah blah is less than epsilon over 2. For n> max{N_1,N_2}, blah blah blah < blah = epsilon...

hey everyone: Use a definition to work forward from each of the following statements. b. for functions f anf g the function f + g is convex, where f + g is the function whoes value at any point x is f(x) + g(x). Definition of a convex function...

I've also posted this in the Math forum as it is math as well. --- I want to know if I'm on the right track here. I'm asked to prove the following. a) \nabla \cdot (\vec{A} \times \vec{B}) = \vec{B} \cdot (\nabla \times \vec{A}) - \vec{A} \cdot (\nabla \times \vec{B}) b) \nabla...

I've also posted this in the Physics forum as it applies to some physical aspects as well. --- I want to know if I'm on the right track here. I'm asked to prove the following. a) \nabla \cdot (\vec{A} \times \vec{B}) = \vec{B} \cdot (\nabla \times \vec{A}) - \vec{A} \cdot (\nabla \times...

Homework Statement If x and y are arbitrary real numbers with x<y, prove that there is at least one real z satisfying x<z<y Homework Equations The Attempt at a Solution The problem arises from my inexperience in rigorously proving anything. If possible a general explanation of...

I'm not sure if it goes here or the section beyond calculus, so I'm just putting it here because it doesn't involve any calculus. Homework Statement Suppose that (a,b)=1 [Greatest Common Divisor=1] and (a,c)=1. Does (bc, a)=1? Homework Equations (a,b)=d=au+bv, where u and v are...

I have a 23 problem assignment due at the end of the week, and although I'm going to have a chance to talk to my teacher about the questions I have, I'd like to go ahead and get going on the problems. I've successfully completed 21 of them, but the last two are stumping me. I'm submitting them...

Hey everybody... I have a few quick questions concerning sets and functions for the experts... I've been trained in applied mathematics, so I'm not really used to this kind of formalism. 1. Can somebody look at my "proposed proof" of this elementary theorem for me? I have a feeling that it...

Philosophy of basic set theory proofs involving "or". Hey! I'm working through an Introduction to Analysis text, and I'm currently on the first chapter, which covers set theory. In one of the end-of-chapter problems, I'm asked to prove a basic theorem which leads to the following statement...

Hi, I finished calculus 1 in college this past year, and I was reviewing it in the summer to make sure I understand it and have a solid foundation for when continue taking math classes this upcoming year. My math has always been lacking a little from my high school past where I never paid...

My adviser asked me to study the first 50 pages of a book so I'm working the exercises. But they are all proofs, so I have no idea if I'm doing them correctly. I can't find any answers online, and even if I did, that would just tell me one way of proving it-- and there are, of course, many...

i've texed up three proofs in from elementary topology. can someone please check them? actually i'll just retype them here for convenience 8.2.5 Let f: X_{\tau} \rightarrow Y_{\nu} be continuous and injective. Also let Y_{\nu} be Hausdorff. Prove : X_{\tau} is Hausdorff...

Homework Statement For U(w)=sqrt(w), prove that U(pie(x)+(1-pie)y) > pie*U(x)+(1-pie)U(y) Homework Equations sqrt(x)=x^(1/2) The Attempt at a Solution I have: sqrt(pie(x)+(1-pie)y) > pie*sqrt(x)+(1-pie)sqrt(y) so... (pie(x)+(1-pie)y)^(1/2) > pie*(x)^1/2+(1-pie)(y)^1/2...

Hey Im trying to study abstract algebra, set theory and group theory, on my own. I have trouble understanding how to construct mathematical proofs though. Some of the things the excercises tells me to prove, seems so intuitively clear and obvious that I don't know what's left to prove. For...

I've been working on these problems and unfortunately i can't make heads or tails of these two. Any insight where to start the proof would be great. 1)Let A, B and C be sets. Show that if A~B⊆C, then A~C⊆B holds. What I got so far: Is it correct to state that A~B = A⋂B' and A~C = A⋂C'...

Would taking an intro to logic course help me prepare for the abstract proof writing skills that I'll need in upper division math?

In my self-study Calculus book I finished with the 'intuitive' definition of the limit and the text directed me to the 'formal' definition of the limit. After reading the section covering it a few times I think I comprehended the details of the rigorous rules dictating it - but obviously not...

Homework Statement I am having trouble understanding the proofs in Marion and Thorton [Newest Edition]. The section where he goes through proof of products in tensor notation. An example is page 26 example 1.6. I don't get the switching of the indices on the very last part. Also can someone...

I'm trying to understand \epsilon-\delta proofs, but I'm having some trouble. For example, if we want to prove that \lim_{x\rightarrow2}x^3=8, starting from |x^3-8| we get to something like |x-2||x^2+2x+4| And this is what confuses me: we conjecture that |x-2|<1, then |x|<3, so we get...

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[SOLVED] Couple of Proofs (Regular Induction / Well Ordering) Hi there everyone, I've been having a bit of trouble of solving these questions, so any help would be greatly appreciated: Homework Statement 1: Prove, via regular induction, that it is possible to draw a line-segment of length...

Hi First of all, I would like to mention that I can do proofs that involve algebraic manipulations (in a field i.e.) pretty well, or proofs that involve epsilon-delta arguments or mathematical induction. However, at the moment I'm reading "Principles of mathematical analysis" and I have a hard...

I searched around and I found some books on how to write proofs. There are so many of them that got good review and I have no idea which to choose. Here are some books I am considering: How to Solve It, by Polya An Introduction For Mathematical Reasoning, by Eccles The Nuts and Bolts of...

Homework Statement I need to prove two things about the Catalan numbers. The first is that Cn is odd iff n=(2^k)-1 for some positive integer k. The second is that given the matrix A defined by the rule a(i,j)=C(i+j), prove that det A=1. I have not covered determinants in my linear class...

Homework Statement ......A..... ....../\..... ....../..\... ...../...\.... ..../...\... .../...\... .../...\... ...../...\..... ..../...\.../.\.... .../...\.../...\... .../...\..F./...\... .../...5..x..6...\... .../.../...\...\... .../.../...\...\.. .../.../....\...\...

Homework Statement The problem comes with a diagram but I'll use the wikipedia diagram because it's nice and pretty and I'll just rearrange the letters to suit it. http://upload.wikimedia.org/wikipedia/commons/9/9d/Circle-trig6.svg Just in case the image doesn't load in the page...

Homework Statement 1. A function f(x) is said to be monotonic increasing in A if for all x1, x2 ∈ A, x1≤x2 implies f(x1)≤f(x2). Prove that if f(x) is monotonic increasing in R [f: R→R] and c is a cluster point of R then the limit of f(x) as x→c^{-} exists (might be +∞). 2. s(δ) =...

[SOLVED] Some Compostion Proofs Homework Statement Prove: 1.) The composition of subjective functions is subjective 2.) The composition of injective functions is injective Homework Equations Subjective: A function f: A->B is surjective iff For all members of B, there exists a...

Hi there everyone, I have the basic idea of what to do, its just trying to show the cases work is where the problems occurs. Anyways for the first one: Homework Statement Prove that if x is any positive integer, then ⌈x/2⌉ ≤ (x + 1)/2. (Here, for any real number r, ⌈r⌉ is the smallest...

hello, i am having some trouble with a few induction problems. 1) Prove by induction that for all natural numbers n, n^2 + 3 < 2^n + 5. 2) Prove by induction that for all natural numbers n, (1-1/2)(1-1/4)...(1-1/(2^n)) > or equal to 1/4 + 1/(2^(n+1)). i got started on these but ran into...

Homework Statement Prove 1^3 + 2^3 + ... + n^3 = (1 + 2 + ... + n)^2 for all natural numbers n. Homework Equations The Attempt at a Solution Well, this seems like the typical induction proof, so I start by testing the hypothesis at 1: 1^3 = 1^2 = 1. Then I assume that the...

More Trig Identity Proofs ... Homework Statement 1. cot^2x - 1 = cot2x ----------- 2cotx2. tanx + cotx = 2csc2x3. cos(A+B) = 1-tanAtanB --------- ---------- cos(A-B) 1+tanAtanB Homework Equations The Attempt at a Solution...

im just starting to write proofs and it's going well but some things aren't immediately obvious to me. for example it is not immediately obvious to me why \forall_i ~ p_i \vee q_i \Leftrightarrow (\forall_i p_i ) \vee (\forall_i q_i) isn't a tautology and it wasn't immediately obvious...

Hey guys, I've got two sets of questions here both requiring proofs. Here is a little progress I made with Question Two part c) f=({a,1},{b,1},{c,2},{d,2}) , A={a,c} & B={b,d} f(A)/f(B) = emptyset & f(A/B)={1,2} Any help with the other two parts to Question Two/Three would be great...

Question 1 Let u, v1,v2 ... vn be vectors in R^{n}. Show that if u is orthogonal to v1,v2 ...vn then u is orthogonal to every vector in span{v1,v2...vn} My attempt if u is orthogonal to v1,v2 ...vn then (u.v1)+(u.v2)+...+(u.vn)=0 Let w be a vector in span{v1,v2...vn} therefore...

Homework Statement Question 1: A) Show that if A is diagonalizable then A^{T} is also diagonalizable. The Attempt at a Solution We know that A is diagonalizable if it's similar to a diagonal matrix. So A=PDP^{-1} A^{T}=(PDP^{-1})^{T} which gives A^{T}=(P^{-1})^{T}DP^{T} as...

Question 1: Proove that if λ is an eigenvalue of [A], then 1/λ is an eigenvalue of [A]{T} Question 2 Proove that a square matrices [A] and [A]T have the same Eigenvalues. Question 3: Show that |det(A)| is the product of the absolute values of the eigenvalues of [A]...

I am looking for a method to semi-automate mathematical proofs. Precisely, what I would like is that, if for example I define (sorry, I have never used latex in this forum, and I still do not know how to do it yet): f[A] := {y|\existsx\inA (y=f(x))} then, if I have f[A\cupB] what I...

given x<y and x,y,z are elements of R prove there exists at least one z such that x<z<y. proof: x<z<y -> z>x and y>z by the fact that the reals are unbounded there is definitely at least one z such that z>x now either z>y,z<y, or z=y by the order axioms. so... do i just let z<y...

im just starting to work through vol 1 of apostol and these questions are kind of dumbfounding? #3 on page 15 Let A={1}, B={1,2} Discuss the validity of the following statements (prove the ones that are true) (a) A\subset B (b) A\subseteq B (c) A \in B (d) 1 \in A (e) 1 \subseteq B (f) 1...

What does QED stand for behind every mathematical proof? i can't seem to find out what it stands for thanks.

http://www.geocities.com/asdfasdf23135/advcal15.JPG Well...I have no idea about this question. I don't even know where to start, and I am having terrible panic on these types of proof. Can someone please explain and guide me through? Believe it or not, my textbook (which is horrible) has...

Homework Statement L is the midpoint of line JN, line PJ congruent line QN, line PL congrent to LINe ql, angel pkj and angle omn are ryte angels. prove: triangle PKJ congruent to TRiangle QMN Homework Equations it mite be line segment, because it has a line on top of it.. no arrows...

Homework Statement prove: sin3x = sinx (3-4sin^2x) tanx+sinx/2tanx = cos^2(x/2) cot2x = (cot^2 x-1)/(2cotx)

We all know the standard proof that the square root of two is irrational, and it's easily extended to all integers that are not perfect squares, but It just striked me yesterday that I have only seen one proof (which really is enough, but still =]). One of the lecturers at the University of...

say function f is continuous on (-\infty,\infty). show that f can be written as f = g + h, where g is an even function and h is an odd function. help pleaseee!

All my classes are just profs doing proofs. Great. Too bad the tests requires us to use what we prove to calculate something. For example, my prof spent the first two lectures proving cross and dot product identities (the ones found on the inside covers of many math or physics books). Why...

Is there any experiment that shows the increase in mass ( or kinetic energy ) of a moving body when seen from an observer at rest ? I know that sincrotons ( particles accelerators ) must change the frequency bla bla .. But once that these particles have been accelerated and hit a target...

(a)Let u be a nonzero vector in R^{n}. For all v\epsilonR^{n}, show that proj_{u}(proj_{u}(v)) = proj_{u}(v) and proj_{u}(v - proj_{u}(v)) = \vec{0} (b) An alternate proof of the Cauchy-Schwarz inequality. For v,w \epsilonR^{n}, consider the function q: R -> R defined by q(t) =...

Homework Statement Proof 1: Show that S= {v1, v2, ... vp} is a linearly independent set iff Ax = 0 has only the trivial solution, where the columns of A are composed of the vectors in S. Be sure to state the relationship of the vector x to the vectors in S 2. The attempt at a solution As far...