Proofs Definition and 671 Threads

Earlier today, I was thinking about the statement that "there exists no greatest natural number" and immediately, two proofs sprang to my mind. Since my question depends on these, I'll write them out below . . . Proof 1: Let n \in \mathbb{N}. Clearly n+1 \in \mathbb{N} and n < n+1. Since...

I'm looking for book about making proof. Is this kind of book even required to understand proofs? Is there some special theory behind proofs, or books about proofs just provide examples, and are more like "math for dummies" ? I'm not sure if it's proper to use that kind of book, should i...

\sum_{n=1}^{\infty} n \sin(\frac{1}{n}) I rewrote the sum as \sum_{n=1}^{\infty} \frac{\sin(\frac{1}{n})}{\frac{1}{n}} Then I applied the Nth term test and used L'Hoptials rule so \lim_{n\to\infty} \frac{\cos(\frac{1}{n})\frac{-1}{n^2}}{\frac{-1}{n^2}} The \frac{-1}{n^2} cancel...

Homework Statement Which of the following sequences have a convergent subsequence? Why? (a) (-2)n (b) (5+(-1)^n)/(2+3n) (c) 2(-1)n Homework Equations Cauchy Sequence Bolzano-Weirstrass Theorem, etc. The Attempt at a Solution (a) The sequence I get is...

Homework Statement Show that for all integers n,m where 0 ≤ m ≤ n The sum from k=m to n of {(nCk)*(kCm)} = (nCm)*2^(n-m) The Attempt at a Solution So for the proof, I have to use a real example, such as choosing committees, binary sequences, giving fruit to kids, etc. I have been...

Homework Statement (Give a combinatorial proof of each of the following identities. In other words, describe a collection of combinatorial objects and then explain two different methods for counting those objects. Leave each identity in the form given. Do not rearrange terms or use any other...

I've got two calculus proofs that I can't seem to get! I was wondering if you guys could help me out a bit... 1. Homework Statement Suppose x_{n} is the sequence defined recursively by x_{1}=0 and x_{n+1}=\sqrt{5 + 2x_{n}} for n=0, 1, 2, 3, ... Prove that x_{n} converges and find...

Hi, I am having trouble understanding this proof. Statement If pn is the nth prime number, then pn \leq 22n-1 Proof: Let us proceed by induction on n, the asserted inequality being clearly true when n=1. As the hypothesis of the induction, we assume n>1 and the result holds for all...

Homework Statement Two problems. (1) Prove that if two homogeneous systems of linear equations in two unknowns have the same solutions, then they are equivalent. (2) Can you prove that the matrix A = [1 1/2 ... 1/n 1/2 1/3 ...1/(n+1) ... 1/n 1/(n+1)...1/(2n-1)]...

As a prospective physics major, I would like to know if physics is as "proof heavy" as math is, outside of the math prerequisites. Thanks!

Does anyone know of any sources which provide a proof, or outline of, the Coleman-Mandula theorem and the Haag-Lopuszanski-Sohnius Theorem?

Hi, would someone be able to check my proofs for me and tell me if they are right and if not what is wrong please? So for the first one I said let u=p(x) and v=b(x) T(u+v)=p(x)+b(x)=p(5)x2+b(5)x2=Tu+Tv and T(ku)=(kp)(x)=kp(5)x2=kTu So it is a linear transformation. For the second I said...

I'm going to fail my first proofs class. How do people even learn this!? A bit of cheese with my whine perhaps but I'm more frustrated because I don't know what to do. I am going to fail this class unless I figure it out before our first exam. I read the chapters, take notes, try to understand...

So I am looking at the following two proofs via induction, but I have not a single idea where to start. The First is: 1. Suppose hat F1=1, F2=1, F3=2, F4=3, F5=5 where Fn is called a Fibonacci number and in general: Fn=Fn-1+Fn-2 for n>/= 3. Prove that: F1+F2+F3+...+Fn=(Fn+2)-1 Secondly...

I have two proofs that I am uneasy about and one I'm having trouble with so hopefully I can figure out where I'm going wrong if I am. Ignore the weird numbers, its to help me organize the problems. 14) Let G be a group with the following property: Whenever a, b and c belong to G and ab=ca, then...

Hi there, I'm having trouble understanding how to prove things using the \delta \epsilon definition. I have read a few other threads and sites, but I can't seem to put it together. For example, I came across this problem, if given limx-->af(x) = L, how would I prove (using delta-epsilon and...

Homework Statement Prove that A \oplus B' \oplus C = (A \oplus B \oplus C)'Homework Equations The Attempt at a Solution I tried to use A \oplus B' \oplus C = ABC' + A'B'C' + A'BC + AB'C But i am not sure how to proceed further from there... Please could someone give me a little bit of help ?

I enjoy math, but when i have to proof something - i find it confucing. I don't know how strict proof should be. If proof ends with 17<18 , it this really end? Why shouldn't i proof that 17<18 is actually true? I never know what sentences can i treat as obvious, and which one i have to proof...

Here are some that I'm stuck on. Pg. 56, #12 Prove by induction on n that, for all positive integers n, 3 divides 4^n + 5 Of course, the base case it is P(1) = (4^1 + 5) / 3 = 9/3 = 3...TRUE! I just can't see the trick here. P(K+1)= (4^(K+1) + 5) / 3 = ((4)(4^K) + 5)/3= ... not...

Homework Statement Let A and B be any sets. 1: Prove A is the disjoint union of A\B and A intersect B. 2: Prove A U B is the disjoint union of A\B, A intersect B, and B\A. Homework Equations ? The Attempt at a Solution I understand most of the basic terminology used. I know disjoint...

Have been revising geometry today and have came across some proofs that I can't seem to find in books, but I can't get through either. Any help would be great. Let A be a 3x3 orthogonal matrix and let x and y be vectors in R^3 a) Show that detA = +/- 1 b) Show that the length of Ax is...

Homework Statement I am home schooled and I am having a really hard time with proofs! I keep going back over stuff and reviewing but every time I just feel more lost than before! Any advice or help?

i know how to do basic proofs, but some proofs on the actual limit theorems confuse me. my textbook's choices for delta are very obscure and i have no idea how they even came up with them. for the proof of the limit theorem where the limit of a product of 2 functions is equal to the product...

Hey, I have been studying differential equations a bit and was wanting some help on some proofs. There are 3 laplace transforms I would like proofs for. Not really sure where to get started or if someone could lead me to place that has these proofs I would greatly appreciate it. Thank you.

Homework Statement Ok so I am stick on three proofs for my linear algebra final adn help on any of all of them would really help with my studying For the first 2 assume that A is an nxn matrix 1.If the collumns of A span Rn then the homogenous system Ax = 0 has only the trivial solution...

let { v_1, v_2, v_3, ... v_m } (m > 3) be linearly independent vectors in a vector space V. Prove that the set { v_1 - 2v_2 + v_3, v_2, v_3, ..., v_m } is also Linearly. I did this: a_1( v_1 - 2v_2 + v_3) + a_2(v_2) + a_3(v_3) + ... + a_m(v_m) = a_1(v_1) - a_1(2v_2) + a_1(v_3)...

I wish to prove \lim_{x \to 0} \frac{\sin x}{x} = 1 using L'Hôpital's rule. The problem with this is, even though the result after applying the rule is 1 (the correct answer), the limit itself was assumed to be correct in order to calculate the derivative of sinx. This constitutes circular...

Homework Statement Let E and F be two subspaces of R^n. Prove the following statements: (n means "intersection") If EnF = {0}, {u1, u2, ..., uk} is a linearly independent set of vectors of E and {v1, v2,...vk} is a linearly independent set of vectors Note: Above zero denotes the...

So lately I've been trying to start practicing proving things without being given a prompt (as in actually finding something out, not answering the question: "prove so and so"), which I'd never done before. -- But then the bummer is not knowing the name of what you've just proved :rolleyes...

1) prove that for any five vectors (x1, ..., x5) in R3 there exist real numbers (c1, ..., c5), not all zero, so that BOTH c1x1+c2x2+c3x3+c4x4+c5x5=0 AND c1+c2+c3+c4+c5=0 2)Let T:R5-->R5 be a linear transformation and x1, x2 & x3 be three non-zero vectors in R5 so that T(x1)=x1...

Just wondering. I haven't been having problems with proofs, so far, but I'm interested in how people think about proofs. I feel I'm still far from ideal. There are some standard proofs, like when proving uniqueness which have all looked the same so far. There are also counting proofs, in which I...

1. Hi! I was wondering if anyone could help me to solve the following problem! Let L : [R][n] ->[R][m] and M :[R][m]-> [R][m] be linear mappings. Prove that if M is invertible, then rank (M o L) = rank (L) thanks! :)

My prof gave us twelve basic properties of numbers, and I think I'm supposed to use those in my proof, but I'm not sure how to incorporate them. The properties are: P1 Associative law for addition P2 Additive identity P3 Additive inverse P4 Commutative law for addition P5 Associative law...

Here's an example that helps illustrate my question: Prove: A sequence in R can have at most one limit. Proof: Assume a sequence {xn}n\inN has two limits a and b. By definition: -For any \epsilon>0, there exists an N\inN such that n\geqN implies that |xn-a| < \epsilon/2. -A...

So how should I approach more difficult math? On a lower math level it is possible to really understand a certain property, relation, formula, ... you can see where it's coming from: intuitive and/or by quickly derivating it in your head. But as the math becomes more difficult, this...

Like many people on this forum, i am seemingly having a lot of trouble grasping the concepts of Epsilon Delta proofs and the logic behind them. I have read the definition and i realize for e>0 there is a d>0 such that... 0<sqrt((x-1)^2 - (y-b)^2) < d then f(x,y) - L <e (excuse my use of...

I really, [SIZE="4"]really suck at proving things. Do you know of any free online resource one could use to master proving things? I know enough math up to about one semester of calculus, a bit about sentential logic and a little bit of set theory, but even still, past finding a trig identity...

What are some classic, good books on learning mathematical language and writing proofs? (to gain facility with mathematical language and method of conjecture, proof and counter example, with emphasis on proofs. Topics: logic, sets, functions and others.)

Homework Statement Statements: 1. Line GB congruent to Line GD 2. Angle BGE congruent to Angle DGE 3. Line GE congruent to Line GE 4. Triangle BGE congruent to Triangle DGE Homework Equations The Attempt at a Solution...

looking for proofs of the following integrals integral ( e^(au)sin(bu)du ) and integral(e^(au)cos(bu)du) and integral (sec^3u du)

Homework Statement The maximum of 2 numbers x and y is denoted by max(x,y) and the minimum of 2 numbers x and y is denoted by min(x,y). Prove that max(x,y) = (x + y + l y - x l) / 2 and min(x,y) = (x + y - ly - xl ) / 2. Homework Equations The Attempt at a Solution Theorem...

Amateur question ahead, be warned. I really dislike proofs that something cannot be done. My first gripe is that they limit the areas we are "allowed to think about", so to speak. But more importantly, I have this feeling that any proof of impossibility is unavoidably flawed because it cannot...

formal proofs, where...? In which fields of maths are formal proofs used often?

I understand most of the logic behind the formal definition of a limit, but I don't understand the the logic behind an epsilon delta proof. The parts I'm having trouble with are these: 1. How does proving that, the distance between the function and the limit is less than epsilon whenever the...

Is there a point to geometric proofs? The ones that I have encountered are so obvious that it almost seems useless to have to use them for anything.

I've noticed in my physics textbooks that every time the author wishes to prove something, he'll go for a direct proof/derivation. Is there any particular reason for this? I think that I have yet to see any proofs by contradiction/proving the contrapositive/mathematical induction in any of my...

Does anyone know of any good precaluclus-level books that are good for learning how to write basic proofs? I have a lot of very nice 1960-early 70s era texts that ask for proofs in the majority of the exercises but the problem is that not only do I not know how to write proofs, but there are no...

1. Let O be an open cover for [a,b] and let x=sup{d|[a,d] can be covered by finitely many elements of O}. Clearly x>a. If x<b or x=b, then there is an element O_1 of O which is a neighborhood of x, and there is an \epsilon>0 such that x-\epsilon is an element of O_1, and since there is a...

I am giving a short presentation on Fermat's polygonal number theorem (any number may be written as the sum of n n-gonal numbers). I need books that provide some exposition/history on the theorem as well as a proof. I acquired Nathanson's Additive Number Theory from my university's library, but...

Homework Statement 1) (AB)^-1=A^-1B^-1 A and B are nonsingular nxn 2) If A is nonsingular then A^-1 is nonsingular also and then (A^-1)^-1=A The Attempt at a Solution 1) I do know that I have to multiply but I don´t know why. Can you tell me why?? This is how I do it...