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. M

    Proof: Twin Primes Always Result in Perfect Squares

    Proof: Suppose ## p ## and ## p+2 ## are twin primes. Then we have ## p(p+2)+1=p^2+2p+1=(p+1)^2 ##. Thus, ## (p+1)^2 ## is a perfect square. Therefore, if ## 1 ## is added to a product of twin primes, then a perfect square is always obtained.
  2. chwala

    Proof involving ##ω(ξ,n)=u(x,y)## - Partial differential equations

    I am going through this page again...just out of curiosity, how did they arrive at the given transforms?, ...i think i get it...very confusing... in general, ##U_{xx} = ξ_{xx} =ξ_{x}ξ_{x}= ξ^2_{x}## . Also we may have ##U_{xy} =ξ_{xy} =ξ_{x}ξ_{y}.## the other transforms follow in a similar manner.
  3. mopit_011

    Doubt In Explanation of Proof of Chain Rule

    In Chapter 3 of Thomas’s Calculus, they give the following proof of the Chain Rule. After the proof, the text says that this proof doesn’t apply when the function g(x) oscillates rapidly near the origin and therefore leads delta u to be 0 even when delta x is not equal to 0. Doesn’t this proof...
  4. chwala

    Proof of the trig identities for half-angles

    I was just checking this out the sin##\frac {A}{2}## property, in doing so i picked a Right-Angled triangle, say ##ABC##, with ##AB=5cm##, ##BC=4cm## and ##CA= 3cm##. From this i have, ##s=6cm## now substituting this into the formula, ##sin\frac {A}{2}##= ##\frac {1×3}{5×3}##=##\frac...
  5. E

    Proof of Schrodinger equation solution persisting in time

    I've started reading Introduction to Quantum Mechanics by Griffiths and I encountered this proof that once normalized the solution of Schrodinger equation will always be normalized in future: And I am not 100% convinced to this proof. In 1.26 he states that ##\Psi^{*} \frac{\partial...
  6. MevsEinstein

    B This CAN'T be true (Is my proof that 1=0 correct?)

    After learning about this formula for the sum of increasing powers - ##1+p+p^2+p^3+...=1/(1-p)## - I decided to differentiate both sides of the equation, getting: ##1+2p+3p^2+4p^3+...=1/((1-p))^2##. Substituting ##1## for ##p##, I get: ##1+2+3+4+...=1/0##. But Ramanujan said that...
  7. M

    Given that p is a prime? (Review/verify this proof)?

    Proof: Suppose that p is a prime and ##p \mid a^n ##. Note that a prime number is a number that has only two factors, 1 and the number itself. Then we have (p*1)##\mid##a*## a^{(n-1)} ##. Thus p##\mid##a, which implies that pk=a for some k##\in\mathbb{Z}##. Now we have ## a^n ##=## (pk)^n ##...
  8. Math Amateur

    MHB Proving Lemma 3.3 of L&S: Further Aspects of the Proof

    I am reading Chapter 3: Jordan Measure ... of Miklos Laczkovich and Vera T Sos's book "Real Analysis: Series, Functions of Several Variables, and Applications" (Springer) ... I need help with some further aspects of the proof of Lemma 3.3 ... ... in order to fully understand the proof ... The...
  9. B

    Rational epsilon-delta limit proof questions

    Summary:: Good afternoon. I have more questions about the details of epsilon-delta proofs. Below is a simple, rational limit proof example with questions at the end. The scratch work and proof are a bit pedantic but I don't follow proofs very well which omit a lot of details, including scratch...
  10. M

    Please review/verify this proof of assertion (Number Theory)

    Proof: Suppose that all primes except for 3 must have remainder of 1 or 2 when divided by 3. Then we have the form 3p+1 or 3p+2. Note that the product of integers of the form 3p+1 also have the form...
  11. M

    Can anyone please review/verify this proof of assertion?

    Proof: Suppose that any prime of the form 3n+1 is also of the form 6m+1. Note that 2 is the only even prime number and it is not of the form 3n+1. This means any prime of the form 3n+1 must be odd...
  12. M

    Can anyone please verify/review this proof about primes?

    Proof: Suppose that there are infinitely many primes of the form n^2-2. Then we have n^2-2=2^2-2=2, n^2-2=3^2-2=7, n^2-2=5^2-2=23, n^2-2=7^2-2=47...
  13. Jehannum

    I Standard way of expressing 'no proof given'?

    In a proof of a theorem or in mathematical writing generally, if there is a statement of a sub-theorem, does a proof always need to be given if 'obvious' or if obtained by inspection? Is there a way of saying "I got this by trying some numbers in a calculator and the pattern was clear"? The...
  14. Math Amateur

    MHB Checking Proof of Theorem 6.2.8 Part (ii)

    I have completed a formal proof of D&K Theorem 6.2.8 Part (ii) ... but I am unsure of whether the proof is correct ... so I would be most grateful if someone could check the proof and point out any errors or shortcomings ... Theorem 6.2.8 reads as follows: Attempted Proof of Theorem 6.2.8...
  15. K

    B Inductive proof for multiplicative property of sdet

    Hello! Reading Roger's book on supermanifolds one can find sketch of the proof for multiplicative property of super determinant. Which looks as follows All the words sounds reasonable however when it comes to the direct computation it turns out to be technical mess and I am about to give up. I...
  16. Math Amateur

    MHB How can we prove the inequality for the supremum and infimum of f*g and f*g?

    I am reading J. J. Duistermaat and J. A. C. Kolk: Multidimensional Analysis Vol.II Chapter 6: Integration ... I need help with the proof of Theorem 6.2.8 Part (iii) ...The Definition of Riemann integrable functions with compact support and Theorem 6.2.8 and a brief indication of its proof...
  17. guyvsdcsniper

    What is the Proof for Newton's Law of Cooling Formula Using Quadratic Equation?

    I have went about this problem many different ways but cannot seem to come up with the answer. I am essentially trying to prove the formula provided in the ss of the problem.Could someone help me and tell me if I am approaching this wrong?
  18. shivajikobardan

    Comp Sci Is my resolution tree proof correct?

  19. diazdaiz

    B Help Me Determine Validity of Time Dilation Formula

    I recently trying to learn General Relativity by first scraping the surface on ScienceClic's general relativity playlist, and then I stumbled upon a video where it said that we actually move through spacetime on a constant speed of c, and then I remember about time dilation because how speed on...
  20. Math Amateur

    I Duistermaat & Kolk .... Vol II .... Proof of Proposition 6.1.2

    I am reading Multidimensional Real Analysis II (Integration) by J.J. Duistermaat and J.A.C. Kolk ... and am focused on Chapter 6: Integration ...I need some help with the proof of Proposition 6.1.2 ... Proposition 6.1.2 reads as follows: Definitions and text preliminary to the Proposition...
  21. G

    A What assumptions underlie the proof that singularities are inevitable?

    Poking around on the internet has not helped me. Penrose references Hawking and his 1996 book and I have ordered that, but I suspect my progress through that book will be slow. I have read that the assumptions include an energy condition, which I assume is expressed as a restriction on the...
  22. M

    Can anyone please review/verify this proof of greatest common divisor?

    Proof: Suppose gcd(a, b)=d. Then we have d##\mid##a and d##\mid##b for some a, b##\in## ##\mathbb{Z}##. This means a=md and b=nd for some m, n##\in## ##\mathbb{Z}##. Now we have lcm(a, b)=##\frac{ab}{gcd(a, b)}##...
  23. M

    Can anyone please review/verify this proof of a nonzero integer a?

    Proof: First, we will show that gcd(a, 0)=abs(a). Suppose a is a nonzero integer such that a##\neq##0. Note that gcd(a, 0)##\le##abs(a) by definition of the greatest common divisor. Since abs(a) divides both a and 0, we have that...
  24. M

    Can anyone please review/verify/check this number theory proof?

    Proof: Suppose for the sake of contradiction that gcd(a, b) \neq 1. Then there exists a prime number k that divides both a+b and ab. Note that k divides either a or b. Since k divides a+b, it follows that k divides b. Thus, this is a...
  25. S

    MHB Perfecting My Proof of Generalized Vandermonde's Identity

    My tests are submitted and marked anonymously. I got a 2/5 on the following, but the grader wrote no feedback besides that more detail was required. What details could I have added? How could I perfect my proof? Beneath is my proof graded 2/5.
  26. 1

    I Why did I lose 60% on my proof of Generalized Vandermonde's Identity?

    My tests are submitted and marked anonymously. I got a 2/5 on the following, but the grader wrote no feedback besides that more detail was required. What details could I have added? How could I perfect my proof? Beneath is my proof graded 2/5.
  27. Physics Slayer

    B What proof do we have of wave functions?

    How can we be sure that a system on the scale of atoms can be described by a single scalar field or the wave function ##\psi##. I don't just want to do shut up and calculate, maybe using a wave function and then putting it through the time evolution of the Schrödinger equation works, but why...
  28. P

    I Proof that two linear forms kernels are equal

    Attempt of a solution. By the Rank–nullity theorem, $$ \dim V=\dim Im_{F}+\dim\ker\left(F\right) \Rightarrow n=1+\dim\ker\left(F\right) \Rightarrow \dim\ker\left(F\right)=n-1. $$ Similarly, it follows that $$\dim\ker\left(G\right)=n-1.$$ This first part, for obvious reasons, is very clear. The...
  29. chwala

    Solve the proof in problem involving logarithms

    In my approach, i made use of change of base; i.e $$x-y=\frac {log_b n}{log_b a} -\frac {log_b n}{log_b c}$$ $$x-y=\frac {log_b c ⋅log_b n - log_b n ⋅logba}{log_b a ⋅log_bc}$$ and $$x+y=\frac {log_b n}{log_b a} +\frac {log_b n}{log_b c}$$ $$x-y=\frac {log_b c ⋅log_b n + log_b n ⋅logba}{log_b a...
  30. mcastillo356

    Anything missing or redundant about this one-sided limit proof?

    Hi, PF In a Spanish math forum I got this proof of a right hand limit: "For a generic ##\epsilon>0##, in case the inequality is met, we have the following: ##|x^{2/3}|<\epsilon\Rightarrow{|x|^{2/3}}\Rightarrow{|x|<\epsilon^{3/2}}##. Therein lies the condition. If ##x>0##, then ##|x|=x##...
  31. Mikaelochi

    I Understanding the Role of the Identity Map in Fundamental Group Theory

    So, this problem I sort of get conceptually but I don't know how I can possibly rewrite (idX)∗ : π1(X) → π1(X). Does this involve group theory? It's supposed to be simple but I honestly I don't see how. Again, any help is greatly appreciated. Thanks.
  32. B

    MHB Proof of a set union and intersection

    Hello! Lately, I've been struggling with this assignment. (angle brackets represent closed interval) I figured out that: a) union = R intersection = {0} b) union = (0, 2) intersection = {1} I asked my prof about this and she explained to me that it should be shown that if a set is an...
  33. Eclair_de_XII

    B I want this short proof of the Bolzano-Weierstrass Theorem checked please

    Let ##X## be a bounded subset of ##\mathbb{R}## with infinite cardinality. We consider a countably-infinite subset of ##X##. We write this set as a sequence to be denoted ##\{a_n\}_{n\in\mathbb{N}}##. Now define ##A## to be the set of points in the sequence with the property that for each...
  34. J

    I Proof of average height of half circle

    I need proof how find average height of half circle? Lets say pressure distribution is half circle with Pmax = radius,I must find average/resultant pressure..
  35. mcastillo356

    B Another proof of the existence of extreme values on open intervals

    Hello, PF This is Theorem 8 of Chapter 4 of the ninth edition of Calculus, by Robert A. Adams: "Existence of extreme values on open intervals". I have an alternative and easier proof, based on epsilon-delta arguments, but it's not mine, and I don't understand it completely. The fact is that...
  36. H

    A quite verbal proof that if V is finite dimensional then S is also....

    If a linear space ##V## is finite dimensional then ##S##, a subspace of ##V##, is also finite-dimensional and ##dim ~S \leq dim~V##. Proof: Let's assume that ##A = \{u_1, u_2, \cdots u_n\}## be a basis for ##V##. Well, then any element ##x## of ##V## can be represented as $$ x =...
  37. M

    Can anyone please check my work of this proof? (Number Theory)

    This is my work.
  38. DaTario

    I A proof of Archimedes' Principle

    Hi All, is there a proof to Arquimedes Principle and the expression for the buoyancy force? In case there is a proof, may we refer to this as Arquimedes theorem? (Buoyancy force = density of the fluid x acceleration of gravity x submerged volume) Best Regards, DaTario
  39. M

    Can anyone please check my proof for this number theory problem?

    Please view the picture of my work which I've uploaded.
  40. Leo Liu

    I A question about a small step in the proof of RSA encryption

    From the paper https://people.csail.mit.edu/rivest/Rsapaper.pdf Can someone explain the green highlight to me please? Sorry that I can't type much because this is the final week. Thanks.
  41. R

    B Proof involving two linear equations

    Given ## a,b,c,d,e,f \in \mathbb {R}, ad - bc \neq 0 ##, if ##(x_1,y_1)## and ##(x_2,y_2)## are pairs of real numbers satisfying: ## ax_1 + by_1 = e, cx_1 + dy_1 =f ## ## ax_2 + by_2 = e, cx_2 + dy_2 = f ## then ## (x_1,y_1) = (x_2,y_2). ## Here is my attempt at a proof, I have gotten stuck...
  42. P

    Proof that given function is convex

    Part 1 ##\left\| \vec{y} \right\|^2 \leq \left\| \vec{y} \right\|^2## and since ##\lambda \in \left[ 0,1 \right] \Rightarrow \lambda^2 \leq \lambda## So ##\lambda^2 \left\| \vec{y} \right\|^2 \leq \lambda \left\| \vec{y} \right\|^2 ## Part 2 ##\left\| \vec{x} \right\|^2 \leq \left\| \vec{x}...
  43. S

    I Physical proof of a simulation?

    The above article gives lots of evidence to support the claim we are living in a simulation. I know this is usually considered hypothetical, but in the article they give physical explanations that fit topics discussed in this forum. Please read and give your opinion
  44. FMJalink

    I Proof of Special Relativity w/ Michelson–Morley Experiment

    Dear readers, Maybe someone can enlighten me on the understanding of the proof given by the Michelson–Morley experiment on the special relativity. Just as introduction to detail the setting: There are 2 coordinate systems A and B. A stands still and B moves with the velocity v along one of...
  45. Leo Liu

    I [Congruence class] Proof of modular arithmetic theorem

    Could someone explain why ##[a][x_0]=[c]\iff ax_0\equiv c\, (mod\, m)##? My instructor said it came from the definition of congruence class. But I am not convinced.
  46. mcastillo356

    I Understanding a proof of inexistence of max nor min

    Although a function cannot have extreme values anywhere other than at endpoints, critical points, and singular points, it need not have extreme values at such points. It is more difficult to draw the graph of a function whose domain has an endpoint at which the function fails to have an extreme...
  47. shivajikobardan

    MHB Halting problem is undecidable proof confusion-:

    https://slideplayer.com/slide/10708471/ This is the context I am talking about. What contradiction occur here? We begin by telling that there is a Turing machine H that solves the halting problem. So how does this contradicts? Can you tell me about that? What contradiction occur here? We...
  48. shivajikobardan

    Comp Sci "Halting problem is undecidable" -- proof confusion

    https://slideplayer.com/slide/10708471/ This is the context I am talking about. What contradiction occur here? We begin by telling that there is a Turing machine H that solves the halting problem. So how does this contradicts? Can you tell me about that?
  49. shivajikobardan

    MHB Pumping Lemma proof for L=0^p 1^p 0^p 1^p-: Review my proof please-:

    Please review my proof. Is this correct or not?
  50. mcastillo356

    I Extreme value nonexistence proof

    Hi, PF "It is more difficult to draw the graph of a function whose domain has an endpoint at which the function fails to have an extreme value", states my textbook, "Calculus: A Complete Course" A function with no max or min at an endpoint Let...
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