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. S

    I Analytical proof of LASER production

    Is there analytical proof that a photon Pe will be emitted by an excited atom Ae when another photon Pp of the same frequency is passing by Ae in LASER production? I tried using Feynman diagram to show a high probability of this event. I failed (most likely because I am not an expert in QFT)...
  2. S

    Limit Definition of Derivative as n Approaches Infinity

    ##f'(x_0)## is defined as: $$f'(x_0)=\lim_{h \rightarrow 0} \frac{f(x_0+h)-f(x_0)}{h}$$ or $$f'(x_0)=\lim_{x \rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0}$$ I can imagine that as ##n \rightarrow \infty## the value of ##f(b_n)## and ##f(a_n)## will approach ##f(x_0)## so the value of the limit will...
  3. hilbert2

    A Does there exist a proof for these conjectures?

    Does anyone know if a proof exists for these statements about 1d quantum mechanics? 1. If the potential energy where a particle moves is of the form ##V(x) = c_2 x^2 + c_4 x^4 + c_6 x^6 + \dots## or ##V(x) = c_2 x^2 + c_3 |x|^3 + c_4 x^4 + c_5 |x|^5 + c_6 x^6 + \dots## with ##c_j \geq 0##...
  4. elcaro

    I Is this a proof of the Collatz Conjecture?

    Note as soon as the term 3N+1 become divisible by a power of 2 we can repeatedly divide by 2. For the proof below we rearrange the sequence so it becomes: First step: If N is odd, multiply by 3 and add 1. Each next step: - Repeatedly divide by 2, as many times as the number k, which is...
  5. M

    B Proof of inverse square law for gravitation?

    Newton arrived at "there is a force that drives a planet around the star by examining kepler's laws but how did he arrive to inverse square law by kepler's third law (##T^2=\frac {4\pi r^3}{GM}##)? Thank you.
  6. A

    Prove by the principle of induction

    (expression given to be proven) check for p(1)... 2=2 substitute (n+n) to And here is the problem, I just can't find a way to continue solving this problem
  7. M

    Determine (with proof) the set of all prime numbers

    Proof: Let ## p ## be the prime divisor of two successive integers ## n^{2}+3 ## and ## (n+1)^{2}+3 ##. Then ## p\mid [(n+1)^{2}+3-(n^{2}+3)]\implies p\mid (2n+1) ##. Observe that ## p\mid (n^{2}+3) ## and ## p\mid (2n+1) ##. Now we see that ## p\mid [(n^{2}+3)-3(2n+1)]\implies p\mid...
  8. P

    A Question regarding proof of convex body theorem

    Hello, I am currently working on the proof of Minkowski's convex body theorem. The statement of the corollary here is the following: Now in the proof the following is done: My questions are as follows: First, why does the equality ##vol(S/2) = 2^{-m} vol(S)## hold here and second what...
  9. A

    Series inequality induction proof

    My first attempt was ##... + n^{2} + (n+1)^{2} > \frac {1}{3} n^{3} + (n+1)^{2}## then we must show that ##\frac {1}{3} n^{3} + (n+1)^{2} > \frac {1}{3} (n+1)^{3}## We evaluate both sides and see that the LHS is indeed bigger than RHS. However, this solution is inconsistent so I am asking for...
  10. P

    Is this computability theory proof correct?

    I am proposing a new theorem of computability theory: THEOREM 1: There are numbers k and s and a program A(n,m) satisfying the following conditions. 1. If A(n,m)↓, then C_n(m)↑. 2. For all n, C_k(n) = A(n,n) and C_s(n) = C_k(s). 3. A(k,s)↓ and for all n, A(s,n)↑. Here C_n(∙) is a program with...
  11. Z

    I Spivak, Ch. 20: Understanding a step in the proof of lemma

    In Chapter 20 of Spivak's Calculus is the lemma shown below (used afterward to prove Taylor's Theorem). My question is about a step in the proof of this lemma. Here is the proof as it appears in the book My question is: how do we know that ##(R')^{n+1}## is defined in ##(2)##? Let me try to...
  12. I

    B Proving the Existence of Particles: An Exploration

    I studied physics in University a bit out of interest. Curious on how exactly one proves the existence of particles. If I look it up, often the most basic example would be the cathode ray experiment. It seems pretty simple to me, but in my eyes it does not prove the existence of particles...
  13. M

    Proof: Palindromes Divisible by 11

    Proof: Suppose ## N ## is a palindrome with an even number of digits. Let ## N=a_{m}10^{m}+\dotsb +a_{2}10^{2}+a_{1}10+a_{0} ##, where ## 0\leq a_{k}\leq 9 ##, be the decimal expansion of a positive integer ## N ##, and let ## T=a_{0}-a_{1}+a_{2}-\dotsb +(-1)^{m}a_{m} ##. Note that ## m ## is...
  14. M

    Proof That 6 Divides Any Integer N

    Proof: Suppose that ## 6 ## divides ## N ##. Let ## N=a_{m}10^{m}+\dotsb +a_{2}10^{2}+a_{1}10+a_{0} ##, where ## 0\leq a_{k}\leq 9 ##, be the decimal expansion of a positive integer ## N ##. Note that ## 6=2\dotsb 3 ##. This means ## 2\mid 6 ## and ## 3\mid 6 ##. Then ## 2\mid...
  15. cianfa72

    Formal proof of Thevenin theorem

    Hi, I am looking for a formal proof of Thevenin theorem. Actually the first point to clarify is why any linear network seen from a port is equivalent to a linear bipole. In other words look at the following picture: each of the two parts are networks of bipoles themselves. Why the part 1 -- as...
  16. R

    B Vacuously true statements and why false implies truth

    We say that an implication p --> q is vaccuously true if p is false. Since now it's impossible to have p true and q false. That is we can't check anymore whether the contrary, p being true and q being false,can be.Since p being true is non-existent. So we take the implication as true. For eg...
  17. H

    I Proof that if T is Hermitian, eigenvectors form an orthonormal basis

    Actual statement: Proof (of Mr. Tom Apostol): We will do the proof by induction on ##n##. Base Case: n=1. When ##n=1##, the matrix of T will be have just one value and therefore, the characteristic polynomial ##det(\lambda I -A)=0## will have only one solution. So, the Eigenvector...
  18. P

    I Is this proof of cp - cv correct

    cp=(dU/dT)P+P(dv/dT)P cv=(dU/dT)V cp-cv=(dU/dT)P+P(dv/dT)P- (dU/dT)V=(dU/dV)T(dV/dT)P+P(dv/dT)P- (dU/dV)T(dV/dT)V since dV is zero (dU/dV)T(dV/dT)V is zero. Hence cp-cv=(dU/dV)T(dV/dT)P+P(dv/dT)P I expanded both dU/dT and since one of them has no change in volume it is zero. is it acceptable...
  19. M

    Proof: Divisibility of Integers by 4

    Proof: Let ## N ## be an integer. Then ## N=a_{m}10^{m}+a_{m-1}10^{m-1}+\dotsb +a_{1}10+a_{0} ## for ## 0\leq a_{k}\leq 9 ##. Note that ## 10^{k}\equiv 0\pmod {4} ## for ## k\geq 2 ##. Thus ## 4\mid N\Leftrightarrow N\equiv 0\pmod {4}\Leftrightarrow a_{1}10+a_{0}\equiv 0\pmod {4} ##...
  20. M

    Proof: Integer Divisibility by 3 via Polynomials

    Proof: Let ## P(x)= \Sigma^{m}_{k=0} a_{k} x^{k} ## be a polynomial function. Then ## N=a_{m}10^{m}+a_{m-1}10^{m-1}+\dotsb +a_{1}10+a_{0} ## for ## 0\leq a_{k}\leq 9 ##. Since ## 10\equiv 1\pmod {3} ##, it follows that ## P(10)\equiv P(1)\pmod {3} ##. Note that ## N\equiv (a_{m}+a_{m-1}+\dotsb...
  21. M

    Proof That an Integer is Divisible by 2

    Proof: Suppose ## N ## is the integer and ## x ## is the units digit of ## N ##. Then ## N=10k+x ## for some ## k\in\mathbb{Z} ## where ## x={0, 1, 2, 3, 4, 5, 6, 7, 8, 9} ##. Note that ## 10k\equiv 0\pmod {2}\implies N\equiv x\pmod {2} ##. Thus ## 2\mid N\implies N\equiv 0\pmod {2}\implies...
  22. H

    I ##\epsilon - \delta## proof and algebraic proof of limits

    It occurred to me that I should ask this to people who passed the stage in which I’m right now, being unable to find anyone in my milieu (maybe because people around me have expertise in other fields than mathematics) I reckoned to come here. Let’s see this sequence: ## s_n =...
  23. T

    B Proof of the existence of atoms

    It is said that some physicists doubted the existence of atoms in 1900 until Einstein proved their existence a few years later. Did Mendeleev's creation of the periodic table in the 1870s already prove the reality of atoms by giving the known elements atomic masses?
  24. J

    Clarification on Proof by Contradiction

    This is more a general question that this problem spurred and this is what I came up with. I do not feel it is acceptable but would like clarification moving forward. My text states the format for proof by contradiction is as follow; Proposition: P PF: Suppose ~P. ...a little math and...
  25. chwala

    Show the proof involving multiples of numbers

    Let the first multiple of ##4=x##, then it follows that; ##x+(x+4)+(x+8)+(x+12)=4x+24=4(x+6)## ...where ##4## is a multiple of ##8##
  26. J

    Induction Proof, Apostol Calc Vol I, I.4.4.7

    For calculating ##n_1## I had no problems as ##n_1 = 3##. PF: Show ##(1+x)^n > 1 + nx + nx^2 ## is true for all ##x \ge 3##. Let ##n = 3, (1+x)^3 > 3x^2 + 3x + 1 ## ##x^3 +3x^2 +3x + 3 > 3x^2 + 3x + 1 ## is true. Assume ## n = k ## such that ##(1+x)^k > 1+kx + kx^2, k \ge 3## is true. We...
  27. Jamister

    I Proof about a positive definite matrix

    I need to prove the following: A symmetric real matrix ##A## with positive elements ##a_{i,j}\geq 0## can’t be definite positive matrix (i.e. with only positive eigenvalues) if the following condition holds: $$\sum_{i=1}^{N-1}a_{i,i+1}>\frac{1}{2}\sum_{i=1}^{N}a_{i,i}=\frac{1}{2}\text{Tr}(A)$$...
  28. benorin

    This is for an Insights article: Bivariate induction proof using Calc3

    Link to my insight Article it's right where I need you to start checking, read the above boxes to, check out the picture to see examples of the kind of sequence of sets we are dealing with. I need you to read the section jusr below the first picture entitled "3.0.2 Lemma 2.1: Nesting Property of...
  29. H

    Proof of ##g(A_1, A_2, \cdots A_n) = c g (I_1, \cdots I_n)##.

    How can we prove that $$ g(A_1, \cdots A_n)= c g(I_1 \cdots I_n)$$? From the those three axioms we can prove a property of g that if any of two vectors in domain exchange their respective places the sign of output of g will be changed. Now, do we have to argue that any matrix can be changed...
  30. e2m2a

    B Difference between like powers proof

    This may seem like a trivial question but I don't know if there is a formal proof for this. Is the following expression never true? a^n-b^n =1, where a >b, a,b,n are positive integer numbers. Was this known since ancient times? Or is there a modern proof for this?
  31. e2m2a

    I Proof of Differences of Odd Powers

    I am interested in finding any proofs that exist which demonstrates that the difference between two odd powered integers can never be equal to a square? Has there been any research in this? For example, given this expression a^n -b^n = c^2, where a,b,c are positive integers and a>b, n = odd...
  32. H

    I Proof of Induction Principle starting from N, not from 1

    In this video lecture (though I have linked the video at "current time", in case it doesn't; work please see the video at 19:16), the lecturer just works out (he is not explaining anything) the proof of Induction Principle starting from ##N##. Let me give out here what he did: Statement: Let N...
  33. PaxFinnica96

    Engineering Fluid Dynamics: Proof of the Static Pressure Head equation

    I am trying to mathematically prove the Static Pressure Head equation: H = p/ρg How can I prove this equation and thus determine the nature of the relationship between these variables?
  34. J

    I Proof of Q=CV for arbitrarily shaped capacitors

    What is a proof of the formula Q=CV for a capacitor with arbitrary but unchanging shape where C is a constant?
  35. shivajikobardan

    MHB Few confusions about halting problem is unsolvable proof-:

    https://lh6.googleusercontent.com/xFVXel6_szTL_WVir-dz4SpFIGkHqMY9428mA3HRY2Nl06Ez9Wt9N3RZ8U0Jmsshwnl7ekEQX31ccWBXBNW5XUhRwVQafqZOPHpmy3fY4L94b4UmqGR-N7IIO8Ep1wpWg4BmDebD How can same machine take same machine as input(is it same machine taking encoding of itself as input—if that’s the case...
  36. 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.
  37. 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.
  38. 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...
  39. 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...
  40. 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...
  41. 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...
  42. 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 ##...
  43. 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...
  44. 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...
  45. 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...
  46. 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...
  47. 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...
  48. 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...
  49. 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...
  50. 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...
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