Proof Definition and 999 Threads

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. K

    Proof -- motion under a central force in text Symon Mechanics

    1. The derivation In a 3-dim space,a particle is acted by a central force(the center of the force fixed in the origin) .we now take the motion entirely in the xy-plane and write the equations of the motion in polar coordinate how can i derive from these equation that T(kinetic...
  2. Valour549

    A Trying an alternate Proof of the Fundamental Theorem

    The proofs of the Fundamental Theorem of Calculus in the textbook I'm reading and those that I have found online, basically show us: 1) That when we apply the definition of the derivative to the integral of f (say F) below, we get f back. F(x) = \int_a^x f(t) dt 2) That any definite integral...
  3. RoboNerd

    Question on showing general formula of solution

    Homework Statement show that the general solution of the differential equation d^2/dt^2 + 2 *alpha * dr/dt + omega^2 * r = 0, where alpha and w are constant and R is a function of time "t" is R = e^(-alpha * t) * [ C1*sin( sqrt(omega^2 - alpha^2) * t) + C2*cos( sqrt(omega^2 - alpha^2) * t)...
  4. T

    I I'm having trouble seeing the big picture of this proof.

    I don't see how it proves that (n-r+1, r) is the number of r-combinations of X which contain no consecutive integers.
  5. Mr Davis 97

    I Proving propositions involving "if and only if"

    I am usually pretty good about interpreting what a question is asking when it is in the form, "prove that if p, then q," where p and q are statements. However, I cannot seem to understand how to interpret when it is in the form "prove that p if and only if q." The statement I am working with...
  6. SuperSusanoo

    Proof about symmetric groups and generators

    Homework Statement Let n>=2 n is natural and set x=(1,2,3,...,n) and y=(1,2). Show that Sym(n)=<x,y> Homework EquationsThe Attempt at a Solution Approach: Induction Proof: Base case n=2 x=(1,2) y=(1,2) Sym(2)={Id,(1,2)} (1,2)=x and Id=xy so base case holds Inductive step assume...
  7. P

    Proof of Kinematics Equation: Eliminating Time from the Equation

    Homework Statement Eliminate t from the equation (x-xi)=vi(t)+1/2(a)t^2 using the kinematic equation v=vi+at to get v^2=vi^2+2a(x-xi) The Attempt at a Solution I wind up with (x-xi)=vi(v-vi/a) + 1/2(v^2-vi^2/a). If the first term on the right side didn't exist, I could see what the solution...
  8. Mr Davis 97

    B Proof that exterior angles of a triangle sum to 360

    So I am working on this simple proof, but am confused about the term "external angle." The problem says that if ##a##, ##b##, and ##c## are external angles to a triangle, then ##a + b + c = 360##. However, is seems that the vertex of each triangle has two possible external angles, since there...
  9. Mr Davis 97

    B Verification of correct solution to quadratic problem

    The problem statement: Show that if ##r_1## and ##r_2## are the distinct real roots of ##x^2 + px + 8 = 0##, then ##r_1 + r_2 > 4 \sqrt{2}##. We start by noting that ##r_1 r_2 = 8##. Using this relation, we'll find the minimum value of ##r_1 + r_2##. To minimize ##r_1 + r_2##, we need to...
  10. KT KIM

    Proof of angular momentum conservation

    This is from text [Introduction to Lagrangian and Hamiltonian Mechanics] on NTNU opencourse. Annnnd... I don't use english as my primary language, so sorry for poor sentences. I can't get two things in here. First, at (1.12) I can't understand how L dot derivated like that. Since I know...
  11. Twigg

    Proof and Examples of Conservation of Etendue

    Hi all, Can you guys provide a proof of the conservation of etendue (simple/memorable is preferred, if possible!) and a few realistic, practical examples just so I can get the hang of the ideas and the calculations? Much appreciated.
  12. H

    [Statistics] Factorisation theorem proof

    Hello. I have a question about a step in the factorization theorem demonstration. 1. Homework Statement Here is the theorem (begins end of page 1), it is not my course but I have almost the same demonstration : http://math.arizona.edu/~jwatkins/sufficiency.pdf Screenshot of it: Homework...
  13. F

    MHB Proof Quest: Non-Equilateral Triangle Enclosing Point Z

    I am struggling with this question, it would be easy enough if the triangle was equilateral but that is not necessarily the case. Let (ha, hb, hc) be heights in the triangle ABC, and let Z be a point inside the triangle. Further to this, consider the points P, Q, R on the sides AB, BC and AC...
  14. M

    MHB Geometry proof Mid point theorem

    Hi,I have been stuck on this problem The midpoints of the sides AB and AC of the triangle ABC are P and Q respectively. BQ produced and the straight line through A drawn parallel to PQ meet at R. Draw a figure with this information marked on it and prove that, area of ABCR = 8 x area of APQ. I...
  15. M

    MHB Problem on Parallelogram proof

    Can anyone solve the following question Please try to make you answer detail as possible :)
  16. S

    Would this be sufficient as a proof?

    Homework Statement 2. Homework Equations Vieta's formula, quadratic discriminant 3. The Attempt at a Solution
  17. person_random_normal

    I Understanding Ideal Gas Equation in Arbitrarily Shaped Containers

    While deriving ideal gas equation - we take gas molecules to be contained in a cubical container (convinent shape) , but how do we derive it for a gas inside some arbitarily shaped container ? i think this has 2 answers 1) Using maths - but it will be mostly impossible 2) or it will be a...
  18. PhysicsBoyMan

    B Is 2ab Always Less Than or Equal to a² + b² for All Integers a and b?

    a and b are integers Prove that: 2ab <= a2 + b2 I have tested various values for a and b and determined that the statement seems to be generally true. I'm having a hard time though constructing a formal proof. It will not do to suppose the statement is wrong and then provide a counterexample...
  19. Derek Hart

    Adequate proof? Spivak's Calculus ; Dense sets

    Homework Statement Let A be a dense set**. Prove that if f is continuous and f(x) = 0 for all x in A, then f(x) = 0 for all x. **A dense set is defined, in the book, as a set which contains a point in every open interval, such as the set of all irrational or all rational numbers.Homework...
  20. parshyaa

    Is there a mathematical proof for the existence of black holes?

    Is there a mathematical proof for the existence of black hole.
  21. M

    B Is the intersection of two planes a line?

    This is not a homework question. School year has ended for me and I'm doing some revision on my own. I want to proof the following because in an exercise I had to find the equation of the line that passed through a given point and 2 given lines. If a line r intersects with 2 given crossing...
  22. K

    Is the Proof for Normalization in Quantum Mechanics Valid?

    Homework Statement In Griffiths Introduction to Quantum Mechanics textbook, he shows that for any wave function that is time-dependent (which implies that the state of any particle evolves with time), the wave function will stay normalized for all future time. There is a step in the proof that...
  23. Zeeree

    Epsilon-delta proof for limits (multivariable)

    Homework Statement : the question wants me to prove that the limit of f(x,y) as x approaches 1.3 and y approaches -1 is (3.3, 4.4, 0.3). f(x,y) is defined as (2y2+x, -2x+7, x+y). [/B] The attempt at a solution: This is the solution my lecturer has given. it's not very neat, sorry...
  24. K

    I Understanding the Heine Borel Theorem: An In-Depth Analysis

    Hello, I have a question about Heine Borel Theorem. First, I am not sure why we have to show "gamma=Beta" gamma is the supremum of F(which is equivalent to H_squiggly_bar in the text ), and it has to be greater than beta. Otherwise, S contains H_squiggly_barSecond, for the case 1, why...
  25. A

    I Solving Mass-Speed Relation Proof Challenges

    I have a hard time understanding the variation of mass with velocity, more precisely the proof. In almost every material I've found, the author analyses 2 bodies colliding. The idea of looking at the collision is not hard to grasp and by considering one of the velocities equal zero, you get a...
  26. K

    MHB Proof: K is a Root Field for Every Irreducible Polynomial with a Root in K

    Suppose [K:F]=n, where K is a root field over F. Prove K is a root field over F of every irreducible polynomial of degree n in F[x] having a root in K. I don't believe my solution to this problem because I 'prove' the stronger statement: "K is a root field over F for every irreducible...
  27. JasMath33

    A Is This Proof Correct? Ask & Discuss Here

    I was reading this book yesterday and looking at this proof/justification. I was thinking it is possibly incorrect, but wanted to get some other opinions. Here is the example they gave in the book with the work attached.
  28. Alfreds9

    B Visual proof of being at altitude?

    Hi, this may seem like an odd questions to most of you but I'd still like to ask what could be some visual proofs of being at high altitude, say 10,000 feet above sea level. While any said proof is not extremely rigorous or untamperable and probably little more than a showy capture to add to...
  29. EternusVia

    How Does Sum of Consecutive Numbers Relate to Cubes of Integers?

    Homework Statement Consider the table: 1 = 0 + 1 2 + 3 + 4 = 1 + 8 5 + 6 + 7 + 8 + 9 = 8 + 27 10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64 Guess the general law suggested by these examples, express it in suitable mathematical notation, and prove it. Homework Equations [/B] It's clear that if...
  30. P

    Tough geometry problem about triangles, proof

    Homework Statement let be ABC a generic triangle, build on each side of the triangle an equilater triangle, proof that the triangle having as vertices the centers of the equilaters triangles is equilater Homework Equations sum of internal angles in a triangle is 180, rules about congruency in...
  31. Math Amateur

    The Weak Nullstellensatz .... aspects of proof by Cox et al

    Homework Statement I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition) by David Cox, John Little and Donal O'Shea ... ... I am currently...
  32. Math Amateur

    MHB Understand Theorem 1: Weak Nullstellensatz Proof by Cox et al - Exercise 3(a)

    I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition) by David Cox, John Little and Donal O'Shea ... ... I am currently focused on Chapter 4...
  33. SamRoss

    I Proof of double angle formulas using Euler's equation

    Hi all, I'm slowly working through "Mathematical Methods in the Physical Sciences" by Mary Boas, which I highly recommend, and I'm stumped on one of the questions. The problem is to prove the double angle formulas sin (2Θ)=2sinΘcosΘ and cos(2Θ)=cos2Θ-sin2Θ by using Euler's formula (raised to...
  34. JasMath33

    Help Solve Calculus Limit Proof Homework Statement

    Homework Statement I am posting this for another student who I noticed did not have the proof in the problem. Here is what she said. Let's try and help her out. I have been working on the problem below and I am stuck. I am stuck primarily because of the part where is says x=0. If x-0, it...
  35. kyphysics

    Why Do People Say Accounting is Recession Proof? And is it?

    Very curious. Is there a supply and demand imbalance? When there is a recession and businesses are stagnant and new ones aren't starting up, how do accountants still get good work?
  36. D

    I Proving Isometries: A Step-By-Step Guide

    Hello, i'm trying to prove this statements, but I'm stuck. Be ##V=R^n## furnished with the standard inner product and the standard basis S. And let W ##\subseteq## V be a subspace of V and let ##W^\bot## be the orthogonal complement. a) Show that there is exactly one linear map ##\Phi:V...
  37. J

    Help with Epsilon Delta Proof of Multivariable Limit

    Homework Statement Hey guys. I am having a little trouble answering this question. I am teaching myself calc 3 and am a little confused here (and thus can't ask a teacher). I need to find the limit as (x,y) approaches (0,1) of f(x,y) when f(x,y)=(xy-x)/(x^2+y^2-2y+1). Homework Equations...
  38. J

    I Help With Epsilon Delta Proof of multivariable limit

    I realized I placed this in the wrong forum... I will put it in coursework help
  39. A

    Is there any absolute proof that photons exist?

    I mean, look this stupidity: [Mentor's note - link to crackpot site deleted] This guy denies that light photons exist, and that we are 'magically creating it' like cyclops X-Men This is worst than flat-earthers, I wonder If there is some evidence or is it unfalsifiable, like solipsism? Because I...
  40. R

    Deriving the Fourier Transform of the Signum Function and Proving its Properties

    Homework Statement The signum function is defined by$$sgn(t)=\left\{\begin{matrix}-1, \ t<0\\0, \ t=0 \\ 1, \ t>0 \end{matrix}\right.$$It has derivative$$\frac{d}{dt} sign(t) = 2 \delta(t)$$Use this result to show that ##j2\pi \nu S(\nu)=2,## and give an argument why ##S(0)=0.## Where...
  41. F

    B How Do You Prove This Vector Calculus Identity from Turbulence?

    The following identity is found in a book on Turbulence: Can someone provide a proof of this identity? It isn't listed in the list of vector calculus identities on Wiki. Thanks
  42. Q

    I What is the proof for 1+4+9+16+....=0?

    Hello, I started to learn divergent series/sums, to practice I calculated some basic ones, you know: 1+2+3+4+5+6...= -1/12, but I really had problems when i tried to demonstrate that 1+4+9+16+...= 0(the sum of squares of natural numbers), I've tried to add, subtract etc, but I couldn't prove it...
  43. F

    A Proof for Close Packing of Congruent Identical Spheres

    I developed two algorithms for calculating the density of close packed congruent identical spheres in two different arrangements: A tetrahedron with four equilateral triangular faces, and A square pyramid with a square base and four equilateral triangular faces, as shown below. Figure...
  44. S

    MLE of Bivariate Vector Random Variable: Proof & Explanation

    Homework Statement Consider the bivariate vector random variable ##(X,Y)^T## which has the probability density function $$f_{X,Y}(x,y) = \theta xe^{-x(y+\theta)}, \quad x\geq 0, y\geq 0 \; \; \text{and} \; \; \theta > 0.$$ I have shown that the marginal distribution of ##X## is ##f_X(x|\theta)...
  45. Ma Xie Er

    A Why Is the Equality in This Spectral Analysis Proof Correct?

    I'm reading "Time Series Analysis and Its Applications with R examples", 3rd edition, by Shumway and Stoffer, and I don't really understand a proof. This is not for homework, just my own edification. It goes like this: Σt=1n cos2(2πtj/n) = ¼ ∑t=1n (e2πitj/n - e2πitj/n)2 = ¼∑t=1ne4πtj/n + 1 + 1...
  46. ubergewehr273

    Proving 1/3 < log_{34} 5 < 1/2

    Homework Statement Prove that ##{1/3} < log_{34} 5 < {1/ 2}## Homework Equations ##log_b a = {1/ log_a b}## ##logmn = logm + logn##The Attempt at a Solution ##log_{34} 5 = {1/ log_5 34}## ##= 1/(log_5 17 + log_5 2)## ##=1/(1 + log_5 3 + log_5 2 + something)##...
  47. Q

    I Why is E(t) multiplied by e^(-ix) in Plancherel's Theorem proof?

    the first step of the Plancherel's Theorem proof found in: http://mathworld.wolfram.com/PlancherelsTheorem.html, says: let be a function that is sufficiently smooth and that decays sufficiently quickly near infinity so that its integrals exist. Further, let and be FT pairs so that...
  48. V

    I Prove a(b-c)=ab-ac: Is It Enough?

    I am currently working my way though Calculus by Tom Apostol. One of the really early proofs ask the reader to prove: a(b-c)=ab-ac. Here is what I did, I let x=b-c which by the definition of subtraction equals x+c=b. Substituting that value into the right hand side I got...
  49. G

    Proof of image formation property of spherical mirrors

    Hi. I'm trying to proof the image formation property of a concave spherical mirror. I know you can do this easily with a particular choice of rays (namely one that hits the vertex and one that passes through the center of the sphere) but I would like to show that a generic ray yields the same...
  50. T

    Proving theorem for polynomials

    Homework Statement Prove the following statement: Let f be a polynomial, which can be written in the form fix) = a(n)X^(n) + a(n-1)X^(n-1) + • • • + a0 and also in the form fix) = b(n)X^(n) + b(n-1)X^(n-1) + • • • + b0 Prove that a(i)=b(i) for all i=0,1,2,...,n-1,n Homework Equations 3. The...
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