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. BugKingpin

    I Bloch Analysis proof of Theorem 2.5.5 (Definition by recursion)

    Want to understand how set C contains ##N## x H. H is only defined to be a set with element e and as the domain/range of function k. Is this enough information to conclude that the second set in the cartesian product W is H and not a subset of H? My thinking is to show that ##N## and H satisfy...
  2. graphking

    I A strict proof of "why the Earth is a ball"

    "bubbles are ball" is called isoperimetric problem in serious mathematic. In this topic, many essay were written. Here's my serious essay about "why earth ball", which has been rejected by arxiv and my mentors...... I would want to know if physicists are interest? I really think that is...
  3. L

    Show that the limit (1+z/n)^n=e^z holds

    Hi, I have problems proving task d I then started with task c and rewrote it as follows ##\lim_{n\to\infty}\sum\limits_{k=0}^{N}\Bigl( \frac{z^k}{k!} - \binom{n}{k} \frac{z^k}{n^k} \Bigr)=0 \quad \rightarrow \quad \lim_{n\to\infty}\sum\limits_{k=0}^{N} \frac{z^k}{k!} =...
  4. I

    Prove ##(a\cdot b)\cdot c =a\cdot (b \cdot c)## using Peano postulates

    with this background, we proceed to the proof. Let us define a set $$ G = \{ z \in \mathbb{N} | \; x, y \in \mathbb{N}\; (x \cdot y) \cdot z = x \cdot (y \cdot z) \} $$ We want to prove that ##G = \mathbb{N} ##. For this purpose, we will use part 3) of Peano postulates given above...
  5. I

    Prove ##a\cdot b = b \cdot a ##using Peano postulates

    with this background, we proceed to the proof. Let us define a set $$ G = \{ z \in \mathbb{N} | \mbox{ if } y \in \mathbb{N}, y\cdot z = z \cdot y \} $$ We want to prove that ##G = \mathbb{N} ##. For this purpose, we will use part 3) of Peano postulates given above. Obviously, ## G...
  6. I

    Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates

    I want to prove that ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates where ##a,b,c \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch ) defines Peano postulates little differently. Following is a set of Peano postulates I am using. (Axiom 1.2.1...
  7. I

    Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##

    I have to prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch) defines Peano postulates little differently. Following is a set of Peano postulates I am using. (Axiom 1.2.1 in Bloch's book) There exists a set...
  8. I

    Prove ##1 + a=s(a)=a+1## for ##a \in \mathbb{N}##

    I have to prove that ##1 + a = s(a) = a + 1## using Peano postulates if ##a \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch) defines Peano postulates little differently. Following is a set of Peano postulates I am using. (Axiom 1.2.1 in Bloch's book)...
  9. M

    Proof in number theory: the sum of all divisors of n

    let n be a positive integer show that if n is square then σ(n)( the sum of all divisors of n )is odd.
  10. L

    Proving limits for roots and exponents

    Hi I have to prove the following three tasks I now wanted to prove three tasks with a direct proof, e.g. for task a)$$\sqrt[n]{n} = n^{\frac{1}{n}}= e^{ln(n^{\frac{1}{n}})}=e^{\frac{1}{n}ln(n)}$$ $$\displaystyle{\lim_{n \to \infty}} \sqrt[n]{n}= \displaystyle{\lim_{n \to \infty}}...
  11. L

    Induction with binomial coefficient

    Hi, I'm having problems with the proof for the induction of the following problem: ##\sum\limits_{k=0}^{n} \frac{(-1)^k}{k+1} \binom{n}{k}=\frac{1}{n+1}## with ##n \in \mathbb{N}## I proceeded as follows: $$\sum\limits_{k=0}^{n+1} \frac{(-1)^k}{k+1} \binom{n+1}{k}=\frac{1}{n+2}$$...
  12. L

    Proving the Infimum and Supremum: A Short Guide for Scientists

    Hi, I have problems with the proof for task a I started with the supremum first, but the proof for the infimum would go the same way. I used an epsilon neighborhood for the proof I then argued as follows that for ##b- \epsilon## the following holds ##b- \epsilon < b## and ##b- \epsilon \in...
  13. chwala

    Prove that ## 4\tan^{-1}\left[\dfrac{1}{5}\right]- \tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{4}##

    I let, ## 4\tan^{-1}\left[\dfrac{1}{5}\right]- \tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{4}## ##\tan^{-1}\left[\dfrac{1}{5}\right]- \dfrac{1}{4}\tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{16}## Then i let, ##\tan^{-1}\left[\dfrac{1}{5}\right] = α ...
  14. F

    B Have I proved some part of Fermat's last theorem?

    Have I proved Fermat last theorem? X^4 + Y^4 != Z^4 has been proved by Fermat that if X,Y,Z = integer numbers, the formular is fine. Set x=X^2, y=Y^2, z=Z^2, so x, y, z are (some) integer numbers based on X,Y,Z. x^4 + y^4 != z^4 //x, y, z are still integer, would be obey to Fermat's Fermat...
  15. L

    How do you prove that ln(a^x) = xln(a) and a^x = e^xln(a) without using exponent rules?

    In the book "Calculus by Michael Spivak" it says that a^x = e^xln(a) is a definition. And I am not convinced to accept this as true without a proof.
  16. chwala

    Prove that the given inverse trigonometry equation is correct

    Ok in my approach i have, ##2 \tan^{-1} \left(\dfrac{1}{5}\right)= \sin^{-1} \left(\dfrac{3}{5}\right) - \cos^{-1} \left(\dfrac{63}{65}\right)##Consider the rhs, Let ##\sin^{-1} \left(\dfrac{3}{5}\right)= m## then ##\tan m =\dfrac{3}{4}## also let ##\cos^{-1} \left(\dfrac{63}{65}\right)=...
  17. porton

    Check my P=NP proof for errors (based on incompleteness of ZFC)

    Please check for errors my proof of P=NP: PDF file It is based on set theory and logic (incompleteness of ZFC). It uses also inversions of bijections, algorithms as arguments of other algorithms, reduction of SAT to another NP problem. [Moderator's note: link removed.]
  18. chwala

    Understanding the given proof of integers - Ring theory

    My interest is on the highlighted part ... Now to my question, in what cases do we have ##mn<(m,n)[m,n]?## I was able to use my example say, Let ##m=24## and ##n=30## for example, then ##[m,n]=120## and ##(m,n)=6## in this case we can verify that, ##720=6⋅120## implying that, ##mn≤...
  19. S

    I Multivariable fundamental calculus theorem in Wald

    i want to prove that if ##F:\mathbb{R}^n\to\mathbb{R}## is a differentiable function, then $$F(x)=F(a)+\sum_{i=1}^n(x^i-a^i)H_i(x)$$ where ##H_i(a)=\frac{\partial F}{\partial x^i}\bigg|_{x=a}##. the hint is that with the 1-dimensional case, convert the integral into one with limits from ##0## to...
  20. chwala

    I Proof of vector property in space

    My interest is on the associative property; is there anything wrong of showing and concluding proof by; ##c(\vec u⋅\vec v)=(c⋅\vec v)⋅\vec u.## or are we restricted in the prose?
  21. S

    I Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'

    My question is motivated by the proof of TH 5.13 on p 84 in the 2nd edition of Linear Algebra Done Right. (This proof differs from that in the 4th ed - online at: https://linear.axler.net/index.html chapter 5 ) In the proof we arrive at the following situation: ##T## is a linear operator on a...
  22. P

    I Proof regarding congruence relation

    Let ##\Lambda## be a lattice and ##a, b \in \mathbb{R}^n##, then $$a \equiv b \text{ mod } \Lambda \Leftrightarrow a- b \in \Lambda$$ I want to prove the statement. For the left to right direction I would say, ##a \equiv b \text{ mod } \Lambda \Leftarrow a = b +k\Lambda##, where ##k \in...
  23. Feynstein100

    B Proof that pattern recognition is unending?

    So I've thought of an admittedly crude proof that the process of pattern recognition i.e. finding patterns, be they linguistic, mathematical, artistic, whatever, is a process that can never end. It goes like this: Imagine we find all patterns, and I mean ALL of them, and we list them on a...
  24. H

    A How Do We Prove ##L / Z(L)## is Nilpotent in Engel's Theorem?

    in the Proof of Engel's Theorem. (3.3), p. 13: please, how we get this step: ##L / Z(L)## evidently consists of ad-nilpotent elements and has smaller dimension than ##L##. Using induction on ##\operatorname{dim} L##, we find that ##L / Z(L)## is nilpotent. Thanks in advance,
  25. hjam24

    I Write probability in terms of shape parameters of beta distribution

    Assume that players A and B play a match where the probability that A will win each point is p, for B its 1-p and a player wins when he reach 11 points by a margin of >= 2The outcome of the match is specified by $$P(y|p, A_{wins})$$ If we know that A wins, his score is specified by B's score; he...
  26. Shreya

    The Definition of Torque - a proof

    I have been trying to understand this proof from the book 'Introduction to classical mechanics' by David Morin. This proof comes up in the first chapter of statics and is a proof for the definition of torque. I don't understand why the assumption taken in the beginning of the proof is...
  27. M

    Proof of ##M^n## (matrix multiplication problem)

    For, Does anybody please know why they did not change the order in the second line of the proof? For example, why did they not rearrange the order to be ##M^n = (DP^{-1}P)(DP^{-1}P)(DP^{-1}P)(DP^{-1}P)---(DP^{-1}P)## for to get ##M^n = (DI)(DI)(DI)(DI)---(DI) = D^n## Many thanks!
  28. mattTch

    I Proof of Column Extraction Theorem for Finding a Basis for Col(A)

    Theorem: The columns of A which correspond to leading ones in the reduced row echelon form of A form a basis for Col(A). Moreover, dimCol(A)=rank(A).
  29. C

    I Is an algorithm for a proof required to halt?

    I know that when giving an algorithm to prove something we need to prove two things about the algorithm ( there’s another option which is to show time-complexity but that’s optional since it’s irrelevant to the proof): 1. Correctness 2. That it halts But there are also algorithms/procedures...
  30. Vanilla Gorilla

    B Attempted proof of the Contracted Bianchi Identity

    My Attempted Proof ##R^{mn}_{;n} - \frac {1} {2} g^{mn} R_{;n} = 0## ##R^{mn}_{;n} = \frac {1} {2} g^{mn} R_{;n}## So, we want ##2 R^{mn}_{;n} = g^{mn} R_{;n} ## Start w/ 2nd Bianchi Identity ##R_{abmn;l} + R_{ablm;n} + R_{abnl;m} = 0## Sum w/ inverse metric tensor twice ##g^{bn} g^{am}...
  31. VX10

    I A question about Young's inequality and complex numbers

    Let ##\Omega## here be ##\Omega=\sqrt{-u}##, in which it is not difficult to realize that ##\Omega ## is real if ##u<0##; imaginary, if ##u>0##. Now, suppose further that ##u=(a-b)^2## with ##a<0## and ##b>0## real numbers. Bearing this in mind, I want to demonstrate that ##\Omega## is real. To...
  32. M

    Proof of angle in path difference formula for two slits

    For this I am trying to prove that angle theta between PQ and QO is equal to theta highlighted so that I know I can use theta is the path difference formula. I assume that the rays ##r_1## and ##r_2## are parallel since ##L >> d## My proof gives that the two thetas are equal, however I am...
  33. M

    Power Rule Proof: Get Help with Line 3 to Line 4

    For this proof, I am unsure how they got from line 3 to line 4. If I simplify and collect like terms for line 3 I get ##f'(a) = 4a^{n-1}## Would some please be able to help? Many thanks!
  34. V

    Logical Proof: Theorem (Truths of Logic) A iff ~~A

    My thought was to break up the sentence into its equivalent form: (A ->~~A) & (~~A -> A) From there I assumed the premise of both sides to use indirect proofs, so: 1. ~(A -> ~~A) AP 2. ~(~A or ~~A) 1 Implication 3. ~~A & ~~~A 2...
  35. E

    I Proof of Lorentz Gauge Existence: Help Understanding Schutz 8.3

    In Schutz 8.3, while proving that a Lorentz gauge exists, it is stated that $$\bar h^{(new)}_{\mu\nu} = \bar h^{(old)}_{\mu\nu} - \xi_{\mu,\nu} - \xi_{\nu,\mu} + \eta_{\mu\nu}\xi^\alpha_{,\alpha}$$ where ##\bar h## is the trace reverse and ##\xi^\alpha## are the gauge functions. Then it follows...
  36. 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...
  37. 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)##.
  38. 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...
  39. 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##...
  40. J

    Can a Function Have Two Different Tangent Lines at the Same Point?

    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...
  41. 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 ##...
  42. 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...
  43. 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?
  44. 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 <...
  45. 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...
  46. 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.
  47. 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...
  48. 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...
  49. 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}...
  50. 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?
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