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. J

    I Show ##sup\{a \in \mathbb{Q}: a^2 \leq 3\} = \sqrt{3}##

    I would wish to receive verification for my proof that ##sup\{a \in \mathbb{Q}: a^2 \leq 3\} = \sqrt{3}##. • It is easy to verify that ##A = \{a \in \mathbb{Q}: a^2 \leq 3\} \neq \varnothing##. For instance, ##1 \in \mathbb{Q}, 1^2 \leq 3## whence ##1 \in A##. • We claim that ##\sqrt{3}## is an...
  2. H

    I Lars Olsen proof of Darboux's Intermediate Value Theorem for Derivatives

    Here is Lars Olsen's proof. I'm having difficulty in understanding why ##y## will lie between ##f_a (a)## and ##f_a(b)##. Initially, we assumed that ##f'(a) \lt y \lt f'(b)##, but ##f_a(b)## doesn't equal to ##f'(b)##.
  3. M

    My proof of the Geometry-Real Analysis theorem

    Consider a convex shape ##S## of positive area ##A## inside the unit square. Let ##a≤1## be the supremum of all subsets of the unit square that can be obtained as disjoint union of finitely many scaled and translated copies of ##S##. Partition the square into ##n×n## smaller squares (see...
  4. chwala

    Show the proof by induction in the given problem

    My interest is solely on the highlighted part in red...hmmmmmmm :cool: taken a bit of my time to figure that out...but i got it. Looking for any other way of looking at it; I just realised that the next term would be given by; ##\dfrac{1}{4}(k+1)^2(k+2)^2-\dfrac{1}{4}k^2(k+1)^2##...
  5. J

    Proof for Real Analysis

    Proof: Suppose f is a function and x is in the domain of f s.t. there is a derivative at the point x and sppse. there are two tangent lines at the point (x,f(x)). Let t1 represent one of the tangent lines at (x,f(x)) and let t2 represent the other tangent line at (x,f(x)) s.t. the slopes of t1...
  6. M

    Proof about two disjoint non-empty sets ## S ## and ## T ##

    Proof: Let ## p ## be an odd prime and ## G=\left \{ 1, 2, ..., p-1 \right \} ## be the set which can be expressed as the union of two nonempty subsets ## S ## and ## T ## such that ## S\neq T ##. Observe that ## p-1=22\implies p=23 ##. Let ## g\in G ##. Since ## g ## is either an element of ##...
  7. Expiring

    I Is My Proof of the Chain Rule Correct?

    I am currently self-studying Taylor and Mann's Advanced Calculus (3rd edition, specifically). I stumbled across their guidelines for a proof of the chain rule, leaving the rest of the proof up to the reader to complete. I was wondering if someone could look over my proof, and point out any...
  8. S

    B Is there any practical proof that quantum entanglement really works?

    What i mean if we change state/spin at one end it will immediately effect the other. Can we see that live using two camera which may be 10 meter apart so that minium time delay. Is there any video proof exist such kind?
  9. murshid_islam

    I Is there any way to simplify my proof by induction?

    I'm trying to prove the statement ##n^2 + 1 < n!## for ##n \geq 4##. My proof by induction looks way too contrived. Is there a way to simplify it? Here's what I got. For n = 4, ##n^2 + 1 = 17 < 4!##. So, the statement is true for n = 4. Now let's assume it's true for n = k, that is, ##k^2 + 1 <...
  10. B

    A Why is Tau-AB^2 not t^2 + x^2?

    why is Tau-AB^2 equal to t^2 - x^2 ?It seems it should be t^2 + x^2 according to the geometry of the diagram...
  11. PhysicsRock

    Proof of Inequality: Induction & Derivation Help

    The assignment says proof by induction is possible, I cannot figure out how this is supposed to work out. Does somebody know the name of this by any chance? Seeing a derivation might help come up with an idea for a proof. Thank you everybody.
  12. P

    A Proof of the inequality of a reduced basis

    I would like to show that a LLL-reduced basis satisfies the following property (Reference): My Idea: I also have a first approach for the part ##dist(H,b_i) \leq || b_i ||## of the inequality, which I want to present here based on a picture, which is used to explain my thought: So based...
  13. PhysicsRock

    I Proof about pre-images of functions

    The problem reads: ##f:M \rightarrow N##, and ##L \subseteq M## and ##P \subseteq N##. Then prove that ##L \subseteq f^{-1}(f(L))## and ##f(f^{-1}(P)) \subseteq P##. My co-students and I can't find a way to prove this. I hope, someone here will be able to help us out. It would be very...
  14. greg_rack

    Check on proof for property of the Laplace transform

    Could someone check whether my proof for this simple theorem is correct? I get to the result, but with the feeling of having done something very wrong :) $$\mathcal{L} \{f(ct)\}=\int_{0}^{\infty}e^{-st}f(ct)dt \ \rightarrow ct=u, \ dt=\frac{1}{c}du, \ \mathcal{L}...
  15. WMDhamnekar

    I Proof of ## \displaystyle\sum_{n=0}^\infty \frac{nX^n}{n}= Xe^X##

    I know the Taylor expansion of exponential, ##\exp(x)=\displaystyle\sum_{n=0}^\infty \frac{X^n}{n!}## But if I calculate first and second derivatives of both sides of the above formula, L.H.S and R.H.S remain the same as before i-e ##e^X## So, how can I get the proofs of both series?
  16. 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)...
  17. 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...
  18. 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##...
  19. 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...
  20. 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.
  21. 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
  22. 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...
  23. 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...
  24. 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...
  25. 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...
  26. 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...
  27. 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...
  28. 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...
  29. 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...
  30. 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...
  31. 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...
  32. 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...
  33. 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...
  34. 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} ##...
  35. 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...
  36. 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...
  37. 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 =...
  38. 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?
  39. 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...
  40. 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##
  41. 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...
  42. 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: $$\sum_{i=1}^{N-1}a_{i,i+1}>\frac{1}{2}\sum_{i=1}^{N}a_{i,i}=\frac{1}{2}\text{Tr}(A)$$...
  43. 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...
  44. 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...
  45. 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?
  46. 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...
  47. 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...
  48. 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?
  49. 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?
  50. 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...
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