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. C

    Capacitance vs. Resistance Proof

    Can somebody confirm if this is correct? I'm trying to use a wye-delta transformation on capacitors to solve for equivalent capacitance, but to be super-precise, I want to put capacitance in terms of resistance. I = C*(dV/dt) V = IR, so I = V/R V/R = C*(dV/dt) (V*dt) = R*C* dV Integrate both...
  2. M

    Ordered set proof review request

    Homework Statement , relevant equations, and the attempt at a solution are all in the attached file. I was reading through Invitation to Discrete Mathematics and attempted to solve an exercise that involved a proof. I've typeset everything in LaTeX and made a PDF out of it so that it does not...
  3. Memocyl

    Mathematical Proof of Kepler's First Law of Orbits

    Hello friends (I hope :biggrin:), For a maths project I am working on, I need to be able to prove the equation for an elliptical orbit, related to Kepler's first law: and p = a(1-e2) (or should be as p can be replaced by that value) Where: r = distance from sun to any point on the orbit p =...
  4. Danielm

    Proving the Bijectivity of a Function: σ : Z_11 → Z_11 | Homework Solution

    Homework Statement Let σ : Z_11 → Z_11 be given by σ([a]) = [5a + 3]). Prove that σ is bijective. Homework EquationsThe Attempt at a Solution I am just wondering if I can treat σ as a normal function and prove that is bijective by using the definitions of one to one and onto.
  5. Danielm

    Proving R = I_X: Equivalence Relation and Function Homework Solution

    Homework Statement Let X be a set and R ⊂ X × X. Assume R is an equivalence relation and a function. Prove that R = I_X, the identity function. Homework EquationsThe Attempt at a Solution Proof We know that R has to be reflexive, so for all elements b in X, bRb but b can't be related to any...
  6. Biker

    Electric Field Proof: Can Point A Have a Higher Field Than Point B?

    I had a thought about electric fields created by charges Look at this picture: Point ##B## is at the half the distance between ##q## and ##2q##. What I am trying to prove/disprove That there might be actually a point (##A##) near of charge ##2q## that might have an electric field stronger than...
  7. entropy1

    I Proof of unitarity of time evolution in Susskind's book

    In "The Theoretical Minimum" of Susskind (p.98) it says that if we take any two basisvectors |i \rangle and |j \rangle of any orthonormal basis, and we take any linear time-development operator U, that the inner product between U(t)|i \rangle and U(t)|j \rangle should be 1 if |i \rangle=|j...
  8. Danielm

    Proving Linear Independence and Spanning in Vector Spaces

    Homework Statement Prove the following: Let V be a vector space and assume there is an integer n such that if (v1, . . . , vk) is a linearly independent sequence from V then k ≤ n. Prove is (v1, . . . , vk) is a maximal linearly independent sequence from V then (v1, . . . , vk) spans V and is...
  9. Danielm

    Proof: Linear Dependence of Vectors in a Vector Space

    Homework Statement Prove the following theorem: Let (v1, . . . , vk) be a sequence of vectors from a vector space V . Prove that the sequence if linearly dependent if and only if for some j, 1 ≤ j ≤ k, vj is a linear combination of (v1, . . . , vk) − (vj ). Homework EquationsThe Attempt at a...
  10. D

    Mathematical proof (Drawing a help line)

    Homework Statement I'm doing quite a strict proof in school. Where we should proof something and use mathematical language and symbols. Homework Equations The Attempt at a Solution To proof what I have to proof I need to draw some help lines. As for instance the "red" one I did from A to B...
  11. B

    Postmodernists: proof of points of triviality and sophistry

    In Charles Murray's book Real Smart: Four Simple Truths For Bringing America's Schools Back To Reality, Murray writes about the postmodernists in literary criticism. His description really gets my interest. I think it would be interesting and perhaps amusing (I have a strange sense of humor)...
  12. D

    So the shortest title I can come up with is: Complex Numbers and Real Solutions

    Homework Statement z1, z2 are complex numbers. If z1z2 =/= -1 and |z1| = |z2| = 1 then number : z1 + z2 ________ 1 + z1z2 is real. Homework EquationsThe Attempt at a Solution z1 = (a+bi), z2 = (c+di)[/B] Should i use this extended form or is there a shorter...
  13. Danielm

    Counting Reflexive and Anti-Symmetric Relations on a Finite Set

    Homework Statement Let X = {1, 2, 3, 4, 5, 6}. Determine the number of relations on X which are reflexive and anti-symmetric Homework EquationsThe Attempt at a Solution This problem looks a little bit hard. Approach: consider R={(x,x),... } If there is just one pair in the relation in the...
  14. S

    B Proof that non-integer root of an integer is irrational

    I have been looking at various proofs of this statement, for example Proof 1 on this page : http://www.cut-the-knot.org/proofs/sq_root.shtml I'd like to know if the following can be considered as a valid and rigorous proof: Given ##y \in \mathbb{Z}##, we are looking for integers m and n ##\in...
  15. I

    Limit points, closure of set (Is my proof correct?)

    Homework Statement Let ##E'## be the set of all limit points of a set ##E##. Prove that ##E'## is closed. Prove that ##E## and ##\bar E = E \cup E'## have the same limit points. Do ##E## and ##E'## always have the same limit points? Homework Equations Theorem: (i) ##\bar E## is closed (ii)...
  16. edguy99

    A Is there proof that black holes really exist?

    Intriguing and informative story on gravity wave detection. Are gravastars an alternative to black holes? Is it possible the there are NO black holes? The collapse of mass into a ball of energy that presses out and stabilizes the incoming mass is a thought provoking alternative to the common...
  17. L

    Can Fermat's Little Theorem Help Disprove This Statement?

    Homework Statement . Disprove the following statement: There exists integers a, b, c, none divisible by 7, such that 7|a^3 + b^3 + c^3 Homework EquationsThe Attempt at a Solution if 7|a^3 + b^3 + c^3, then a^3 + b^3 + c^3 is congruent to 0(mod 7) if a,b,c are none divisible by 7 then I just...
  18. L

    Proving Even Integer Coefficients in Quadratic Polynomials - Homework Question

    Homework Statement Let f(x) = ax^2 + bx + c be a quadratic polynomial. Either prove or disprove the following statement: If f(0) and f(1) are even integers then f(n) is an integer for every natural number n. Homework EquationsThe Attempt at a Solution I tried different approaches such as...
  19. S

    The principle of least Action proof of minimum

    Homework Statement Reading Feynman The Principle of Least Action out of The Feynman Lectures on Physics, Vol 2. Link to text http://www.feynmanlectures.caltech.edu/II_19.html So I'm having a problem proving that, section 19-2 5th paragraf, that "Now the mean square of something that deviates...
  20. M

    MHB Can Logarithmic Functions Be Expressed as Infinite Series?

    Hi, I'm stuck on the following proof: \log[3] = \frac 1{729} \sum_{k=0}^\infty \frac 1{729^k} \left[\frac{729}{6k+1}+\frac{81}{6k+2}+\frac{81}{6k+3}+\frac 9{6k+4}+\frac 9{6k+5}+\frac 1{6k+6}\right] Manipulating and converting summands to integrals of the form $x^{-(6k+n)}$ over {x,0,3} seems...
  21. L

    Is 3^n Greater Than n^4 for All n>=8?

    Homework Statement Prove that 3^n>n^4 for all n in N , n>=8 Homework Equations The Attempt at a Solution Base case: 3^8>8^4 Inductive step Assume 3^n>n^4. Show 3^n+1>(n+1)^4 I tried a lot of approaches to get from the inductive hypothesis to what I want to show Ex: 3^n>n^4 3^n+1>3n^4...
  22. e2theipi2026

    MHB Equivalent Goldbach proof impossible question.

    Prove that the number of unordered partitions of an even number 2n into 2 composites is greater than the number of unordered partitions of an odd number 2n+1 into 2 composites for n>1 and n\ne p prime.
  23. 5

    Proving Vector Dot Products: AC · BP = 0 | Step-by-Step Guide

    Homework Statement Homework Equations let the point where the line through B and X intersects with AC be P The Attempt at a Solution [/B] I know that ACdotBP = 0 AC = AD+DC BP = PC+ CD Therefore (AD+DC)dot(PC+CD)=0 I also know that: ECdotCE = 0 BCdotDA=0 However I am stuck on...
  24. R

    Hilbert Transform Homework: Show Envelope is |m(t)|

    Homework Statement For a real, band-limited function ##m(t)## and ##\nu_v > \nu_m,## show that the Hilbert transform of $$h(t) = m(t) cos(2\pi \nu_c t)$$ is $$\hat{h}(t) = m(t) sin(2 \pi \nu_c t),$$ and therefore the envelope of ##h(t)## is ##|m(t)|.## Homework Equations Analytic signal...
  25. L

    Disproving a Polynomial with Integer Coefficients: Elementary Math Proof

    Homework Statement Disprove the following: There exists a polynomial f(x) with integer coefficients such that f(1) is even and f(3) is odd. Homework EquationsThe Attempt at a Solution It's a little bit intuitive. Proof 1 and 3 have the same parity. They are both odd so if(odd)=odd then...
  26. L

    Proving 2/5*(2^0.5)-1/7 is Irrational: Elementary Math Proof

    Homework Statement Proof 2/5*(2^0.5)-1/7 is irrational Homework EquationsThe Attempt at a Solution I did this by splitting the expression and setting contradictions 2/5->rational 2^0.5->irrational Proof first rational times irrational is irrational Proof by contradiction Assume the product...
  27. P

    Explicit proof of the Jacobian inverse

    Homework Statement Given the transformations ##x^2+y^2=2*r*cos(theta)## and ##x*y=r*sin(theta)## prove the Jacobian explicitly The question then goes on to ask how r and theta are related to the cylindrical coordinates rho and phi. I think ##r=1/2*(x^2+y^2)## and hence ##r=1/2 rho## but I am...
  28. L

    Exploring the Optimality Proof of Earliest Finish Time Algorithm

    Homework Statement I am trying to understand the optimality proof of the earliest finish time algorithm. I have attached the pdf which I am reading. It's just 2 pages. I don't understand what they mean with solution still feasible and optimal (but contradicts maximality of r). An explanation...
  29. C

    Proof about min spanning tree property

    Homework Statement (a) If e is part of some MST of G, then it must be a lightest edge in some cutset of G. Homework Equations Cut property The Attempt at a Solution When the cutset has just one edge then yes it's true obviously. I am think I can do this by contradiction. Assuming e_i is part...
  30. SpartaBagelz

    Geometry Proof: Tips & Theorems to Solve It

    Mod note: Member warned that homework questions must be posted in the Homework & Coursework sections http://imgur.com/zGB2dnY Was given this problem a few weeks ago and I'm not sure how I should be approaching it. Please let me know which theorems I should look into in order to solve the problem.
  31. weezy

    I Doubt regarding proof of Clausius Inequality.

    I have attached two images from my textbook one of which is a diagram and the other a paragraph with which I am having problems. The last sentence mentions that due to violation of 2nd law we cannot convert all the heat to work in this thermodynamic cycle. However what is preventing the carnot...
  32. M

    I De Broglie Matter Waves: Where Does the Fraction Come From?

    In de Broglie's original proof of the theorem of phase harmony, the frequency of the moving wave of energy mc^2 (not the internal periodic phenomenon wave) is multiplied by the following term ##freq * ( t - \frac{\beta * x}{c} ) ## Does anyone have an idea where the fraction comes from? All...
  33. RJLiberator

    Two conjugate elements of a group have the same order PROOF

    Homework Statement Let x and y be conjugate elements of a Group G. Prove that x^n = e if and only if y^n = e, hence x and y have the same order. Homework Equations Conjugate elements : http://mathworld.wolfram.com/ConjugateElement.html The Attempt at a Solution Since y is a conjugate of x...
  34. S

    A Proof of Lorentz invariance of Klein-Gordon equation

    I would like to prove the Lorentz invariance of the Klein-Gordon equation by proving the invariance of the action ##\mathcal{S} = \int d^{4}x\ \mathcal{L}_{KG}## under a Lorentz tranformation. I would like to do this by first proving the Lorentz invariance of the ##\mathcal{L}_{KG}## and then...
  35. TheMathNoob

    Proving a/b+b/a >= 2 using Mathematical Proof | Homework Help

    Homework Statement Let a,b be in the positive reals. Prove a/b+b/a is >=2 Homework EquationsThe Attempt at a Solution I have no idea. Maybe add the two ratios: (a^2+b^2)/a*b and then try to analyze separately the numerator and denominator?
  36. K

    Courses How to succeed in upper-level math

    This question will essentially be more of a how-to plea or general help request. I'm currently studying math and I'm at the point where I've transitioned into upper-division classes, most if not all of which are proof based. To be blunt, I currently feel discouraged at the prospect of being...
  37. TheMathNoob

    Proof of Partition Property for Subset A in Universal Set U

    Homework Statement Assume {B, C, D} is a partition of the universal set U, A is a subset of U and A is not a subset of B complement, A is not a subset of C complement, A is not a subset of D complement. Prove that {A ∩ B, A ∩ C, A ∩ D} is a partition of A. Homework EquationsThe Attempt at a...
  38. S

    Proof of optimality of algorithm

    Homework Statement Given an array of positive integers A[1, . . . , n], and an integer M > 0, you want to partition the array into segments A[1, . . . , i1], A[i_1 + 1, . . . , i2], . . . , A[i_k−1 + 1, . . . , ik], so that the sum of integers in every segment does not exceed M, while...
  39. Dusty912

    How Can Phasors Represent the Function g(t) in Complex Form?

    Homework Statement Given a function g(t)=acosωt + bsinωt, where a and b are constants, show that g(t) is the real part of the complex function: keiΦeiωt for some k and Φ Remark: the complex expression keiΦ is called a phasor. If we know that g(t) has the form kcos(ωt+Φ) then we need know only...
  40. Dusty912

    Why Are Eigenvectors with Complex Eigenvalues Linearly Independent?

    Homework Statement Suppose the matrix A with real entries has the complex eigenvalue λ=α+iβ, β does not equal 0. Let Y0 be an eigenvector for λ and write Y0=Y1 +iY2 , where Y1 =(x1, y1) and Y2 =(x2, y2) have real entries. Show that Y1 and Y2 are linearly independent. [Hint: Suppose they are...
  41. I

    Proving Equivalence of 1-1 Function Statements

    Homework Statement Let ##f:S\to T## be a given function. Show the following statements are equivalent: a) ##f## is 1-1 b) ##f(A\cap B) = f(A) \cap f(B),\; \forall A,B \in S## c) ##f^{-1}(f(A)) = A,\; \forall A \subseteq S.## Homework Equations Definition: ##f## is 1-1 of ##A## into ##B##...
  42. Q

    Euler's Totient Function, Proof

    Homework Statement Suppose m, n are relatively prime. In the problem you will prove the key property of Euler’s function that φ(mn) = φ(m)φ(n). (a) Prove that for any a, b, there is an x such that x ##\equiv## a (mod m), (1) x ##\equiv## b (mod n). (2) Hint: Congruence (1) holds iff x...
  43. I

    Rational numbers, supremum (Is my proof correct?)

    Homework Statement Show that the set ##\{x \in \mathbf Q; x^2< 2 \}## has no least upper bound in ##\mathbf Q##; using that if ##r## were one then ##r^2=2##. Do this assuming that the real field haven't been constructed. Homework Equations N/A The Attempt at a Solution Attempt at proof: ##r\in...
  44. Alpharup

    Proving that a function can't take exactly same value twice

    Homework Statement Prove that there does not exist a continuous function f, defined on R which takes on every value exactly twice. Homework Equations It uses this property: 1... If f is continuous on [a,b], then there exists some y in [a,b], such that f(y)≥f(x), for all x in [a,b]The Attempt...
  45. I

    Supremum, Infimum (Is my proof correct?)

    Homework Statement Let ##A## be a nonempty set of real numbers which is bounded below. Let ##-A## be the set of all numbers ##-x##, where ##x \in A##. Prove that ##\inf A = -\sup(-A)##. Homework Equations Definition: Suppose ##S## is an ordered set, ##E\subset S##, and ##E## is bounded above...
  46. S

    Proof derivative of a vector following precession motion

    I do not get some points of this proof about the time derivative of a unit vector $\hat{u}$ (costant magnitude) which is following a precession motion. The picture is the following. I want to prove that $$\frac{d\hat{u}}{dt}=\vec{\Omega}\wedge \hat{u}.$$ I'm ok with almost all the proof except...
  47. W

    I How Do Supremum and Infimum Relate When s < t for All s in S and t in T?

    Let S and T be subsets of R such that s < t for each s ∈ S and each t ∈ T. Prove carefully that sup S ≤ inf T. Attempt: I start by using the definition for supremum and infinum, and let sup(S)= a and inf(T)= b i know that a> s and b< t for all s and t. How do i continue? , do i prove it...
  48. JulienB

    Proof that the definite integral of 1/ln(x) doesn't exist

    Homework Statement Hi everybody! I'm having a hard time to find a way to cleanly prove that ∫(1/ln(x)) dx between 1 and 2 doesn't exist. At first I thought it was because it's not bounded (Riemann criterion I believe), but then I looked at another unbounded definite integral such as ∫lnx dx...
  49. TheMathNoob

    Why does this sorting algorithm properly sort the array?

    Homework Statement Consider the following sorting algorithm for an array of numbers (Assume the size n of the array is divisible by 3): • Sort the initial 2/3 of the array. • Sort the final 2/3 and then again the initial 2/3. Reason that this algorithm properly sorts the array. What is its...
  50. S

    MHB Proof of Fredholm-Volterra Equation Convergence

    Does there exist a proof of the following: It is well known that Picard successive approximations on the Fredholm-equation (1) $y(x)=f(x)+{\lambda}_{1}\int_{a}^{b} \,k(x,s)y(s)ds$ written in operator form as $y=f+{\lambda}_{1} Ky$ converges if (2) $|{\lambda}_{1}|. ||K||<1$ where $K$...
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