# What is Proof: Definition and 1000 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. ### 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...
2. ### 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.
3. ### 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}...
4. ### 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.]
5. ### 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≤...
6. ### 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...
7. ### 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?
8. ### 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...
9. ### 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...
10. ### 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...
11. ### A About Proof of 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,
12. ### 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 >= 2 The 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...
13. ### 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...
14. ### 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!
15. ### 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).
16. ### 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...
17. ### 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}...
18. ### 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...
19. ### 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...
20. ### 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!
21. ### 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...
22. ### 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...
23. ### 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...
24. ### 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)##.
25. ### 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...
26. ### 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##...
27. ### 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...
28. ### 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 ##...
29. ### I Proof of the Chain Rule

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...
30. ### 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?
31. ### 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 <...
32. ### 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...
33. ### 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.
34. ### 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...
35. ### 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...

40. ### 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##...
41. ### 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...
42. ### 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.
43. ### 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
44. ### 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...
45. ### 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...
46. ### 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...
47. ### 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...
48. ### 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...
49. ### 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...
50. ### 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...