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. P

    Is this proof of an ##\infty## norm valid?

    I am trying to prove ##||A||_{\infty} = max_i \sum_{j} |a_{ij}|## which reads as the ##\infty## norm is the max row sum of matrix A. ##i## is the row index and ##j## is the column index. Here is what I thought of: ##||A||_{\infty} = sup_{x\neq 0} \frac{||Ax||_{\infty}}{||x||_{\infty}}## The...
  2. S

    Proving Lorentz Transform Without Light Signal

    The thought experiment used to prove Lorentz transform uses a light signal as an assumption. What if there was something other than the light signal then Lorentz transformation would have totally different term in place of 'c'(speed of light).
  3. A

    Proof that COF depends only on asperities

    Homework Statement I am trying to craft a hypothesis regarding factors that affect the coefficient of friction. I know that it is determined by the triboforces and asperity interactions at the interface between the materials (among other factors, but right now I'm just going to focus on this)...
  4. C

    Proof about successor function

    Homework Statement Successor of a set x is defined as S(x)=x \cup {x} Prove that if S(x)=S(y) then x=y Our teacher gives us a hint and says use the foundation axiom. The Attempt at a Solution if S(x)=S(y)=x \cup {x}=y \cup {y} I feel like doing a proof by contradiction would work...
  5. RJLiberator

    Relatively prime proof involving a^n and b^n

    Homework Statement Show that if a, b, n, m are Natural Numbers such that a and b are relatively prime, then a^n and b^n are relatively prime. Homework Equations Relatively prime means 1 = am + bn where a and b are relatively prime. gcd(a,b) = 1 We have a couple corollaries that may be...
  6. G

    Mean value theorem variation proof

    Homework Statement Let f is differentiable function on [0,1] and f^{'}(0)=1,f^{'}(1)=0. Prove that \exists c\in(0,1) : f^{'}(c)=f(c). Homework Equations -Mean Value Theorem The Attempt at a Solution The given statement is not true. Counter-example is f(x)=\frac{2}{\pi}\sin\frac{\pi}{2}x+10...
  7. AutumnWater

    Epsilon Delta proof of a 2variable limit. Is my proof valid?

    Homework Statement Use the epsilon delta definition to show that lim(x,y) -> (0,0) (x*y^3)/(x^2 + 2y^2) = 0 Homework Equations sqrt(x^2) = |x| <= sqrt(x^2+y^2) ==> |x|/sqrt(x^2+y^2) <= 1 ==> |x|/(x^2+2y^2)? The Attempt at a Solution This limit is true IFF for all values of epsilon > 0, there...
  8. TheMathNoob

    Graph theory proof (Unlabeled trees)

    Homework Statement A. Show that n^n−2/n! < T(n) by looking at how the symmetric group Sn acts on labelled trees. Use |Sn| = n! T(n) is the number of unlabeled trees on n vertices Homework EquationsThe Attempt at a Solution I can't find any mathematical relation between labelled trees and...
  9. I

    Proof that e is irrational using Taylor series

    Homework Statement Using the equality ##e = \sum_{k=0}^n \frac{1}{k!} + e^\theta \frac{1}{(n+1)!}## with ##0< \theta < 1##, show the inequality ##0 < n!e-a_n<\frac{e}{n+1}## where ##a_n## is a natural number. Use this to show that ##e## is irrational. (Hint: set ##e=p/q## and ##n=q##)...
  10. T

    Understanding proof for Heisenberg uncertainty

    I've uploaded a proof of the Heisenberg uncertainty principle from Konishi's QM. I just don't quite understand one part: what is the significance of the discriminant being less than or equal to 0? Wouldn't this just result in ## \alpha = R \pm iZ ##? Why would this be desired in this proof?
  11. RJLiberator

    Proof: There Exists a prime p such that p=< sqrt(n)

    Homework Statement Question: Let n> 1 be an integer which is not prime. Prove that there exists a prime p such that p|n and p≤ sqrt(n). Homework Equations Fundamental theorem of arithmetic: Every integer n >1 can be written uniquely (up to order) as a product of primes. The Attempt at a...
  12. TheMathNoob

    Proving a Cycle Exists in Finite Graphs with Degree 2+ Vertices

    Homework Statement Exercise 0.1. Suppose that G is a finite graph all of whose vertices has degree two or greater. Prove that a cycle passes through each vertex. Conclude that G cannot be a tree. Homework EquationsThe Attempt at a Solution If every vertex in a graph G has degree two or...
  13. TheMathNoob

    Proving a Tree is a Minimally Connected Graph in Graph Theory

    Homework Statement Let G be a connected graph. We say that G is minimally connected if the removal of any edge of G (without deleting any vertices) results in a disconnected graph. (a) Show that a connected, minimally connected graph has no cycles. (b) Show that a connected graph with no cycles...
  14. H

    Proof of the Rodrigues formula for the Legendre polynomials

    How is the below expression for ##a_{n-2k}## motivated? I verified that the expression for ##a_{n-2k}## satisfies the recurrence relation by using ##j=n-2k## and ##j+2=n-2(k-1)## (and hence a similar expression for ##a_{n-2(k-1)}##), but I don't understand how it is being motivated. Source...
  15. P

    MHB Help with Non-Triangle Proof Needed

    can someone help me with another proof? it isn a triangle proof but this is the closest forum chat i could find
  16. Alpharup

    Is the Limit of a Function at a Point Always Unique?

    Spivak proves that limit of function f (x) as x approaches a is always unique. ie...If lim f (x) =l x-> a and lim f (x) =m x-> a Then l=m. This definition means that limit of function can't approach two different values. He takes definition of both the limits. He...
  17. TheMathNoob

    Does Induction Work for Proving Graph Theory Statements?

    Homework Statement Prove that a complete graph with n vertices contains n(n − 1)/2 edges. Homework EquationsThe Attempt at a Solution The solution gives and inductive proof, but I am just wondering if this works as well. If we have a set of n vertices or points and we try to match all...
  18. TheMathNoob

    Proof about isomorphism (Graph Theory)

    Homework Statement 1. Prove or disprove up to isomorphism, there is only one 2-regular graph on 5 vertices. Homework EquationsThe Attempt at a Solution I am making this thread again hence I think I will get more help in this section old thread...
  19. TheMathNoob

    Proof about isomorphism (Graph Theory)

    Homework Statement 1. up to isomorphism, there is only one 2-regular graph on 5 vertices. Homework EquationsThe Attempt at a Solution I am still working on the problem, but I don't understand what up to isomorphism means. Does it mean without considering isomorphism?. I just need help with...
  20. R

    How Can I Prevent Vibration-Induced Disengagement in a Quadcopter Arm Mechanism?

    Hi All. I am stuck in a problem. Please check the image attached. It's part of a foldable mechanism of a quad copter arm. The red part is fixed to the body, the grey part is fixed to arms. The transparent part is a threaded collar/sleeve. The yellow part is a stopper. The hinge is a connecting...
  21. T

    Proving an equality using induction proof not working

    Homework Statement I work out the problem completely and it does not equal out. Having problems with two variable induction proofs (n and k) in this problem. Below is as far as I got, jpeg below Homework EquationsThe Attempt at a Solution
  22. A

    Understanding Completeness Property Proof

    Homework Statement I'm reading Goldrei's Classic Set Theory, and I'm kind of stuck in the completeness property proof, here is the page from googlebooks...
  23. E

    Proof - Trace-preserving quantum operations are contractive

    Homework Statement Let \mathcal{E} be a trace-preserving quantum operation. Let \rho and \sigma be density operators. Show that D(\mathcal{E}(\rho), \mathcal{E}(\sigma)) \leq D(\rho,\sigma) Homework Equations D(\rho, \sigma) := \frac{1}{2} Tr \lvert \rho-\sigma\rvert We can write...
  24. J

    Proof of time independence for normalization of wavefunction

    Hi pf, I am having trouble with understanding some of the steps involved in a mathematical proof that a normalized wavefunction stays normalized as time evolves. I am new to QM and this derivation is in fact from "An introduction to QM" by Griffiths. Here is the proof: I am fine with most of the...
  25. kaliprasad

    MHB Prove: $(\sin \theta+ i \cos \theta)^8 = \cos 8\theta - i \sin 8\theta$

    Prove that $(\sin \theta+ i \cos \theta)^8 = \cos 8\theta - i \sin 8\theta$
  26. Odious Suspect

    Geometric proof cross product distributes over addition

    If the cross product in ℝ3 is defined as the area of the parallelogram determined by the constituent vectors joined at the tail, how does one go about proving this product to distribute over vector addition? I've attached a drawing showing cyan x yellow, cyan x magenta, and cyan x (magenta +...
  27. A

    Proof of product rule for gradients

    Can someone please help me prove this product rule? I'm not accustomed to seeing the del operator used on a dot product. My understanding tells me that a dot product produces a scalar and I'm tempted to evaluate the left hand side as scalar 0 but the rule says it yields a vector. I'm very confused
  28. P

    "Wave function Collapse" 'proof' articles

    I have seen a number of references to apparent experimental "proof" of wavefunction collapse www.nature.com/articles/ncomms7665 However, I am still seeing propagation of the "Many Worlds" theory, which, and I admit that my understanding is limited, but the MW hass at its very core, a necessary...
  29. F

    How to prove that H_a and H_b are orthogonal?

    1. Okay, so I am going to prove that \int H_a\cdot H_bdv=0 Hint: Use vector Identities H is the Magnetic Field and v is the volume. Homework Equations this this[/B] k_bH_b=\nabla \times E_b k_aH_a=\nabla \times E_a k is the wave vector and E is the electric field The Attempt at a...
  30. Ebenshap

    Archimedes' area of a sphere proof; proposition 2

    Hi, I wanted to see if I could understand Archimedes' proof for the area of a sphere directly from one of his texts. Almost right away I was confused by the language. Archimedes lists a bunch of propositions that eventually lead up to the 25th proposition where the area of the sphere is finally...
  31. moriheru

    Foundations Proof of x^2n beeing even and other fundamental proofs

    Is there a book containing fundamental proofs such as any number of the form x^2n beeing even and such. I know this is very vague, so I must apologize. Thanks for any help.
  32. Alpharup

    Spivak Thomae's Function proof explanation

    I am using Spivak calculus. Now Iam in the chapter limits. In pages 97-98, he has given the example of Thomaes function. What he intends to do is prove that the limit exists. He goes on to define the thomae's function as f(x)=1/q, if x is rational in interval 0<x<1 here x is of the form p/q...
  33. V

    What is the "Book proof" of Euler's formula?

    The eccentric mathematician Paul Erdos believed in a deity known as the SF (supreme fascist). He believed the SF teased him by hiding his glasses, hiding his Hungarian passport and keeping mathematical truths from him. He also believed that the SF has a book that consists of all the most...
  34. S

    How to Derive Raising and Lowering Operators from Ladder Operator Definitions?

    Homework Statement Define n=(x + iy)/(2)½L and ñ=(x - iy)/(2)½L. Also, ∂n = L(∂x - i ∂y)/(2)½ and ∂ñ = L(∂x + i ∂y)/(2)½. with ∂n=∂/∂n, ∂x=∂/∂x, ∂y=∂/∂y, and L being the magnetic length. Show that a=(1/2)ñ+∂n and a†=(1/2)n -∂ñ a and a† are the lowering and raising operators of quantum...
  35. B

    What is different between "holds" and "holds true"?

    Hello! I am currently studying the analysis, and I have a quick question. Whenever i claim (in proof) that a statement P holds for some x in R, can I assume that P holds true for some arbitrary numbers in R but not for all possible numbers in R? What is a difference between the terms "holds"...
  36. B3NR4Y

    Proof that an even degree polynomial has a minimum

    Homework Statement Let $$p(x) = a_{2n} x^{2n} + ... + a_{1} x + a_{0} $$ be any polynomial of even degree. If $$ a_{2n} > 0 $$ then p has a minimum value on R. Homework Equations We say f has a minimum value "m" on D, provided there exists an $$x_m \in D$$ such that $$ f(x) \geq f(x_m) = m $$...
  37. Y

    Proof of Wye-Delta Transform Doesn't Make Sense

    In the picture taken from my book, in the bottom red box, it states that the equivalent resistance seen between terminals 1 and 2 is R1 + R3, implying R1 and R3 are in series. But clearly, there is a third resistor R3 at the same node where R1 and R2 meet. Then that means R1 and R3 cannot be in...
  38. entropy1

    B Proof Photons Exist? Evidence & Facts

    This may seem an odd question, but I'd really like to find out: is there proof that photons actually exist?
  39. O

    Integration Proof with constant

    Homework Statement Prove that if f'(x) = g'(x) for all x in an interval (a,b) then f-g is constant on (a,b) then f-g is constant on (a,b) that is f(x) = g(x) + C Homework Equations Let C be a constant Let D be a constant The Attempt at a Solution f(x) = antiderivative(f'(x)) = f(x) + C g(x)=...
  40. MAGNIBORO

    This Proof is right about sums and limits?

    hello, sorry for bad English, i have a question. if we consider the following equations and we take natural values note that tend 2 x-1=0 -----------------> x = 1 x^2-x-1=0 ----------------->...
  41. J

    Markov Chain - Time Reversibility proof

    Homework Statement Let X = {Xn : n ≥ 0} be an irreducible, aperiodic Markov chain with finite state space S, transition matrix P, and stationary distribution π. For x,y ∈ R|S|, define the inner product ⟨x,y⟩ = ∑i∈S xiyiπi, and let L2(π) = {x ∈ R|S| : ⟨x,x⟩ < ∞}. Show that X is time-reversible...
  42. RJLiberator

    Inner product propety with Scalar Matrix (Proof)

    Homework Statement Let A be an nxn matrix, and let |v>, |w> ∈ℂ. Prove that (A|v>)*|w> = |v>*(A†|w>) † = hermitian conjugate Homework EquationsThe Attempt at a Solution Struggling to start this one. I'm sure this one is likely relatively quick and painless, but I need to identify the trick...
  43. M

    A proof for a given formula of a normalization constant phi4

    Homework Statement Show that phi_n will find the proper phi_4. IE: show that it gives the correct normalization constant. Richard Liboff...chapter 7 Homework Equations A_n = (2^n * n! * pi^1/2)^-1/2 The Attempt at a Solution I don't know where to start really. I tried some things with <...
  44. A

    Proof that in Brayton reverse cycle heat is negative

    Homework Statement I was given the Brayton reverse cycle and asked to prove that the total heat is negative (hence the heat pump cools the system). I was to assume that all the steps are reversible. Homework Equations An ideal diatomic gas. 1st step: adiabatic compression. 2nd step...
  45. T

    MHB Alll Positive Integers proof by contraposition

    For all positive integers $n$, $r$, and $s$, if $rs \le n$ then $r \le\sqrt{n}$ or $s \le \sqrt{n}$ Proof: Suppose $r$ , $s$ and $n$, are integers and $r > \sqrt{n}$ and $ s > \sqrt{e}$. Multiply both sides of the first inequality by $s$. I get $sr > s\sqrt{n} $, but the book gives $rs >...
  46. T

    MHB Prime numbers proof by contradiction

    For prime numbers, $a$, $b$, $c$, $a^2 + b^2 \ne c^2$. Prove this by contradiction. So, I get that $a^2 = c^2 - b^2 = (c - b)(c +b)$ And I get that prime numbers are the product of 2 numbers that are either greater than one, or less than the prime numbers. But I'm unsure how to go from here.
  47. T

    MHB Proof by contraposition for dividing integers

    For all integers $a$, $b$, and $c$, if $a \nmid bc$, then $a \nmid b$ I need to prove this by contraposition. I get that by definition, $b = ak$ for some integer $k$. But I don't get the following step in the textbook: $bc = (ak)c = a(kc)$ I'm guessing there is something very obvious I'm...
  48. T

    MHB Contrapositive Proof: Ints $m$ & $n$ - Even/Odd Combinations

    For all integers $m$ and $n$, if $m+ n$ is even then $m$ and $n$ are both even or both odd. For a contrapositive proof, I need to show that for all ints $m$ and $n$ if $m$ and $n$ and not both even and not both odd, then $ m + n $ is not even. How do I go about doing this?
  49. T

    MHB Sufficient Proof: $4 \nmid n^2 - 2$ for $r=0$

    I would like to prove that this is incorrect: $\exists x \in \Bbb{Z}$ such that $ 4 | n^2 - 2$ I can use the quotient remainder theorem, $n = dq + r$ where $ 0 <= r < d $ and $ d = 4$ For the case $ r = 0$ is this sufficient proof? $n = 4q $ and $4 | n^2 - 2$ thus $4 | 16q^2 - 2$ then...
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