What is Proof: Definition and 999 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. nmsurobert

    How do I prove Lorentz Invariance using 4-vectors?

    Homework Statement I'm asked to prove that Et - p⋅r = E't' - p'⋅r' Homework Equations t = γ (t' + ux') x = γ (x' + ut') y = y' z = z' E = γ (E' + up'x) px = γ (p'x + uE') py = p'y pz = p'z The Attempt at a Solution Im still trying to figure out 4 vectors. I get close to the solution but I...
  2. S

    I Spivak - Proof of f(x) = c on [a, b]

    In Spivak's Calculus, on page 121 there is this theorem Then he generalizes that theorem: I tried proving theorem 4 on my own, before looking at Spivak's proof. Thus I let c = 0 and then by theorem 1, my proof would be completed. Is this a correct proof? Spivak's proof for theorem 4...
  3. S

    B Proof of a lemma of BÉZOUT’S THEOREM

    Hi, One silly thing is bothering me. As per one lemma, If a, b, and c are positive integers such that gcd(a, b) = 1 and a | bc, then a | c. This is intuitively obvious. i.e. Since GCD is 1 'a' does not divide 'b'. Now, 'a' divides 'bc' so, 'a' divides 'c'. Proved. What is bothering me is ...
  4. S

    Proof involving central acceleration and vector products

    Homework Statement Suppose r:R\rightarrow { V }_{ 3 } is a twice-differentiable curve with central acceleration, that is, \ddot { r } is parallel with r. a. Prove N=r\times \dot { r } is constant b. Assuming N\neq 0, prove that r lies in the plane through the origin with normal N. Homework...
  5. M

    I Proving the Implication of p and (p -> q) to q without Truth Tables

    Hello everyone! I want to proof that: ##p \land (p \to q) \Rightarrow q## I know this is a quite trivial problem using truth tables, however, I want to do it without it. As I'm learning this myself, is this the correct approach? ##p \land (p \to q)## ##\iff p\land (\neg p \lor q)## ##\iff (p...
  6. T

    B Some trouble understanding this basic inequality proof

    just starting out with proofs... i tried manipulating the right side of the inequality, but i don't see why it's equal to (n+1)!
  7. J

    I Proof Using Rearrangement Inequality

    The Rearrangement Inequality states that for two sequences ##{a_i}## and ##{b_i}##, the sum ##S_n = \sum_{i=1}^n a_ib_i## is maximized if ##a_i## and ##b_i## are similarly arranged. That is, big numbers are paired with big numbers and small numbers are paired with small numbers. The question...
  8. S

    Prove this is a right triangle in a sphere

    Homework Statement Let P be a point on the sphere with center O, the origin, diameter AB, and radius r. Prove the triangle APB is a right triangle Homework Equations |AB|^2 = |AP|^2 + |PB|^2 |AB}^2 = 4r^2 The Attempt at a Solution Not sure if showing the above equations are true is the...
  9. TheSodesa

    Proving two simple matrix product properties

    Homework Statement Let ##A## be an n × p matrix and ##B## be an p × m matrix with the following column vector representation, B = \begin{bmatrix} b_1 , & b_2, & ... & ,b_m \end{bmatrix} Prove that AB = \begin{bmatrix} Ab_1 , & Ab_2, & ... & , Ab_m \end{bmatrix} If ##A## is represented...
  10. binbagsss

    QM Bra & Ket Linear Algebra Hermitian operator proof -- quick question

    Homework Statement Hi, Just watching Susskind's quantum mechanics lecture notes, I have a couple of questions from his third lecture: Homework Equations [/B] 1) At 25:20 he says that ## <A|\hat{H}|A>=<A|\hat{H}|A>^*## [1] ##<=>## ##<B|\hat{H}|A>=<A|\hat{H}|B>^*=## [2] where ##A## and ##B##...
  11. M

    Understanding the Position Vector in Calculus Problems

    Homework Statement With ##\vec{r}## the position vector and ##r## its norm, we define $$ \vec{f} = \frac{\vec{r}}{r^n}.$$ Show that $$ \nabla^2\vec{f} = n(n-3)\frac{\vec{r}}{r^{n+2}}.$$ Homework Equations Basic rules of calculus. The Attempt at a Solution From the definition of...
  12. M

    MHB Proof That Radius of Melting Snowball Decreases Constantly

    A spherical snowball is melting at a rate proportional to its surface area. That is, the rate at which its volume is decreasing at any instant is proportional to its surface area at that instant. (i) Prove that the radius of the snowball is decreasing at a constant rate. can someone help me?
  13. D

    Linear Algebra with Proof by Contradiction

    This is a linear algebra question which I am confused. 1. Homework Statement Prove that "if the union of two subspaces of ##V## is a subspace of ##V##, then one of the subspaces is contained in the other". The Attempt at a Solution Suppose ##U##, ##W## are subspaces of ##V##. ##U \cup W##...
  14. D

    Contradiction Method: Proving Statements Through Contradiction and Supposition

    Proof by contradiction starts by supposing a statement, and then shows the contradiction. 1. Homework Statement Now, there is a statement ##A##. Suppose ##A## is false. It leads to contradiction. So ##A## is true. My question: There are two statements ##A## and ##B##. Suppose ##A## is true...
  15. A

    Proof for the Hermitian operator

    1. Homework Statement prove the following statement: Hello, can someone help me prove this statement A is hermitian and {|Ψi>} is a full set of functions Homework Equations Σ<r|A|s> <s|B|c>[/B]The Attempt at a Solution Since the right term of the equation reminds of the standard deviation, I...
  16. 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...
  17. 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...
  18. 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)...
  19. 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.
  20. 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...
  21. 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...
  22. 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...
  23. 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...
  24. 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...
  25. 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...
  26. 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.
  27. 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...
  28. 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...
  29. 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...
  30. M

    MHB Problem on Parallelogram proof

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

    Would this be sufficient as a proof?

    Homework Statement 2. Homework Equations Vieta's formula, quadratic discriminant 3. The Attempt at a Solution
  32. 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...
  33. PhysicsBoyMan

    B How to construct a proof?

    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...
  34. 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...
  35. parshyaa

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

    Is there a mathematical proof for the existence of black hole.
  36. 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...
  37. 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...
  38. 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...
  39. 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...
  40. 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...
  41. 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...
  42. 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.
  43. 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...
  44. EternusVia

    Induction Problem (Polya)

    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...
  45. 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...
  46. 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...
  47. 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...
  48. 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...
  49. 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...
  50. 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?
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