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. J

    MHB Understanding Lusin's Theorem for $\mathbb{R}$ and Its Proof

    Problem: Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be measurable. Then there exists a sequence of continuous functions $(g_n)$ such that $limg_n(x)$ exists for all $x \in \mathbb{R}$ and $limg_n(x) = f(x)$ a.e. x. Is this like Lusin's Theorem? Lusin's theorem for the real numbers? If so, how...
  2. S

    Prove that ##\psi## is a solution to Schrödinger equation

    Homework Statement For a wavefunction ##\psi##, the variance of the Hamiltonian operator ##\hat{H}## is defined as: $$\sigma^2 = \big \langle \psi \mid (\hat{H} - \langle\hat{H}\rangle)^2 \psi \big\rangle$$ I want to prove that if ##\sigma^2 = 0##, then ##\psi## is a solution to the...
  3. Fala483

    B Does Squaring 0.999... Always Result in 1?

    I am trying to prove that all numbers of the form 0.999... Squared end in a decimal value of 1. For example 0.99sq = 0.9801 0.999sq = 0.998001 Etc. Is it possible to prove for all 0.999... ?
  4. F

    I Proof that the E field inside a cylindrical resistor is constant

    I am reading a proof for this statement and I don't understand one of the steps. It is stated that since the surrounding medium is nonconductive the flow of charge at the surface has no component along the normal of the surface. From this the conclusion is drawn that the E field along the normal...
  5. M

    Analysis Proof: prove that sqrt(x_n) also tends to 0

    Homework Statement Suppose sequence x_n tends to 0 as n approaches infinity, prove that sqrt(x_n) also tends to 0 x_n is a sequence of non negative real numbers Homework EquationsThe Attempt at a Solution Proof. Let e>0. There exists an N in the naturals such that for n>N Ix_nI < e So if I...
  6. A

    Proof of an inequality with natural numbers

    Homework Statement Prove that ##\forall n \in \mathbb{N}## $$\frac{n}{2} < 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{2^n - 1} \leq n \text{ .}$$ Homework Equations Peano axioms and field axioms for real numbers. The Attempt at a Solution Okay so my first assumption was that this part...
  7. R

    Set theory: Is my proof valid?

    Homework Statement Prove the following for a given universe U A⊆B if and only if A∩(B compliment) = ∅ Homework EquationsThe Attempt at a Solution Assume A,B, (B compliment) are not ∅ if A∩(B compliment) = ∅, x∈A ∨ x∈ (B compliment), but not both If x∈A ∧ x∉(B compliment), then x∈B , because...
  8. binbagsss

    Quick Question- Hamiltonian constant proof

    Homework Statement Show that if the Lagrangian does not explicitly depend on time that the Hamiltonian is a constant of motion. Homework Equations see below The Attempt at a Solution method attached here: Apologies this is probably a bad question, but just on going from the line ##dH## to...
  9. M

    Proof by induction (summation)

    Homework Statement Prove by induction that ##\sum\limits_{k=1}^{2n} \frac{1}{k(k+1)} = \frac{2n}{2n+1}## 2. The attempt at a solution First I showed that it is true for ##n=1## and ##n=2##. Then, assuming it is true for all ##n##, I attempt to show that it is true for ##n+1##...
  10. Biker

    Y-Δ transform proof using superposition

    In the wikipedia page and on every book they proof the transformation by equaling the the equivalent resistance between any pair of terminals while disconnecting the other node.https://en.wikipedia.org/wiki/Y-%CE%94_transform Why this should make the two circuits equal? How can we apply...
  11. W

    I Proof of a Lemma regarding absolute values

    Hi all, There's this proof that I've been trying to wrap my head around but it just doesn't seem to sink in. I've attached a screenshot below. Many thanks in advance! Consider Case 1. There is a step that goes $$\text{Then} \ |r| = r$$ $$Then -|r| \leq |r| \ \text{and} \ r \leq |r|$$ Why is...
  12. H

    MHB Is ψ an Isomorphism from H to G?

    I'm trying to figure out how to prove this, but I'm unsure how to approach it. Let G and H be groups, let ϕ: G → H be an isomorphism, and let ψ be the inverse function of ϕ. Prove that ψ is an isomorphism from H to G. any help? thanks
  13. Marcin H

    Inductance of two-conductor line - Proof Questions

    I am trying to understand how to derive equations for the inductance of an x-conductor line. Any number really. But I want to understand the proof for a two-conductor line first. So to start any of these proofs you first need the equation for the per unit length inductance: I know R is...
  14. Jd_duarte

    I Hermitian Operator Proof - Question

    Hi, I am questioning about this specific proof -https://quantummechanics.ucsd.edu/ph130a/130_notes/node134.html. Why to do this proof is needed to compute the complex conjugate of the expectation value of a physical variable? Why can't we just start with < H\psi \mid \psi > ?
  15. V

    B Proof of the identity A\(A\B)=B

    I'm trying to proof an identity from Munkres' Topology A \ ( A \ B ) = B By definition A \ B = {x : x in A and x not in B} A \( A \ B) = A \ (A ∩ Bc) = A ∩ (A ∩ Bc)c = A ∩ (Ac ∪ B) = (A ∩ Ac) ∪ (A ∩ B) = ∅ ∪ (A ∩ B) = A ∩ B What did I miss?
  16. J

    MHB Find Limit of cos(x) with Inequalities | Part (b) Help

    Need advice on how to find lim of cos(x) using the inequalities provided. Also part (b) for help. Thanks.
  17. J

    MHB Can Dividing by Sin x Help Prove Continuity at x = 0?

    Not sure how to do this question. Help needed. Thanks
  18. K

    I Proof of 'Any Finite Dimensional Unitary Space is Complete'?

    In texts treating Hilbert spaces, it's usually given as an example that "any finite dimensional unitary space is complete", but I've found no proof so far and failed prove it myself.
  19. A

    MHB Help with Logic Proof: Establishing 7.

    Consider the following proof from Copi's "Symbolic Logic", p. 109: 1. (∃x) ¬Fx Assumption 2. ¬Fy Assumption 3. (∀x) Fx Assumption 4. Fy...
  20. F

    Extreme value theorem, proof question

    Homework Statement Why does ##\lim_{n \rightarrow \infty} f(x_n) = f(c)## contradict ##\lim_{n \rightarrow \infty} \vert f(x_n) \vert = +\infty##? edit: where ##c## is in ##[a,b]## Homework Equations Here's the proof I'm reading from Ross page 133. 18.1 Theorem Let ##f## be a continuous real...
  21. O

    Please help prove this fact about a ball falling....

    Why is the time it takes for a vertically thrown ball to reach max height the same as the time it takes for the same ball to fall from max height to ground level? I agree with this logically but I can’t prove it mathematically... Can you please show me the mathematical proof for this fact...
  22. E

    Proof by Induction of shortest suffix of concatenated string

    Homework Statement Wherein α, β are strings, λ = ∅ = empty string, βr is the shortest suffix of the string β, βl is the longest prefix of the string β, and T* is the set of all strings in the Alphabet T, |α| denotes the length of a string α, and the operator ⋅ (dot) denotes concatenation of...
  23. E

    Proof by Induction of String exponentiation? (Algorithms)

    Homework Statement Wherein α is a string, λ = ∅ = the empty string, and T* is the set of all strings in the Alphabet T. Homework Equations (exp-Recursive-Clause 1) : α0 = λ (exp-Recursive-Clause 2) : αn+1 = (αn) ⋅ α The Attempt at a Solution [/B] This one is proving difficult for me. I...
  24. T

    I Proof of The volume under surfaces formula

    Hello everyone, Is there a proof that takes us from the sum idea of the volume: $$\sum_{i=1}^m \sum_{j=1}^n f(x_i,y_j) \Delta x \Delta y$$ To the integral idea: $$\iint_R f(x,y) dxdy$$ Or something that relates the volume to the integral just like The Fundamental Theorem of Calculus?
  25. K

    I Is this a good proof of Schur's Lemma?

    There are plenty of proofs of Schur's lemma on the internet, but I find them hard to follow. Then I came up with my own result, but I'm not sure if it's good enough. Consider ##A v = \kappa v## and ##A v=\kappa v ##. Operating with ##D(g)## the equation then becomes ##D(g)A v = \kappa D(g) v##...
  26. TyroneTheDino

    Proof involving convex function and concave function

    Homework Statement [/B] Let X be a vector space over ##\mathbb{R}## and ## f: X \rightarrow \mathbb{R} ## be a convex function and ##g: X \rightarrow \mathbb{R}## be a concave function. Show: The set {##x \in X: f(x) \leq g(x)##} is convex. Homework Equations [/B] If f is convex...
  27. U

    Why Must n Equal 4q or 4q+2 If It Isn't 4q+1 or 4q+3?

    Homework Statement For any integer n, let A(n) be the statement: “If n 2 = 4k + 1 for some k ∈ Z, then n = 4q + 1 or 4q + 3 for some q ∈ Z.” Use proof by contradiction to show that A(n) is true for all n ∈ Z.The Attempt at a Solution [/B] the answer sheet says that since n !=4q+1 and n !=...
  28. U

    Proof by Contradiction: Converse of A(n) Holds for All n ∈ Z

    Homework Statement “If n = 3q + 1 or n = 3q + 2 for some q ∈ Z, then n 2 = 3t + 1 for some t ∈ Z.” Use proof by contradiction to show that the converse of A(n) is true for all n ∈ Z. For the proof by contradiction, on the answer sheet provided they have assumed n^2 = 3t+1 but n != 3q+1 and n...
  29. R

    B Proving a^0=1: Step-by-Step Guide

    I'm trying to prove that a^0 is = 1 So if I define a^1 to be = (a)(1) and a^n to be = (1)(a)(a)...(a) with the product being taken n times and a^m to be = (1)(a)(a)...(a) with the product being taken m times a^n * a^m would then = (1)[(a)(a)...(a) with the product being taken n times * and...
  30. U

    Proof via mathematical induction

    Homework Statement Use mathematical induction to prove that (8n − 7n − 1) is divisible by 49 for any n ∈ N. Correction by mentor for better readability: ##49\,|\,(8^n-7n-1)## The Attempt at a Solution We can see that the base case is satisfied here: n = 1, 8^1-7*1-1 = 0 and 49 | 0 is true...
  31. U

    Is the Relation Defined by 5 Dividing (2x + 3y) an Equivalence Relation on Z?

    <Moderator's note: Moved from a technical forum and thus no template.> Not sure this should be under Linear and Abstract Algebra, but regardless I need help with a question in my mathematical proofs course. Here it is: Let ∼ be a relation defined on Z by x ∼ y if and only if 5 | (2x + 3y). (a)...
  32. Math Amateur

    MHB Understanding Bland's Proof of Proposition 4.3.14: Primitive Elements of Modules

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need yet further help in order to fully understand the proof of Proposition 4.3.14 ... ... Proposition 4.3.14 reads as follows: In the above proof by...
  33. M

    Why must q be the least element for (q+1)a to be greater than b?

    Homework Statement Let a, b be natural numbers then there exists a unique pair (q,r) that are elements of the non-negative integers such that b=aq+r and 0 is less than or equal to r which is less than a I have a question regarding the existence part of the proof, now if I assumed a is less...
  34. Krushnaraj Pandya

    Proof of an inverse trigonometric identity

    Homework Statement Show that ##\arcsin 2x \sqrt{1-x^2} = 2 \arccos{x}## when 1/√2 < x < 1 Homework Equations All trigonometric and inverse trigonometric identities, special usage of double angle identities here The Attempt at a Solution I can get the answer by puting x=cosy, the term inside...
  35. Matt Chu

    How Do You Prove the Fourier Transform Definition Using Integral Evaluation?

    Homework Statement Given a continuous non-periodic function, its Fourier transform is defined as: $$f(x) = \int_{-\infty}^\infty c(k) e^{ikx} dk, \ \ \ \ \ \ \ \ \ \ \ \ \ c(k) = \frac{1}{2\pi} \int_{-\infty}^\infty f(x) e^{-ikx} dx$$ The problem is proving this is true by evaluating the...
  36. M

    Proof of Subgroup Property for Cyclic Group G: Homework Help

    Homework Statement Let G be a group. Assume a to be an element of the group. Then the set <a> = {ak I k∈ℤ} is a subgroup of G. I am confused as to why the proof makes the assumption that <a> is a subset of the set G. Homework EquationsThe Attempt at a Solution The proof I think is like the...
  37. M

    I Question regarding a sequence proof from a book

    I have a Dover edition of Louis Brand's Advanced Calculus: An Introduction to Classical Analysis. I really like this book, but find his proof of limit laws for sequences questionable. He first proves the sum of null sequences is null and that the product of a bounded sequence with a null...
  38. T

    I Proof Explanation: Showing an extension to a continuous function

    I am reading Kaplansky's text on metric spaces and this part seems redundant to me. It was stated below (purple highlight) that we need to show that the convergence of ##(f(a_n))## to ##c## is independent of what sequence ##(a_n)## converges to ##b##, when trying to prove the claim ##f(b)=c##...
  39. B

    Mathematical Analysis Proof: |x-y|<= |x|+|y|

    Homework Statement 1. Show that for all real numbers x and y: a) |x-y| ≤ |x| + |y| Homework Equations Possibly -|x| ≤ x ≤ |x|, and -|y| ≤ y ≤ |y|? The Attempt at a Solution I tried using a direct proof here, but I keep getting stuck, especially since this is my first time ever coming...
  40. E

    I Spivak's proof of Cauchy Schwarz

    I was browsing through Spivak's Calculus book and found in a problem a very simple way to prove the cauchy schwarz inequality. Basically he tells to substitute x=xᵢ/[√(x₁²+x₂²)] and similarly for y (i=1 and 2), put into x^2 + y^2 >= 2xy. Add the two cases and we get the result. The problem is...
  41. B

    Courses Applied vs Proof Based Linear Algebra

    Hi, I’m going to be entering my first year of University this fall to study physics. In my second semester I will have to take a linear algebra course; however, my school has two different lower level linear algebra courses, and I must choose one. One course is focused more on applications of...
  42. T

    I Proof that cube roots of 2 and 3 are irrational

    Proof by contradiction that cube root of 2 is irrational: Assume cube root of 2 is equal to a/b where a, b are integers of an improper fraction in its lowest terns. So the can be even/odd, odd/even or odd/odd. The only one that can make mathematical sense is even/odd. That is...
  43. S

    I What is the proof that the divergence is normal to the surface?

    If I am given a function f( x , y , z , ...) = C then the normal direction to it is simply the (unit vector of the) divergence of the function. How has this been proven?
  44. S

    Studying Physics students and proof based calculus

    Hey, I have been told to study calculus following Spivak's book. I was in an Engineering program and I have moved to a Physics one, and I want to retake calculus to really get good at it. The problem is, Spivak's seems to me like it's very proof based, and I'm having a hard time even with the...
  45. Math_Maniac

    Moment of Inertia of Solid Sphere - Proof

    So I have been having a bit of trouble trying to derive the moment of inertia of a solid sphere through its center of mass. Here is my working as shown in the attached file. The problem is, I end up getting a solution of I = (3/5)MR^2, whereas, in any textbook, it says that the inertia should...
  46. S

    I Difference between Constructive proof and Existential Generalization?

    What is the difference between Constructive Proof of existence and Existential generalization? Logically they seem to be the same because, for a given predicate and specific member of the predicate's domain, you are concluding the general statement about the predicate.
  47. I

    [Linear Algebra] Linear transformation proof

    Homework Statement Let ##V## and ##W## be vector spaces, ##T : V \rightarrow W## a linear transformation and ##B \subset Im(T)## a subspace. (a) Prove that ##A = T^{-1}(B)## is the only subspace of ##V## such that ##Ker(T) \subseteq A## and ##T(A) = B## (b) Let ##C \subseteq V## be a...
  48. Mathmellow

    MHB Understanding induction proof of pigeonhole principle

    I am struggling to understand the induction proof of the pigeonhole principle in my textbook. The theorem and the proof, from Biggs Discrete Mathematics, is pasted below, and I will explain further (see bold text) what I am having trouble with. Theorem. Let m be a natural number. Then the...
  49. T

    I Zorn's Lemma: Need help finding errors in proof

    Proposition(Zorn's Lemma): Let ##X\neq\emptyset## be of partial order with the property that ##\forall Y\subseteq X## such that ##Y## is of total-order then ##Y## has an upperbound, then ##X## contains a maximal element. Proof: Case 1: ##B\neq\emptyset## such that ##B##=##\{####b\in X##: ##b##...
  50. R

    Intro Math What to read after "Book of Proof?"

    Hi since U.S. education is shite, I've decided that I'm going to learn math from the ground up by myself. My goal is to reach graduate level mathematics in 2-3 years. I'm currently reading Book of Proof, what should I read after this? My end goal is to be proficient in applied math/ physics.
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