What is Proofs: Definition and 698 Discussions

A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

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  1. C

    Algebraic proofs for probability equations?

    Hello So I'll be going back to school for math next semester, so I might not know the answer to this because I haven't taken combinatorics. I only really know algebra one and two, calc one and some trig. Anyway, currently studying for my cfa, and it's easy enough to plug the formulas in...
  2. K

    Can Trivial Conditions Validate an Implication in Mathematical Proofs?

    Homework Statement Let ##n\in N ##. Prove that if ##|n-1|+|n+1|\leq 1,## then ## |n^{2}-1|\leq 4## Homework Equations The Attempt at a Solution I am trying to show by a counter example that this statement is not true. Consider this statement: ##|n-1|+|n+1| \leq 1## Assume ...
  3. A

    MHB How to Prove Certain Matrix Operations and Understand Their Properties?

    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...
  4. L

    Simple Proofs for Matrix Algebra Properties: A Beginner's Guide

    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...
  5. J

    Number Theory: Why always elementary proofs?

    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!
  6. tom.stoer

    A question about a common pattern in mathematical proofs

    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...
  7. polygamma

    MHB Integral Proofs: $|a| \le \frac{\pi}{2}$ and $|a| \le \pi$

    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 $|a| \le \pi$, $$ \int_{0}^{\infty} \frac{\sin (\pi x) \sin(ax)}{1-x^{2}} \ dx = \frac{\pi}{2} \sin a $$
  8. J

    Explanatory Vs non-exp. proofs for a given theorem. Any examples?

    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.
  9. Ascendant78

    Method for proofs involving vectors and dot products?

    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...
  10. binbagsss

    Basic epsilon and delta proofs, limits, quick questions.

    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...
  11. M

    Understanding Set Relations: Exploring Principles and Notations

    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...
  12. A

    Cantor set ℵ , inductive proofs by openly counting.

    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...
  13. 4

    Is R Transitive if R^2 is a Subset of R?

    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...
  14. T

    What's the point of inductive proofs?

    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...
  15. Mandelbroth

    Lazy Group Proofs and Efficiently Using Categories

    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...
  16. L

    Is x^2 Not Uniformly Continuous on the Real Numbers?

    Homework Statement Show that the function ##x^2## is not uniformly continuous on ##\mathbb{R}## Homework Equations Delta - Epsilon Definition: ##\exists \epsilon > 0, \ \forall \delta >0, \exists x \in S [|x-x_0|< \delta \text{and} |x^2 - x_0^2| \ge \epsilon ].## The Attempt at a...
  17. P

    MHB Proving Singular Matrix and Non-Zero Solutions: A Tutorial

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

    Can you like math without liking proofs?

    (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...
  19. L

    The usefulness of proofs to a physicist: eg The Schwarz Inequality

    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...
  20. TheFerruccio

    Question regarding epsilon-delta proofs.

    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...
  21. Z

    Can a Natural Number Indivisible by 3 Have a Square Divisible by 3?

    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...
  22. J

    Please explain the proofs for sin(90 - θ)

    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...
  23. G

    Starting Out with Proofs: Seeking Advice from Experienced Math Students

    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...
  24. skate_nerd

    MHB Tensor notation for vector product proofs

    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...
  25. C

    Question about trigonometry proofs

    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...
  26. skate_nerd

    MHB Proofs regarding uniform circular motion in vectors

    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...
  27. N

    Limit Proofs: Understanding Epsilon-Delta Proofs for Calculus

    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...
  28. 462chevelle

    Mastering the Art of Proofs: Tips for Struggling Learners

    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...
  29. A

    MHB Using isomorphism and permutations in proofs

    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...
  30. D

    Proofs, derivations, or both? Feel I've learned math/physics wrong

    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...
  31. J

    What is the proof for the sum formula for sines?

    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...
  32. A

    Take Analysis concurrently with Proofs Course? Smart or Stupid Idea?

    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...
  33. A

    How Do Combinatorial Proofs Work?

    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.
  34. 4

    Started learning proofs - need some feedback

    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...
  35. C

    MHB Proving Definition of Continuous Function: Hi, Carla!

    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...
  36. C

    Matrix Proofs Homework: Q1 & Q2

    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...
  37. D

    Status and proofs on Superstring Unified Field Theory ?

    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...
  38. D

    Matrix manipulation/arithmetic in equations for proofs

    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...
  39. V

    Base Case in Strong Induction Proofs

    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)...
  40. T

    Problem with set of inside proofs

    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)...
  41. B

    Where can I find proofs of these theorems?

    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...
  42. SrVishi

    What are proofs in mathematics like?

    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...
  43. I

    Proofs about invertible linear functions

    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...
  44. nomadreid

    Measures beyond Lebesgue: are Solovay's proofs extendible to them?

    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...
  45. I

    MHB Proofs about invertible linear functions

    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...
  46. S

    Proving Statements: Square Roots, Even Numbers, and Multiples of 6

    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...
  47. M

    MHB Set Theory Proofs: A, B, and C - Solving for Set Equality and Complements

    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...
  48. S

    Good introductory book on mathematical proofs?

    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...
  49. T

    MHB How Can You Prove Standard Limits Without Using L'Hopital's Rule?

    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...
  50. B

    Proving Curvature Formulas: V X A / l V3l and a(t) * N(t) / l V(t) I2

    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...
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