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. 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...
  2. 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...
  3. 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...
  4. kaliprasad

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

    Prove that $(\sin \theta+ i \cos \theta)^8 = \cos 8\theta - i \sin 8\theta$
  5. 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 +...
  6. 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
  7. 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...
  8. 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...
  9. 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...
  10. 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.
  11. 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...
  12. 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...
  13. S

    Proof of ladder operator identity

    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...
  14. 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"...
  15. 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 $$ f(x) \geq f(x_m) = m $$...
  16. 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...
  17. 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?
  18. 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)=...
  19. 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 ----------------->...
  20. 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...
  21. 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...
  22. 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 <...
  23. 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...
  24. 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 >...
  25. 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.
  26. 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...
  27. 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?
  28. 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...
  29. AlexOliya

    Proving an Inequality: A Scientific Approach

    Homework Statement Homework Equations With the regards to posting such a incomplete equation, I will soon put in the updated one Thank you The Attempt at a Solution visual graph... didn't help
  30. S

    MHB Solve Proving 2 Triangles Cong. w/ Diam & Eq. Triangle (2-Col Proof)

    Not sure how to solve considering nothing is given as perpendicular or bisected. Is anyone aware on how to solve this problem? ~S.R.K.
  31. M

    MHB Proof that the solutions are algebraic functions

    Hey! :o I am looking at the following: I haven't really understood the proof... Why do we consider the differential equation $y'=P(x)y$ ? (Wondering) Why does the sentence: "If $(3)_{\mathfrak{p}}$ has a solution in $\overline{K}_{\mathfrak{p}}(x)$, then $(3)_{\mathfrak{p}}$...
  32. TheMathNoob

    Easy proof about spanning trees (for you)

    Homework Statement Let T be any spanning tree of an undirected graph G. Suppose that uv is any edge in G that is not in T. The following proofs are easy by using the definitions of undirected tree, spanning tree and cycle a)Let G1 be the subgraph that results from adding uv to T. Show that G1...
  33. ArcanaNoir

    Proof by combinatorial argument

    Homework Statement Give a combinatorial proof that (n-r)\binom{n+r-1}{r} \binom{n}{r}=n\binom{n+r-1}{2r} \binom{2r}{r} Homework EquationsThe Attempt at a Solution I interpreted the right side of the equation as: There are n grad students and r undergrads. First, from the n grad students...
  34. L

    Need help with Simplification step in Inductive Proof

    Homework Statement Let P (n) be the statement that 1^3 + 2^3 + · · · + n^3 = (n(n + 1)/2)^2 for the positive integer n. Prove inductively. Homework EquationsThe Attempt at a Solution [/B] I am skipping a few steps...I just need help here: 1/4K^2(k + 1)^2 + (k + 1)^3 Since I have access to...
  35. S

    Proof than an equation is Lorentz invariant

    In Peskin and Schroeder page 37, it is written that Using vector and tensor fields, we can write a variety of Lorentz-invariant equations. Criteria for Lorentz invariance: In general, any equation in which each term has the same set of uncontracted Lorentz indices will naturally be invariant...
  36. L

    Riemann's proof of the existence of definite integrals

    Hello, Since it was mentioned in my textbook, I've been trying to find Riemann's proof of the existence of definite integrals (that is, the proof of the theorem stating that all continuous functions are integrable). If anyone knows where to find it or could point me in the right direction, I...
  37. BiGyElLoWhAt

    Prove A~B=>f(A)~f(B) for a continuous f:X->Y

    So proofs are a weak point of mine. The hint is that a composite of a continuous function is continuous. I'm not really sure how to use that. What I was thinking was something to the effect of an epsilon delta proof, is that applicable? Something to the effect of: ##A \sim B\text{ and let } f...
  38. 0

    Is this a correct proof of a function's continuity?

    Hello, 1. Homework Statement 1) Let f(x) continuous for all x and (f(x)2)=1 for all x. Prove that f(x)=1 for all x or f(x)=-1 for all x. 2) Give an example of a function f(x) s.t. (f(x)2)=1 for all x and it has both positive and negative values. Does it contradict (1) ? 2. The attempt at a...
  39. B

    Is there any algebraic proof for Thevenin's theorem?

    Is there any algebraic proof for Thevenin's theorem?
  40. Z

    Proof: Every convergent sequence is Cauchy

    Hi, I am trying to prove that every convergent sequence is Cauchy - just wanted to see if my reasoning is valid and that the proof is correct. Thanks! 1. Homework Statement Prove that every convergent sequence is Cauchy Homework Equations / Theorems[/B] Theorem 1: Every convergent set is...
  41. T

    MHB Proof of Equality for Odd Integers with Floor Function

    I need to prove or disprove that $$ \left\lfloor{\frac{n}{2}}\right\rfloor= \left\lfloor{ \frac{n - 1}{ 2}}\right\rfloor$$ where n is an odd integer.I start with something like, $$\left\lfloor{\frac{2k + 1}{2}}\right\rfloor$$ and then $$\left\lfloor{k + \frac{1}{2}}\right\rfloor$$ which...
  42. Z

    Proof: A point is a limit point of S is a limt of a sequence

    Hi guys, I attempted to prove this theorem, but just wanted to see if it a valid proof. Thanks! 1. Homework Statement Prove that x is an accumulation point of a set S iff there exists a sequence ( s n ) of points in S \ {x} that converges to x Homework Equations N * ( x; ε ) is the x -...
  43. Q

    B Prove Bisecting Angle Theorem - 5 Min Exercise

    Hi, guys, i made an exercise, can you prove this? m(∠EAD)=[m(∠ABC)-m(∠ACB)]/2 If you have 5 free minutes, try it, i hope you'll like it! It's my first own exercise, so I would like some feedback, too. AD= bisecting(splits angle in 2 equal sides)
  44. M

    MHB How to prove an inequality with a direct proof?

    Hello, I'm having trouble with an assigned problem, not really sure where to begin with it: Prove that if a \in R and b \in R such that 0 < b < a, then {a}^{n} - {b}^{n} \le {na}^{n-1}(a-b), where n is a positive integer, using a direct proof. Pointers or the whole proof would be appreciated...
  45. T

    Proof of |2^N x 2^N| = |2^N| with N the natural numbers

    Hello, At my exam I had to proof the title of this topic. I now know that it can easily be done by making a bijection between the two, but I still want to know why I didn't receive any points for my answer, or better stated, if there is still a way to proof the statement from my work. My work...
  46. Z

    Real Analysis - Infimum and Supremum Proof

    Hi Guys, I am self teaching myself analysis after a long period off. I have done the following proof but was hoping more experienced / adept mathematicians could look over it and see if it made sense. Homework Statement Question: Suppose D is a non empty set and that f: D → ℝ and g: D →ℝ. If...
  47. J

    Proof of sin(θ-Φ)=sinθcosΦ-cosθsinΦ using vector algebra

    Homework Statement given two unit vectors a= cosθi + sinθi b=cosΦi+sinΦj prove that sin(θ-Φ)=sinθcosΦ-cosΦsinθ using vector algebra[/B]Homework Equations sin(θ-Φ)=sinθcosΦ-cosΦsinθ[/B]The Attempt at a Solution axb= (cosθsinΦ-cosΦsinθ)k and I'm guessing that the change in sign has...
  48. S

    Proof that SU(n) is a group

    Homework Statement Show that the set of all ##n \times n## unitary matrices with unit determinant forms a group. 2. Homework Equations The Attempt at a Solution For two unitary matrices ##U_{1}## and ##U_{2}## with unit determinant, det(##U_{1}U_{2}##) = det(##U_{1}##)det(##U_{2}##) = 1...
  49. S

    Unitary Matrices as a Group: Proof and Properties

    Homework Statement Show that the set of all ##n \times n## unitary matrices forms a group. Homework Equations The Attempt at a Solution For two unitary matrices ##U_{1}## and ##U_{2}##, ##x'^{2} = x'^{\dagger}x' = (U_{1}U_{2}x)^{\dagger}(U_{1}U_{2}x) =...
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