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. SamRoss

    B Don't follow one small step in proof

    https://www.wyzant.com/resources/lessons/math/calculus/derivative_proofs/inverse_trig The above link is just a proof for the derivative of arcsinx. Going from line 3 to 4, how does d/dx siny become dy/dx cosy ?
  2. M

    Prove ##5^n+9<6^n## for ##n\epsilon N|n\ge2## by induction

    Homework Statement Prove ##5^n+9<6^n## for ##n\epsilon \mathbb{N}|n\ge2## by induction. Homework Equations None The Attempt at a Solution The base case which is when ##n=2##: ##5^2+9<6^2## ##34<36## Thus, the base case is true. Now for the induction step. Induction hypothesis: Assume...
  3. B

    B Understanding Invertible Matrices and Homogenous Systems

    For a ##n\times n## matrix A, the following are equivalent. 1) A is invertible 2) The homogenous system ##A\bf X = 0## has only the trivial solution ##\mathbf X = 0## 3) The system of equations ##A\bf X = \bf Y## has a solution for each ##n\times 1 ## matrix ##\bf Y##. I have problem in third...
  4. B

    B Proof of elementary row matrix operation.

    Prove that interchange of two rows of a matrix can be accomplished by a finite sequence of elemenatary row operations of the other two types. My proof :- If ##A_k## is to be interchanged by ##A_l## then, ##\displaystyle \begin{align} A_k &\to A_l + A_k \\ A_l &\to - A_l \\ A_l &\to A_k + A_l...
  5. F

    Proving the Evenness of Elements Not Equal to Their Own Inverse in Finite Groups

    Homework Statement Prove in any finite group G, the number of elements not equal to their own inverse is an even number. Homework Equations if ab = ba = e, then a = b-1 and b = a-1 The Attempt at a Solution Let S, A, B, be subsets of G where S = A + B. Let a ∈ A s.t. there exists a unique b...
  6. stevendaryl

    I Prove Inequality: A,A', B, B' in [0,1]

    I'm pretty sure that the following is true, but I don't see an immediate compelling proof, so I'm going to throw it out as a challenge: Let A,A', B, B' be four real numbers, each in the range [0,1]. Show that: AB + AB' + A'B \leq A' B' + A + B (or show a counter-example, if it's not true)...
  7. PHstud

    Proof of fundamental thermodynamics equation for open systems

    Hi ! I'm having a bit of trouble understanding something. Let 'u' be internal energy, 'h' enthalpy, 'e' work and 'q' heat. ('r' are dissipations and 'S' entropy) From a book , i read that de+dr=PdV= -du + TdS This seems to stand for closed cycle. Yet, my teacher uses the formula de+dr=vdP=...
  8. K

    B What is the role of tautology in an axiomatic system and proof?

    Hi, I have a question about axiomatic system and proof. Let's say we have a finite sequence of propositions ai, which is an axiomatic system. To prove a proposition P that is a finite sequence of propositions qi with axiomatic system {ai}, we can take 3 methodologies. (A) qi itself is...
  9. S

    I Proof of Harmonic Function Infinitely Differentiable

    Hello! I have this Proposition: "A harmonic function is infinitely differentiable". The book gives a proof that uses this theorem: "Suppose u is harmonic on a simply-connected region G. Then there exists a harmonic function v in G such that ##f = u + iv## is holomorphic in G. ". In the proof...
  10. B

    Solving Symmetric Group Induction Proof: Hints for Double Induction

    Homework Statement Consider the symmetric group ##S_n##. I am trying to establish that ##(i,i+1)=(1,2,...,n)(i-1,i)(1,2,...,n)^{-1}## Homework EquationsThe Attempt at a Solution I am trying to decide whether I need double induction or not. I have done several calculations to see whether I can...
  11. Adgorn

    Proof regarding transpose mapping

    Homework Statement Suppose T:V→U is linear and u ∈ U. Prove that u ∈ I am T or that there exists ##\phi## ∈ V* such that TT(##\phi##) = 0 and ##\phi##(u)=1. Homework Equations N/A The Attempt at a Solution Let ##\phi## ∈ Ker Tt, then Tt(##\phi##)(v)=##\phi##(T(v))=0 ∀T(v) ∈ I am T. So...
  12. SherLOCKed

    A Help with proof of eq. 2.64 of Intro. to Quantum Mechanics

    I am self studying the Book- Introduction to Quantum Mechanics , 2e. Griffith. Page 47. While the book has given a proof for eq. 2.64 but its not very ellaborate Integral(infinity,-infinity) [f*(a±g(x)).dx] = Integral(infinity,-infinity) [(a±f)* g(x).dx] . It would be great help if somebody...
  13. P

    Need help understanding the proof of Thevenin's theorem

    Hello. Uh, I'm trying to undestand how to prove Thevenin's theorem. The Sadiku book puts an independent current source where the load used to be in order to reach the equation: V = Vth + I*Rth. I do understand how he reaches that conclusion after putting the source, what I don't understand is...
  14. G

    I Is the Proof of Geometric Progression in Probability Common Sense?

    My Statistics textbook does not prove this. The author think it is commons sense. I am not sure about this proof. Thank you.
  15. Adgorn

    Proof regarding linear functionals

    Homework Statement Let V be a vector space over R. let Φ1, Φ2 ∈ V* (the duel space) and suppose σ:V→R, defined by σ(v)=Φ1(v)Φ2(v), also belongs to V*. Show that either Φ1 = 0 or Φ2 = 0. Homework Equations N/A The Attempt at a Solution Since σ is also an element of the duel space, it is...
  16. P

    MHB Visual proof log(ab) = log a + log b

    Hi, I'm looking for a visual proof log(ab) = log a + log b I've seen diagrams where the values are measured out, but it's not immediately obvious why this holds. Is there an intuitive way to illustrate this? Also, are there other functions with this property.
  17. shihab-kol

    Simple proof of Snell's law without calculus

    Well, I have checked out the ones with calculus but I was just wondering if there was one without calculus I tried it but could not do it I think Fermat's principle can be used to do it but I am not being successful So, anyone please help
  18. N

    MHB Prove Probability: Step-by-Step Guide

    Prove the following I literally have no idea where to start or what to do.
  19. R

    Question about Carnot theorem proof

    Homework Statement Carnot theorem states that no engine working between two temperatures T1 of source and T2 of sink can have a greater efficiency than that of the Carnot engine. Second law of thermodynamics:it is impossible for a self acting machine to transfer heat from a body at a higher...
  20. G

    I Bertrand's Postulate and Erdős' Proof

    Hello. Is there a quick proof for showing that the next prime is within twice the current prime? Edit: Never mind. Erdős had given a proof of this (of Bertrand's postulate to be precise) at a fairly young age. http://www3.nd.edu/~dgalvin1/pdf/bertrand.pdf
  21. Faisal Moshiur

    I Proof of some identities regarding spin angular momentum.

    If we define Si=(1/2)× (reduced Planck's const)×sigma Then what will be (sigma dot vect{A})multiplied by (Sigma dot vect{B}) Here (sigma)i is Pauli matrix. Next one is, what will we get from simplifying <Alpha|vect{S}|Alpha> where vect{S} is spin vector & |Apha>is equal to " exp[{i×(vect{S} dot...
  22. N

    I Looking for a proof that u(x) du(x)/dx = 0.5 d(u(x)^2)/dx

    Can anyone help with a proper proof for the following relation, please? u(x) \frac{\partial u(x)}{\partial x} = \frac{1}{2} \frac{\partial u(x)^2}{\partial x} From simple calculations I agree that it's true, but it's been annoying me for a while that I can't find a proper mathematical proof...
  23. T

    Proof of trace theorems for gamma matrices

    Hi, I'm currently going through Griffith's Particle Physics gamma matrices proofs. There's one that puzzles me, it's very simple but I'm obviously missing something (I'm fairly new to tensor algebra). 1. Homework Statement Prove that ##\text{Tr}(\gamma^\mu \gamma^\nu) = 4g^{\mu\nu}##...
  24. K

    Greatest common divisor proof

    Hi, I need opinion about this problem. ================================================== question :Prove: If(a,b)= l and if ( "(a,b)=1" mean greatest common divisor of integers and b is 1 ) c|a (c divides a) and d|b (d divides b ) then (c,d)= 1. ( "(c,d)=1" mean...
  25. gelfand

    Show that potential energy is conserved

    Homework Statement potential energy function of : $$ U(x) = 4x^2 + 3 $$ And have to i) Work out the equation of motion ii) Prove explicitly that the total energy is conservedHomework Equations$$ F = \frac{dU}{dt} $$ The Attempt at a Solution I'm not too sure how to go about this...
  26. N

    Euclidean and non Euclidean geometries problems

    So I was reading this book, "Euclidean and non Euclidean geometries" by Greenberg I solved the first problems of the first chapter, and I would like to verify my solutions 1. Homework Statement Homework Equations [/B] Um, none that I can think of? The Attempt at a Solution (1) Correct...
  27. J

    A General relativity -- Proof of energy measured by observer

    I want to prove that ##E = -g_{\mu \nu}u^\mu p^\nu## is the energy measured by an observer with velocity ##u^\mu## of an object with momentum ##p^\mu##. My reasoning is that in special relativity we know that ##\gamma m = E##. We can transform to coordinates where ##u'^\mu = (1,\vec{0})##. Since...
  28. H

    I Contradiction in an absolute value property?

    An absolute value property is $$\lvert a \rvert \geq b \iff a\leq-b \quad \text{ or } \quad a\geq b,$$ for ##b>0##. Is this true for the case ##a=0##? I mean if ##a=0, \lvert a \rvert =0## so ##0 \geq b##. But ##b## is supposed to be ##b>0##, so we have a contradiction. How can this property...
  29. davidge

    I Proving Stokes' Theorem: General Cases and the Fundamental Theorem of Calculus

    How would one prove the Stokes' theorem for general cases? Namely that $$ \int_{\partial M} W = \int_M \partial W$$ where ##M## is the manifold.
  30. digogalvao

    Proof of expectation value for a dynamic observable

    Homework Statement Show that: d<A(q,p)>/dt=<{A,H}>, where {A,H} is a Poisson Bracket Homework Equations Liouville theorem The Attempt at a Solution <A>=Tr(Aρ)⇒d<A>/dt=Tr(Adρ/dt)=Tr(A{H,ρ}) So, in order to get the correct result, Tr(A{H,ρ}) must be equal to Tr({A,H}ρ), but I don't think I can...
  31. F

    Prove Induction: u_n < 4 for All n ≥ 1

    Homework Statement The sequence of positive numbers ##u_1,u_2,u_3...## is such that ##u_1<4## and ##u_{n+1}= \frac{5u_n+4}{u_n+2} ## i. By considering ##4-u_{n+1} ##, prove by induction that ##u_1<4## for ##n\geq 1## Mod note: The above is incorrect. In a later post the OP revised this to The...
  32. Eclair_de_XII

    Is my short induction proof correct?

    Homework Statement "Prove: ##∀n∈ℕ##, ##3^n>n^2## Homework EquationsThe Attempt at a Solution (1) We will prove that ##3^n>n^2## at ##n=1## ##3=3^1>1=1^2## (2) Now assume that ##3^k>k^2## for some ##k>1## (3) We will prove that ##3^{k+1}>(k+1)^2## or ##3⋅3^k>k^2+2k+1## Note that...
  33. N

    Superposition Proof: Understanding Angle of Sin

    I don't get the first part. why did he make the angle of sin equal to n pi.
  34. A

    I Another negative one equals one proof

    Hey guys! I need help proving why this proof is wrong. I know it's wrong, but I can't figure out why. Anyway: i = sqrt -1 i^4 = 1 1^4 = 1 Substution: i^4 =1^4 i = 1 1 = sqrt -1 1^2 = -1 1 = 1^2 1= -1 If you have any questions, feel free to ask.
  35. koustav

    Are Spacelike and Timelike Orthogonal: Mathematical Proof Explained

    are spacelike and timelike orthogonal?what is the mathematical proof
  36. J

    MHB Proving an Integral with a Direct Proof & Epsilon Argument

    Okay, these are my last questions and then I'll get out of your hair for a while. For 1, I have already done a proof by contradiction, but I'm supposed to also do a direct proof. Seems like it should be simple? For 2, this seems obvious because it's the definition of an integral. My delta is...
  37. S

    A Proof - gauge transformation of yang mills field strength

    In Yang-Mills theory, the gauge transformations $$\psi \to (1 \pm i\theta^{a}T^{a}_{\bf R})\psi$$ and $$A^{a}_{\mu} \to A_{\mu}^{a} \pm \partial_{\mu}\theta^{a} \pm f^{abc}A_{\mu}^{b}\theta^{c}$$ induce the gauge transformation$$F_{\mu\nu}^{a} \to F_{\mu\nu}^{a} -...
  38. J

    MHB Real Analysis - Riemann Integral Proof

    I have no idea how to incorporate the limit into the basic definitions for a Riemann integral? All we have learned so far is how to define a Riemann integral and the properties of Riemann integrals. What should I be using for this?
  39. P

    Explosion and conservation of momentum problem

    Note: Please only give hints please! No answers because I want the satisfaction of solving it. 1. Homework Statement A mass M at height h above flat round and falling vertically with velocity v breaks up explosively into 2 parts. The kinetic energy given to the system in the explosion is E...
  40. T

    Relation closures proof

    Homework Statement Suppose R1 and R2 are relations on A and R1 ⊆ R2. Let S1 and S2 be the transitive closures of R1 and R2 respectively. Prove that S1 ⊆ S2. Please check my proof and please explain my mistakes. thank you for taking the time to help. Homework Equations N/A The Attempt at a...
  41. DaniV

    I Proof of series`s tail limit

    {\displaystyle \sum_{n=1}^{\infty}a_{n}} is converage, For N\in \mathbb{N}\sum_{n=N+1}^{\infty}an is also converage proof that \lim_{N\rightarrow\infty}(\sum_{n=N+1}^{\infty}an)=0 {\displaystyle \sum_{n=1}^{\infty}a_{n}} is converage, For N\in \mathbb{N} \sum_{n=N+1}^{\infty}an is...
  42. B

    Trying to understand a proof about ##\lim S##

    Homework Statement I am trying to understand the proof that ##\lim S## is a closed set in the metric space ##M##, where ##\lim S = \{ p \in M ~|~ p \mbox{ is a limit point of } S\}##. Here is the definition of a limit point: ##p## is a limit point of ##S## if and only if there exists a...
  43. lfdahl

    MHB Proving Inequality for Variables with Constraints

    Let $0 \le a,b,c \le 1.$ Prove the inequality:$\sqrt{a(1-b)(1-c)}+ \sqrt{b(1-a)(1-c)}+\sqrt{c(1-a)(1-b)} \le 1 + \sqrt{abc}$
  44. F

    MHB Proof of Knaster-Tarski Theorem

    Let $F:P(A)->P(A$) be monotone and $C$ be the union of sets whose image is invariant under F. Prove $F(C)=C$ https://i.stack.imgur.com/3Wjdg.png
  45. F

    MHB How to Prove a Set Theory Ordinal Relationship?

    Let $\beta$ be an ordinal. Prove that $A\cap \bigcup\beta=\bigcup\{A\cap X\mid X \in \beta\}$ I'm not sure on this. It looks a bit like union distributing over intersection. Please help.
  46. M

    Proof by induction, ##(n)^{2} \le (2n)##.

    Homework Statement I need to prove by induction that ##(n!)^{2} \le (2n)!##. I'm pretty sure about my preliminary work, but I just need some suggestions for the end. Homework Equations It is well known from a theorem that if ##a \le b## and ##c \ge 0##, then ##ca \le cb##. The Attempt at a...
  47. A

    I Regarding Cantor's diagonal proof

    I am very open minded and I would fully trust in Cantor's diagonal proof yet this question is the one that keeps holding me back. My question is the following: In any given infinite set, there exist a certain cardinality within that set, this cardinality can be holded as a list. When you change...
  48. M

    Ε-δ proof: lim x->a f(x) = lim h->0 f(a + h)

    This is a simple exercise from Spivak and I would like to make sure that my proof is sufficient as the proof given by Spivak is much longer and more elaborate. Homework Statement Prove that \lim_{x\to a} f(x) = \lim_{h\to 0} f(a + h) Homework EquationsThe Attempt at a Solution By the...
  49. S

    A Are there experimental proofs for modern theories

    Quantum theory, although hard to understand with intuition has a lot of experimental proof. Do the more modern theories e.g. String theory, or black hole theories have any experimental proof, or are they theories that the mathematics have led to? Without proof, do they deserve so much credit...
  50. D

    I Proof that parity operator is Hermitian in 3-D

    Hi. I have been looking at the proof that the parity operator is hermitian in 3-D in the QM book by Zettili and I am confused by the following step ∫ d3r φ*(r) ψ(-r) = ∫ d3r φ*(-r) ψ(r) I realize that the variable has been changed from r to -r. In 3-D x,y,z this is achieved by taking the...
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