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.

View More On Wikipedia.org
Recent contents Top users
  1. A

    Understanding Set Operations: A Closer Look at A ∖ (A ∖ B) = A ∩ B

    ##A ∖ B## can't include any elements that are not in ##A##, so it is the same as saying ##A∖(A∩B)##; it's exactly the elements of ##A## except those in ##A∩B##. ##A∖(A∖(A∩B))## is exactly the elements of ##A## except those in (exactly the elements of ##A## except those in ##A∩B##). This is the...
  2. chwala

    Proof of ##\sqrt[n]{n!}## ≤ ##\frac{n+1}{2}##

    ##\sqrt[n]n!## ≤##\frac{n+1}{2}## ##\frac {1}{n}## log [n!]= log##\frac{n+1}{2}##
  3. V

    Unique Identities: Proving O₁ = O₂ in Theorem for Proof of Identity

    Theorem “Identities of + are unique”: O₁ = O₂ Proof: O₁ = Left Identity of + O₁ + x I'm a little confused where to begin this proof, I don't know if that is the first step either I think it is. Proofs are not a strength of mine so I struggle to see how to show that O₁ = O₂. Any guidance would...
  4. MexChemE

    My first proof ever - Linear algebra

    First, a little context. It's been a while since I last posted here. I am a chemical engineer who is currently preparing for grad school, and I've been reviewing linear algebra and multivariable calculus for the last couple of months. I have always been successful at math (at least in the...
  5. M

    Question Error? (epsilon-delta proof of a limit)

    I can not follow this mathematically. I am guessing one of the signs is incorrect? Can anyone verify ?
  6. chwala

    How did my colleague conclude on equation (ii)?

    this is a solution posted by my colleague, i have a problem in understanding how he got to conclude on equation (ii) is this not supposed to be ##(k+2)!≥ 2^{k+1} (k+1)## ##(k+2)(k+1)!≥ 2^{k+1} (k+1)## ?...
  7. H

    Proof of a dot product using sigma notation

    Mentor note: Moved from a technical section, so is missing the homework template. Hi, I'm always not sure how to prove something in math and I'm wondering if this is enough. ##\vec r \cdot (\vec u + \vec v) ## ##\vec u + \vec v = (u_1+v_1, u_2+v_2,u_3+v_3) = \vec s## ##\vec r \cdot (\vec u +...
  8. murshid_islam

    I A question about a proof of the irrationality of pi

    Here is the proof I was reading: https://mathschallenge.net/full/irrationality_of_pi I have a question about this very last inequality at the end: How did they get that "less than 1" bit? .
  9. G

    Constructive Proofs Proof of Correspondence theorem

    Summary:: I'm reading Adkins' book "Algebra. An approach via Module Theory" and I'm trying to prove theorem 3.15 In theorem 3.15 of Adkins' book says: Let ##N \triangleleft G##. The 1-1 correspondence ##H \mapsto H/N## has the property $$H_1 \subseteq H_2 \Longleftrightarrow H_1/N \subseteq...
  10. R

    I Why Torsion = 0 => Planar Curve in this Proof

    I was watching a lecture that made the conclusion about the torsion being equal to zero necessitated that the path was planar. The argument went as follows: -Torsion = 0 => B=v, which is a constant -(α⋅v)'=(T⋅v)'= 0 => α⋅v= a, which is a constant (where α is a function describing the path and...
  11. MidgetDwarf

    Help with this inequality needed for a proof in a textbook

    Note that if we prove problem 4, the proof for problem 5 follows directly. We use properties of logarithms to combine the right hand side of ln into a single logarithm. Then we raise both side of the inequality to a power of e. Which leads us to the desired inequality. But, when I try to be...
  12. karush

    MHB How do I find the value of delta in a delta epsilon proof using a graph?

    Find a graph to a number $\delta$ such that $$\textit{if } |x-1|<\delta \textit{ then } \left|\dfrac{2x}{x^2+4}-0.4\right|<0.1 $$ ok I always had a very hard time doing these I did look at some examples but still ? did a ibispaint drawing to start basically it looks like we are finding the...
  13. fresh_42

    Constructive Proofs (open) Little Fermat Proof

    Show by combinatoric means that ##p\,|\,(a^p-a) ## for a prime ##p \in \mathbb{P}## and a positive integer ##a\in\mathbb{N}##. Hint: e.g. consider chains of ##p## coloured pearls where ##a## is the number of colours.
  14. fresh_42

    Indirect Proof (open) The most basic math proof I've ever seen

    Well, it is what I consider the most basic proof I've ever seen. I like it for several reasons: it is so simple. that you can convince a preschool kid, it uses one of the most fundamental principles at all: you cannot derive false from true, and it is old enough to count as such: No, sorry...
  15. fresh_42

    Math Proof Training Camp and Practice Guidelines

    Math Proof Training Camp and Practice Guidelines We frequently get questions on how and where to learn to prove statements. The vast majority of scientists learned them by doing. It is a try and error approach trained in tutorials and exams. This forum is meant to practice these techniques on a...
  16. U

    I Unclear step in "Change of variable in a multiple integral" proof

    I'm studying the proof of this theorem (Zorich, Mathematical Analysis II, 1st ed., pag.136): which as the main idea uses the fact that a diffeomorphism between two open sets can always be locally decomposed in a composition of elementary ones. As a remark, an elementary diffeomorphism...
  17. U

    I Unclear steps in a Zorich proof (Measurable sets and smooth mappings)

    From Zorich, Mathematical Analysis II, sec. 11.5.2: where as one can read from the statement, the sets could also be unbounded. I do not report here the proof of the fact a), beacuse I have no doubt about it and one can, without the presence of dark steps in the reasoning, assume a) as...
  18. P

    I What proof do we have that the universe is homogenous?

    In the book, it states that a universe is isotropic if it looks the same regardless of which direction you look at large enough scales. This seems fairly easy to prove these days with observations from galaxy surveys and the CMB. However, how can we possibly prove that the university is...
  19. T

    B Any proof of extra dimensions?

    As of 2020 is there any proof of tiny extra dimensions or observations of them?
  20. F

    MHB Proof by induction? No Idea what I should do :(

    find the general rule and prove by induction 1 = 1 1 - 4 = -(1 + 2) 1 - 4 + 9 = 1 + 2 + 3 1 - 4 + 9 -16 = -(1 + 2 + 3 + 4) I created this so far, but don't know if I am even going the correct direction
  21. W

    MHB Can someone help me in the right direction with this proof

    A square table of size 1001x1001 is filled with the numbers 1; 2; 3; ... ; 1001 in such a way that in every row and every column all those numbers appear. If the table is symmetric with respect to one of its diagonals, prove that in this diagonal all of the numbers 1; 2; 3; ... ; 1001 appear.
  22. T

    I Alternative proof to a trivial problem

    Hi.I have this trivial problem for a metric d(x,y) that d((x,y)≥0. My alternative proof is 2d(x,y)=√4d2(x,y)=√d2(x,y)+d2(y,x)+2d(x,y)d(y,x)=√(d(x,y)+d(y,x))2≥d(x,x)=0 .Well it perhaps is a trivial proof but I did not know of this proof so I wanted to post it. Do you know other alternative proofs...
  23. anemone

    MHB Proof: $\cos\dfrac{\pi}{7}=\dfrac{1}{6}+\dfrac{\sqrt{7}}{6}$

    Prove that $\cos\dfrac{\pi}{7}=\dfrac{1}{6}+\dfrac{\sqrt{7}}{6}\left(\cos\left(\dfrac{1}{3}\arccos\dfrac{1}{2\sqrt{7}}\right)+\sqrt{3}\left(\dfrac{1}{3}\cos\dfrac{1}{2\sqrt{7}}\right)\right)$.
  24. C

    MHB Can You Explain the Puzzle Toad Proof?

  25. S

    MHB Logic of Set Operations: Proof

    Given: x\in A\cap B\leftrightarrow x\in A\wedge x\in B x\in A\cup B\leftrightarrow x\in A\vee x\in B x\in A-B\leftrightarrow x\in A\wedge x\notin B A=B\leftrightarrow(\forall x(x\in A\leftrightarrow x\in B)) Then prove using only the above and the laws of logic that: ™ (A\cup B)-(A\cap...
  26. MidgetDwarf

    I Explanation of a Line of a proof in Axler Linear Algebra Done Right 3r

    ∈Was wondering if anyone here could help me with an explanation as to how Axler arrived at a particular step in a proof. These are the relevant definitions listed in the book: Definition of Matrix of a Linear Map, M(T): Suppose ##T∈L(V,W)## and ##v_1,...,v_n## is a basis of V and ##w_1...
  27. F

    Proof of Monotone Classes and Subsets in Measure Theory

    My question is how to show ##\mathcal{F} \subset \sum'##. Here is my work for the problem: Proof of hint: First we'll show ##\sum## is a monotone class. Let ##(A_n)_{n\in\mathbb{N}} \subset \sum## and ##F \in \mathcal{F}##. There are two things to verify. Suppose ##(A_n) \uparrow A =...
  28. CaptainAmerica17

    I Help Finding the Correct Approach to this Proof (Intro Real Analysis)

    Ok, so here is what I have so far: Suppose ##T_1## is infinite and ##\varphi : T_1 \rightarrow T_2## is a bijection. Reasoning: I'm thinking I would then show that there is a bijection, which would be a contradiction since an infinite set couldn't possibly have a one-to-one correspondence...
  29. thaiqi

    I What does this proof mean? (variation of high-order derivative)

    I read in one book proving one nature of variation(variation of high-order derivative). It writes that "##\delta(F^{(n)}) = F^{(n)} - F_0^{(n)} = (F - F_0)^{(n)} = (\delta F)^{(n)}##". But I don't understand where this ##F_0## comes out from.
  30. Rainbow Cupcake

    Proof by mathematical induction

    Summary:: prove that (n 0) + (n 1) + (n 2) + ... + (n n) = 2^n is true using mathematical induction. note that (n n) is a falling factorial Hello! I have trouble dealing with this problem: Mod note: Thread moved from math technical section, so is missing the homework template. Prove that (n...
  31. dk123

    Proof that a projectile comes to rest in a finite distance

    equation of motion : while
  32. U

    I Proof involving functional graphs and the injective property

    Definition: Let ##G## be a graph. ##G## is a functional graph if and only if ##(x_1,y_1) \in G## and ##(x_1,y_2) \in G## implies ##y_1=y_2##. Problem statement, as written: Let ##G## be a functional graph. Prove that ##G## is injective if and only if for arbitrary graphs ##J## and ##H##, ##G...
  33. E

    B Understanding Benford's Law Proof: Scaling and Invariance

    The significant digits of numbers in sets of numerical data supposedly follows "Benford's Law", which asserts that the probability that the first digit in a given data point is ##D## is about ##\log_{10}(1+ \frac{1}{D})##. An upshot is that we expect ~30% of significant digits to be ##1##. The...
  34. Eclair_de_XII

    How to prove divergence of harmonic series by eps-delta proof?

    Set ##\epsilon=\frac{1}{2}##. Let ##N\in \mathbb{N}## and choose ##n=N,m=2N##. Then: ##\begin{align*} \left|s_N-s_{2N}\right|&=&\left|\sum_{l=1}^N \frac{1}{l} - \sum_{l=1}^{2N} \frac{1}{l}\right|\\...
  35. LCSphysicist

    Is there a proof about angular momentum conservation?

    Angular momentum can be exchanged between objects in a closed system, but total angular momentum before and after an exchange remains constant (is conserved). There is a proof about this conservation?
  36. U

    I Proof involving functional graphs and the injective property

    My only qualm is that the statement “Let G be a functional graph” never came into play in my proof, although I believe it to be otherwise consistent. Can someone take a look and let me know if I missed something, please? Or is there another reason to include that piece of information?
  37. CaptainAmerica17

    I Is f(x) an Injective Function? Understanding Proof and Notation

    I typed this up in Overleaf using MathJax. I'm self-studying so I just want to make sure I'm understanding each concept. For clarification, the notation f^{-1}(x) is referring to the inverse image of the function. I think everything else is pretty straight-forward from how I've written it. Thank...
  38. Arman777

    I Proof that two timelike vectors cannot be orthogonal

    For fun, I decided to prove that two timelike never can be orthogonal. And for this, I used the Cauchy Inequality for that. Such that The timelike vectors defined as, $$g(\vec{v_1}, \vec{v_1}) = \vec{v_1} \cdot \vec{v_1} <0$$ $$g(\vec{v_2}, \vec{v_2}) = \vec{v_2} \cdot \vec{v_2} <0$$ And the...
  39. L

    Law of Conservation of Energy Proof

    Although I am not too sure how to answer this quesion I have tried below. I realize that an electromotive force is a supply voltage, the energy transferred per unit charge when one type of energy is converted into electrical energy. However, EMF is not actually a force. It is usually measured...
  40. CaptainAmerica17

    Would someone mind checking my proof? Intro Real Analysis

    Here is my solution. I used mathjax to type it up in Overleaf. I feel like it makes sense, but I also have a feeling I might have "jumped the gun" with my logic. If it is correct, I would appreciate feedback on how to improve it. Thanks!
  41. chwala

    Proof of Trigonometry Identity: a^2b^2=sin^2θcos^2θ

    ##a^2= \frac {sin^4θ} {cos^2θ}## ##b^2=\frac {cos^4θ} {sin^2θ}## therefore ##a^2b^2= sin^2θ. cos^2θ##...1 and ##a^2+b^2+3= \frac {sin^6θ + cos^6θ + 3 sin^2θ cos^2θ} {sin^2θ cos^θ}##...2 multiplying 1 and 2, we get, ##sin^6θ + cos^6θ + 3 sin^2θ cos^2θ## how do i proceed from...
  42. D

    MHB Request to Solve Symbolic Logic Questions Using Strengthened Conditional Proof

    Sir/madam, I request you to solve 2 questions ( q-3 and q-5 ) of symbolic logic ( Strenthened method of conditional proof ). These questions are taken from I.M.Copi's 'symbolic logic' ( edition -5, sec. 3.8, pg- 61 ) File is being attached. thank you yours truly Deep Kumar Trivedi
  43. T

    A Algebraic proof that Euler angles define a proper rotation matrix

    I have asked this question twice and each time, while the answers are OK, I am left dissatisfied. However, now I can state my question properly (due to the last few responses). Go to this page and scroll down to the matrix for sixth row of the proper Euler angles...
  44. O

    How to prove this statement about the derivative of a function

    My try: ##\begin{align} \dfrac{d {r^2}}{d r} \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \tag1\\ \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \dfrac{1}{\dfrac{d r^2}{d r}}=\dfrac{p-a\cos\theta}{r} \tag2\\ \end{align}## By chain rule...
  45. RicardoMP

    Proof of the commutator ## [P^2,P_\mu]=0 ##

    I want to make certain that my proof is correct: Since ## P^2 = P_\nu P^\nu=P^\nu P_\nu ##, then ## [P^2,P_\mu]=[P^\nu P_\nu,P_\mu]=P^\nu[P_\nu,P_\mu]+[P^\nu,P_\mu]P_\nu=[P^\nu,P_\mu]P_\nu=g^{\nu\alpha}[P_\alpha,P_\mu]P_\nu=0 ##, since ## g^{\nu\alpha} ## is just a number, I can bring it...
  46. T

    Vector field equality Curl Proof of Moving Magnet & Conductor Problem

    The moving magnet and conductor problem is an intriguing early 20th century electromagnetics scenario famously cited by Einstein in his seminal 1905 special relativity paper. In the magnet's frame, there's the vector field (v × B), the velocity of the ring conductor crossed with the B-field of...
  47. D

    How Can I Use Axioms to Prove x^2 ≥ 0?

    (x)(x)>0 (D) (x+(-x))(x+(-x)) >0 (A4) x^2 + 2(-x)(x) + (-x)^2 >0 (D) x^2 - 2x^2 + (-x)^2 >0 -x^2 + (-x)^2 >0
  48. S

    I Proof that limit of difference of infinite limits is indeterminate

    (NOTE: I have had a few similar postings lately on this subject, but they were much broader in scope, so I am posting only for this particular case; everything else has been figured out.) If given that limx -> a f( x ) = +∞ limx -> a g( x ) = +∞ what is the epsilon-delta formulation for...
  49. dRic2

    I Proof of the Equipartition Theorem

    The proof for the ET I've found in some of the undergrad books for statistical physics (for example in Reif's "Statistical and Thermal Physics") assumes the form of the Hamiltonian of the system to be: $$H = bp_i^2 + E'(q_1,...,p_f)$$ where ##b## is a constant. My professor in his notes, says...
  50. S

    I Proof of addition of limits when the output value is infinite

    I was looking at some websites that show the proof of addition of limits for a finite output value, but I don't see one for the case of infinite output value, which has a different condition that needs to be met - i.e., | f( x ) | > M instead of | f( x ) - L | < ε...
Back
Top