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. F

    A Is this a correct proof of the Riemann Hypothesis?

    Found an article online detailing a proof of the Riemann Hypothesis: << link deleted by mentor - unacceptable source >>
  2. J

    What is the proof for n(n^4 - 1) = 10Q for some values Q and n being integers?

    For some values Q and n being integers, prove that n(n^4 - 1) = 10Q. So I've tried this with induction, but it gets pretty messy pretty quickly. So I can see that the LHS will be even no matter what, but I'm not sure where to go beyond this.
  3. G

    Checking a proof of a basic property of prime numbers

    Homework Statement Prove: If p is prime and m, n are positive integers such that p divides mn, then either p divides n or p divides m. Is anyone willing to look through this proof and give me comments on the following: a) my reasoning within the strategy I chose (validity, any constraints or...
  4. V

    A Proof of quantum correlation functions

    Reading through David Tong lecture notes on QFT.On pages 76, he gives a proof on correlation functions . See below link: [QFT notes by Tong][1] [1]: http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdfI am following the proof steps to obtain equation (3.95). But several intermediate steps of the...
  5. J

    A Need a Proof that Action-Angle Coordinates are Periodic

    Can someone point me to a proof that Action-Angle coordinates in Hamilton-Jacobi Theory must be periodic. I have looked all over and no one seems to prove it, they just assume it. Thanks.
  6. Dtriction

    I Can this method be used to prove the Collatz Conjecture?

    There is a graph showing n on its x-axis and its total stopping time on its y axis. From here we can see that the points on the graph are not random at all; they have some kind of geometric pattern that is due to the 3x+1 in the odd case and x/2 in the even case. I have seen many attempts to...
  7. A

    Proof of this limit formula for e

    Homework Statement http://prntscr.com/dcfe0u Homework EquationsThe Attempt at a Solution So I'm not really strong in proofs but I think you may be able to do something like this:$$lnL = \frac{ln(1+1/x)}{x}$$ $$lnL = \frac{1/x^2}{1+1/x}$$ and then more simplifying I get something like: $$lnL =...
  8. M

    Is this a valid inductive proof?

    Mentor note: moved to the homework section Claim: all numbers divisible by 4 are divisible by 2. Premise: let p(n) return 4n, I.e., the function covers all numbers divisible by four. Reasoning: Let n equal 1 in the base case P(n) is divisible by 2 P(n+1) is divisible by 2 By induction all...
  9. Z

    I Proof check: S in C Compact implies S is closed and bounded

    I am using Lang's book on complex analysis, i am trying to reprove theorem 4.1 which is a simple theorem: Let Compact(S \in \mathbb{C}) \iff Closed(S) \land Bounded(S) I will show my attempt on one direction of the proof only, before even trying the other direction. Assume S is compact Idea...
  10. H

    I Validity of proof of Cauchy-Schwarz inequality

    Proof: If either x or y is zero, then the inequality |x · y| ≤ | x | | y | is trivially correct because both sides are zero. If neither x nor y is zero, then by x · y = | x | | y | cos θ, |x · y|=| x | | y | cos θ | ≤ | x | | y | since -1 ≤ cos θ ≤ 1 How valid is this a proof of the...
  11. J

    B Simple proof of Bell's theorem

    The thread I wanted to post my question on got closed. Recapitulating: The best (simplest) account I have found to date for the Bell inequality (SPOT stands for Single Photon Orientation Tester): Imagine that each random sequence that comes out of the SPOT detectors is a coded message. When...
  12. G

    I Is Zorn's Lemma Proven? A Closer Look at the Proof

    Salutations, friends from afar. The question I have is mundane, but I felt I should be sure. It is basically to spot the insufficiency in this proof for Zorn's Lemma: If every chain in a partially ordered set M has an upper bound, then M contains a maximal element. Proof: 1. For a set X, take...
  13. H

    Proof that entropy can never decrease

    What is the proof of the fact that for an isolated system, entropy can never decrease?
  14. K

    Math proof: Linear Independence

    Homework Statement How can I show that if a vector (in a vector space V) cannot be written as a linear combination of a linearly independent set of vectors (also in space V) then that vector is linearly independent to the set? Homework Equations To really prove this rigorously it would make...
  15. Nipuna Weerasekara

    How Does the Condition x > -1 Influence the Inequality x² + 1/(x²+1) ≥ 1?

    Homework Statement Let ##x\in \mathbb{R} ## Prove the conditional statement that, if ## x>-1## then ## x^2 + \frac {1}{x^2+1} \geq 1## 2. The attempt at a solution Suppose ## x>-1## is true. Then ## x^2>1## Then ## \frac{1}{2}>\frac {1}{x^2+1}## Then ##x^2+ \frac{1}{2}>x^2+\frac...
  16. Jaroslav

    Prove divisibility, mathematical induction

    I'm still learning English, had to use dictionary and translator, so I'm sorry if its unclear, i will try to explain it more if needed. Homework Statement For n belonging to N when n is even and n > 3, prove that (4^(n-3) + 5^(n-3) + 9) is divisible by 9 Homework Equations 3. The Attempt at...
  17. A

    Mathematical proof for oil drilling

    hi guys, I'm supposed to write a paper and do some research on all aspects of drilling regarding torque and force need to drill an oil well and everything from start to finish and to provide mathematical proof and calculations. I don't know where to start and what are the equations used to...
  18. B

    B Flaw in my proof of something impossible

    Given :- $$g(f(x_1)) = g(f(x_2)) \implies x_1 = x_2$$ Question :- Check whether ##g(x)## is injective or not. Now this is of-course false; counter examples are easy to provide. But I proved that ##g(x)## must be one-one even after knowing the fact it must not. Here is the proof :- Let...
  19. P

    B What's wrong with this proof of sin(i)=0?

    We have, e^(ix)=cosx+isinx So, e^(i*i)=cosi+isini Or e^-1=cosi+isini Or 1/e + 0*i= cosi+isini So, cosi=1/e and sini=0 But that's not the value of sin(i) that I found on the internet. These values are not even satisfying cos^2(x)+sin^2(x)=1. What did I miss?
  20. A

    Torque required to tighten the cap for leak proof

    Hello Anyone, Could you help me in finding the torque req. for a cap to leak proof? My cap (polyproplene) which dia. was 32mm and its detail specs are, thread major dia.- 28.5mm, min. dia. - 26mm, pitch - 3mm, thread angle-45deg which has a EPDM rubber seal placed inside (outer dia 26.5mm &...
  21. bananabandana

    General proof of Arc Length For Parametrised Coodrdinates

    Homework Statement Prove that, given a metric ##g_{ij}## such that ##ds^{2}=g_{ij}dx^{i}dx^{j}##, where ##x^{r} = x^{r}(\lambda)## , we have the following result for the arc length: $$ L(p,q) = \int_{p}^{q} ds = \sqrt{ g_{ij} \frac{dx^{i}}{d \lambda} \frac{ dx^{j}}{d \lambda} } d \lambda $$...
  22. D

    A Proof of expansion of a certain value

    How do I begin proving: sum(k>=1)8/(k^4+4)=pi*coth(pi)-1? I got this from Mathematica. Thanks in advance for any help.
  23. PWiz

    A proof in the Hilbert-style axiom system

    Homework Statement Provide a complete formal proof that ## \vdash ((A \rightarrow B) \rightarrow C) \rightarrow (B \rightarrow C)##. Homework Equations I am only allowed to use modus ponens and these four 'sentential logic' axioms: A1 ## \neg \alpha \rightarrow (\alpha \rightarrow \beta)## A2...
  24. DoobleD

    B Proof / derivation of d'Alembert principle?

    I can't find a derivation of d'Alembert principle. Wikipédia says there is no general proof of it. Same with stackexchange. I find it surprising so I thought I'd come here to check with you guys. D'Alembert principle has indeed no proof ?
  25. N

    Proving Natural Log Proof: ln|1+σx|

    Homework Statement Prove the following statement: ln|1+\sigma x | = \frac{1}{2} ln|1-x^2| + \frac{\sigma}{2} ln| \frac{ |1+x|}{|1-x|} Homework EquationsThe Attempt at a Solution Starting from right to left would be easier: = \frac{1}{2} ln|(1+x)(1-x)| + \frac{\sigma}{2} ln| 1+x| -...
  26. D

    Can the Limit of a Function Exist Despite Contradictory Values?

    Homework Statement Proof that the limit of the function below doesn't exists. limx-->1 1/(x-1)Homework EquationsThe Attempt at a Solution Lets assume that limit L exists. So if (1) 0< |x-1| < δ then (2) |1/(x-1) - L| < ε at the book they gave an example by giving a value...
  27. M

    MHB Proof of Sets: Proving (i) and (ii)

    If $X$ is a set, then the power set $P(X)$ of a set is the set of all subsets of $X$. I need to decide whether the following statements are true or false and prove it: (i) If $Z = X \cup Y$ , then $P(Z) = P(X) \cup P(Y)$. (ii) If $Z = X \cap Y$ , then $P(Z) = P(X) \cap P(Y)$. By examples I...
  28. M

    MHB Proof of Parallelogram ABCD: Midpoint X & Y Show Area $\frac{1}{4}$

    ABCD is a parallelogram . X is the midpoint of AD & Y is the midpoint of BC. Show that the area of $\triangle {ABX}$ is $\frac{1}{4}$ the area of ABCD Can you help me with this proof ? were should i start ? I think It should be by proving $\triangle{DBC} \cong \triangle{DBA} $ using SAS as...
  29. JulienB

    Proof of differentiability for <x,x>

    Homework Statement Hi everybody! I'm struggling to solve the following problem: Let ##< \cdot, \cdot >## be an inner product on the vector space ##X##, and ##|| \cdot ||## is the norm generated by the inner product. Prove that the function ##x \in X \mapsto ||x||^2 \in \mathbb{R}## is...
  30. A

    Solve Set Proof Problems Homework Statement

    Homework Statement Can anyone please help me solve these questions? (1) Prove that (A-B) - (B-C) = A-B (2)Simplify (A-( A N B)) N (B-(ANB)) (3) Simplify ( ( A N ( B U C)) N ( A-B)) N ( B U C') (4)Use element property and algebraic argument to derive the property (A-B) U (B-C) = (A U B) - (B N...
  31. T

    I Motivation and proof behind cross products

    this question is a repost from math stackexchange because that guy worded the question so perfectly the question i really wanted to ask about cross products. *please see image below* as far i can understand, the formula for the cross product is basically that the idea of a cross product is sort...
  32. A

    Prove by Induction: $w_k = w_{k-2} + k$

    Homework Statement Prove by induction $$w_k = w_{k−2} + k$$, for all integers $$k \ge 3, w_1 = 1,w_2 = 2$$ has an explicit formula $$ w_n =\begin{cases} \frac{(n+1)^2}{4}, & \text{if $n$ is odd} \\ \frac n2(\frac n2 + 1), & \text{if $n$ is even} \end{cases}$$ Homework Equations The Attempt...
  33. T

    I Is there a proof for this? n/2

    is there a proof that the number of even/odd permutation matrices of any nxn, where n is greater than 3, is n!/2? basically, i want to understand the derivation of n!/2. thank you!
  34. P

    MHB Is Cantor's second diagonal proof valid?

    Cantor "proved" that if there was a list that purported to include all irrational numbers, then he could find an irrational number that was not on the list. Please consider two scenarios: 1. The list claims to contain all irrationals but doesn't. 2. The list absolutely contains all...
  35. alexmahone

    MHB How do I complete this convergence proof?

    Prove that if a subsequence of a Cauchy sequence converges then so does the original Cauchy sequence. I'm assuming that we're not allowed to use the fact that every Cauchy sequence converges. Here's my attempt: Let $\displaystyle\{s_n\}$ be the original Cauchy sequence. Let $\displaystyle...
  36. M

    I Sum principle proof: discrete mathematics

    Theorem: Let ##A_1, A_2, ..., A_k## be finite, disjunct sets. Then ##|A_1 \cup A_2 \cup \dots \cup A_k| = |A_1| + |A_2| + \dots + |A_k|## I will give the proof my book provides, I don't understand several parts of it. Proof: We have bijections ##f_i: [n_i] \rightarrow A_i## for ##i \in [k]##...
  37. Y

    MHB Formal proof using the deduction theorem

    Hello everyone, I am trying to find a proof for: \[\vdash \left ( \sim \alpha \rightarrow \sim \left ( \sim \alpha \right ) \right )\rightarrow \alpha\] I am using the L inference system, which includes the modus ponens inference rule, and the axioms and statements attached below. That's the...
  38. A

    I Proof to the Expression of Poisson Distribution

    Hello. Given a range of time in which an event can occur an indefinite number of times, we say a random variable X folows a poisson distribution when it follows this statements: X is the number of times an event occurs in an interval and X can take values 0, 1, 2, … The occurrence of one event...
  39. I

    Discrete Math Proof: Necessary Condition for Divisibility by 6

    Homework Statement We have JUST started writing proofs recently, and I am a little bit doubtful in my abilities in doing this, so I just want to verify that my proof actually works. I was expecting this one to be a lot longer since the previous 2 were. I don't see any glaring flaws in it, but...
  40. M

    I Proof that every basis has the same cardinality

    Hello all. I have a question concerning following proof, Lemma 1. http://planetmath.org/allbasesforavectorspacehavethesamecardinalitySo, we suppose that A and B are finite and then we construct a new basis ##B_1## for V by removing an element. So they choose ##a_1 \in A## and add it to...
  41. N

    I How does Bell make this step in his proof?

    From drchinese website http://www.drchinese.com/David/Bell_Compact.pdf on page 406 there are 2 equations at the top. How do you get from the top one to the second one? There is a hint about using (1) but I think it cannot be done. You might be able to do it with other assumptions but I think...
  42. kubaanglin

    Projectile motion equation proof

    Homework Statement Show that the launch angle θ is given by the expression: θ=tan-1(4hmax/R) where hmax is the maximum height in the trajectory and R is the range of the projectile. Homework Equations hmax=vi2sin2(θ)/2g R=vi2sin(2θ)/g The Attempt at a Solution I am trying to understand the...
  43. M

    Proof of A Union of A Intersection B Equals A

    Homework Statement Prove that ##A \cup (A \cap B) = A## Homework Equations In the previous exercise, we proved: Let A, B be sets. Then, the following statements are equivalent: 1) ##A \subseteq B## 2) ##A \cup B = B## 3) ##A \cap B = A## The Attempt at a Solution The proof of ##A \cup (A...
  44. weezy

    Proof of independence of position and velocity

    A particle's position is given by $$r_i=r_i(q_1,q_2,...,q_n,t)$$ So velocity: $$v_i=\frac{dr_i}{dt} = \sum_k \frac{\partial r_i}{\partial q_k}\dot q_k + \frac{\partial r_i}{\partial t} $$ In my book it's given $$\frac{\partial v_i}{\partial \dot q_k} = \frac{\partial r_i}{\partial q_k}$$...
  45. K

    Is the Function f(x) = x^2 Injective?

    Homework Statement Prove that ##f: \mathbb{R}\to\mathbb{R}, f(x) = x^2## is not injective. Homework Equations Definition of an injection: function ##f:A\to B## is an injection if and only if ##\forall a,b \in A, f(a) = f(b) \Rightarrow a = b##. The Attempt at a Solution ##f...
  46. H

    I Proof of convergence & divergence of increasing sequence

    I'm using the book of Jerome Keisler: Elementary calculus an infinitesimal approach. I have trouble understanding the proof of the following theorem. I'm not sure what it means. Theorem: "An increasing sequence <Sn> either converges or diverges to infinity." Proof: Let T be the set of all real...
  47. weezy

    Verifying the Correctness of My Proof

    1. I have to show: 2. Given: 3. My attempt : I just want to verify if what I've done is correct or not. Thanks!
  48. e2m2a

    A Circular reasoning and proof by Contradiction

    I need to understand something about proof by contradiction. Suppose there is an expression "a" and it is known to be equal to expression "b". Furthermore, suppose it is conjectured that expression "c" is also equal to expression "a". This would imply expression "c" is equal to expression...
  49. G

    A Steps in proof for Eotvos' law

    I have purchased an article after recommendation on wikipedia that as far as I am aware proves eotvos law. Here is a quote from wikipedia from this site: https://en.wikipedia.org/wiki/Eötvös_rule: ''John Lennard-Jones and Corner published (1940) a derivation of the equation by means of...
  50. Battlemage!

    Is this a valid proof for n >2^n for all n>3

    Homework Statement Show that n!>2n for all n>3. Homework Equations I will attempt to use induction. The Attempt at a Solution We want to show that n!>2n for all n>3. Consider the case when n=4. 4! = 24 > 2^4 =16. We want to show by way of induction that if the inequality is true for...
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