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.

View More On Wikipedia.org
Recent contents Top users
  1. The-Mad-Lisper

    Proof for Convergent of Series With Seq. Similar to 1/n

    Homework Statement \sum\limits_{n=1}^{\infty}\frac{n-1}{(n+2)(n+3)} Homework Equations S=\sum\limits_{n=1}^{\infty}a_n (1) \lim\limits_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\gt 1\rightarrow S\ is\ divergent (2) \lim\limits_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\lt 1\rightarrow S\ is\...
  2. A

    MHB Master algorithm design and upper bound proof

    Hello, I am currently preparing myself for exams and I have a past exam question which I can't solve. This question concerns online learning and the following picture illustrates it: Is anyone able to help me out and propose a solution to this question?
  3. J

    Proving the Number of Leaves is One More Than Internal Nodes in Binary Trees

    Homework Statement In a binary tree all nodes are either internal or they are leaves. In our definition, internal nodes always have two children and leaves have zero children. Prove that for such trees, the number of leaves is always one more than the number of internal nodes. Homework...
  4. C

    Is decreasing resistance always a valid proof of no resistance increase?

    Is it always true that if I have any system of resistors and I calculate the resistance between two points, when I decrease the resistance of one resistor, then the resistance measured between the same two points as previously will not increase? I.E if i have lots of resistors between two...
  5. J

    Induction Proof for 2^n x 2^n Matrix Using L Transformation

    Homework Statement Attached is the problem Homework EquationsThe Attempt at a Solution The trick to solve this problem is that when we assume that it is true for a 2^n x 2^n matrix and then we expand this matrix with 1's to a 2^n+1 x 2^n+1, we can divide the resulting matrix into 4 submatrices...
  6. RJLiberator

    Does Group Cardinality Determine Element Order?

    Homework Statement If G is a group with n elements and g ∈ G, show that g^n = e, where e is the identity element. Homework EquationsThe Attempt at a Solution I feel like there is missing information, but that cannot be. This seems too simple: The order of G is the smallest possible integer n...
  7. J

    Proof of Partition of a Set with Nested Partitions | Set Theory Proof 2

    Homework Statement . Let A be a set and {B1, B2, B3} a partition of A. Assume {C11, C12} is a partition of B1, {C21, C22} is a partition of B2 and {C31, C32} is a partition of B3. Prove that {C11, C12, C21, C22, C31, C32} is a partition of A. Homework EquationsThe Attempt at a Solution I know...
  8. J

    Proving Set Equality: A Simple and Effective Method

    Homework Statement Attached is the problem Homework EquationsThe Attempt at a Solution So I have to show that each side is a subset of the other side Assume x∈ A ∪ (∩Bi) so x∈A or x∈∩Bi case 1 x∈ ∩ Bi so x∈ (B1∩B2∩B3...∩Bn) which implies x∈B1 and x∈B2 ... and x∈Bn so x∈B1∪A and x∈B2∪A...
  9. Byeonggon Lee

    B 1 = -1, which part of this proof is wrong?

    Of course 1 isn't same as -1. This proof must be wrong but I can't find which part of this proof is wrong. Could you help me with this problem? (1)$$1 = \sqrt{1}$$ (2)$$= \sqrt{(-1)(-1)}$$ (3)$$= \sqrt{(-1)} \cdot i$$ (4)$$= i \cdot i$$ $$=-1$$
  10. TyroneTheDino

    Proving or Disproving f(x) = √x as One-to-One and Onto: Homework Statement

    Homework Statement I am supposed to prove or disporve that ##f:\mathbb{R} \rightarrow \mathbb{R}## ##f(x)=\sqrt{x}## is onto. And prove or disprove that it is one to one Homework EquationsThe Attempt at a Solution I know for certain that this function is not onto given the codomain of real...
  11. a255c

    Show that a sample space is valid by verifying properties

    Homework Statement http://puu.sh/nYQqE/2b0eaf2720.png Homework Equations http://puu.sh/nYSjQ/e48cad3a8b.png The Attempt at a Solution http://puu.sh/nYYjW/174ad8267c.png My main issue is with part b) and part d). I think that part b) is mostly right, but part d) is definitely wrong and...
  12. M

    A Somewhat difficult set theory proof

    I am trying to prove that two definitions of a finite set are equivalent. 1.) A set ##A## is finite if and only if it is equipollent to a natural number ##n##. ( natural number as the set containing all the previous natural numbers including ##0## ) 2.) A set ##A## is finite if and only if...
  13. O

    B Need help with algebra in proving kinetic energy is not conserved

    I'm trying to prove that kinetic energy is not conserved in inelastic collisions using the conservation of momentum. This is the set-up. An object A of momentum ##{m_1}{v_1}## collides inelastically with object B of momentum ##{m_2}{v_2}## using momentum conservation ##P_i = P_f## {m_1}{v_1} +...
  14. M

    Logic: proof that [p -> (q ^ r)] <=> [(p ^ -q) -> r]

    I'm preparing for college on my own. I need to proof that: [p -> (q v r)] and [(p ^ -q) -> r] are logically equivalent. with 1) v "or" 2) ^ "and" 3) -q "negation of q" I did this using truth tables and this perfectly shows that those 2 statements are logically equivalent. Can someone confirm...
  15. Patrick Sossoumihen

    Mathematical Proof: Does Euclidian Geometry Hold in Riemannian?

    Some of proofs without words have been described using Euclidian Geometry, do those proofs still hold alike in Riemmanian Geometry?
  16. Battlemage!

    I Can you use proof by contradiction in the midst of induction

    In the process of doing a proof by induction, can you use a contradiction to show that if P(k) holds then P(k+1) must hold? What I mean is, after establishing that P(0) holds, can I assume that P(k) holds and that P(k+1) does not, and show that a contradiction arises, and thus conclude that if...
  17. NoName3

    MHB Does this count as a proof for the infimum?

    I want to know whether the following the counts as a proof that infimum of the set $S = \left\{2(-1)^n+\frac{5}{n^2+2}: n \in \mathbb{N}^{+} \right\}$ is $\text{inf}(S) = -2$. Let $A \subseteq X$, where $X$ is some ordered field. Then $\text{inf}(A)$ is $m \in X$ such that for any $x \in A$...
  18. W

    I Proof that the general solution of a linear equation is....

    any particular solution plus the general solution to the homogeneous equation. I'm having difficuilty understanding this proof from my lecture notes Theorem : Let T : V → W be a linear transformation. Let w ∈ W and suppose T(u0) = w T(v) = 0. where v ∈ V (the kernel ) to prove: T(u) = w...
  19. Math Amateur

    I Proof of Existence of Tensor Product .... Further Question ...

    I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ...The relevant part of...
  20. Math Amateur

    MHB Proof of Existence of Tensor Product: Cooperstein Theorem 10.1

    I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ... The relevant part of...
  21. pjgrah01

    Can b be an integer if it does not divide k for every natural number k?

    Homework Statement Prove by contradiction that if b is an integer such that b does not divide k for every natural number k, then b=0. Homework EquationsThe Attempt at a Solution I know that proof by contradiction begins by assuming the false statement: If b is an integer such that b does not...
  22. TheSodesa

    A multivariable limit problem (epsilon-delta -proof)

    Homework Statement Find the limit \lim_{(x,y)\to(2,2)}\frac{x^3-y^3}{x-y} Homework Equations \epsilon - \delta, baby: If the limit L exists, \forall \: \epsilon \: \exists \: \delta: 0 < \sqrt{(x-a)^2+(y-b)^2} < \delta \rightarrow |f(x,y)-L| < \epsilon The Attempt at a Solution By...
  23. Math Amateur

    I Proof of Existence of Tensor Product .... Cooperstein ....

    I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with the proof of Theorem 10.1 on the existence of a tensor product ... ...Theorem 10.1 reads as follows: In the above text...
  24. Math Amateur

    MHB Proof of Existence of Tensor Product .... Cooperstein ....

    I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with the proof of Theorem 10.1 on the existence of a tensor product ... ... Theorem 10.1 reads as follows:In the above...
  25. Kartik Yadav

    Proving n log n is Big-Oh of log(n!)

    To prove that n log n is big oh of log(n!), I did: n log n <= C log(n!) n log n/ log(n!) <= C Let k = 1 n > k, so for n = 2 2 log 2 / log 2 <= C 2 <= C C is an element of [2, infinity) Taking C = 2 and k = 1 can we say, n log n <= 2 log(n!) and hence n log n is big oh of log(n!) ?
  26. V

    Proof of points arbitrarily close to supremum

    Homework Statement Let S \subset \mathbb{R} be bounded above. Prove that s \in \mathbb{R} is the supremum of S iff. s is an upper bound of S and for all \epsilon > 0 , there exists x \in S such that |s - x| < \epsilon . Homework Equations **Assume I have only the basic proof...
  27. C

    Proof of Baker-Campbell-Hausdorff involving Bernoulli numbers

    Homework Statement The BCH formula states that the product of two exponentials of non commuting operators can be combined into a single exponential involving commutators of these operators. One may write that ##\ln(e^A e^B) = \sum_{n \geq 1} c_n(A,B)## where $$c_{n+1} = \frac{1}{n+1} \left(...
  28. L

    MHB Conditional proof for multiple quantifier

    Hi, I don't know how to prove ((Ǝx) F(x) →(Ǝx) (G(x)) with conditional proof from: ((Ǝx) F(x) → (∀z) H(z)) H(a) →G(b) Thanks
  29. TheMathNoob

    Proving Graph Theory with Group Permutations | G = Sn and S Set

    Homework Statement The problem is attached. I don't get this part. Let G = Sn be the group of all permutations of S. S is a set, so how can we permute something in a set?. Neither I know if the 4 power in the S is a typo. Homework EquationsThe Attempt at a Solution
  30. TheMathNoob

    Automorphism proof (graph theory)

    Homework Statement The problem is attached and it's part A. There is no need to put problem 4 hence the problem is fully explained in the file attached Homework Equations Zk is mod k basically. The Attempt at a Solution I know that we have to prove that the transformation is onto,one to one...
  31. SrVishi

    Other Proof Tips for Math Majors: Logic & Techniques for Real Analysis

    Every math major eventually learns logic and standard proof techniques. For example, to show that a rigorous statement P implies statement Q, we suppose the statement P is true and use that to show Q is true. This, along with the other general proof techniques are very broad. A math major would...
  32. G

    Proof of NP-completeness via reduction

    Homework Statement You are given some undirected graph G = (V, E), along with a set S which consists of 0 or more pairs of G's edges. As an example, a complete graph on 3 vertices (a triangle, basically) would be described as follows: G = (V, E) = ({v1, v2, v3}, {v1v2, v2v3, v1v3}). A set S...
  33. K

    How Does the Direct Sum Relate to Unique Decomposition in Vector Spaces?

    During lecture, the professor gave us a theorem he wants us to prove on our own before he goes over the theorem in lecture. Theorem: Let ##V_1, V_2, ... V_n## be subspaces of a vector space ##V##. Then the following statements are equivalent. ##W=\sum V_i## is a direct sum. Decomposition of...
  34. I

    MHB Proof that ax^2 + bx + c has No Rational Zeroes if a,b, and c are Odd

    Well, let's look at how this works. Quadratic equations can have either 1, 2, or no zeroes. If it has no real zeroes, the zeroes it DOES have are complex, so that's obviously not it. Let's imagine ax^2 + bx + c = 0 has one zero, call it \alpha (Cuz it looks pretty). Then that means ax^2 +...
  35. Superposed_Cat

    Proof of expanded divided difference?

    Hey all, since I was programming a polynomial interpolater i found it easier to use the expanded divided difference $$ f[x_0 ,...,x_n] = \sum_{j=0}^{n} \frac{f(x_j)}{\Pi_{k}^{n,j \neq k} (x_j - x_k)} $$ , it works, but I can find no proof, any help/ references appreciated. Second question: how...
  36. B

    Proof that disk of charge = point charge when very far?

    Homework Statement Take the expression 21.11 (pictured below, specifically the bottom one) for the electric field above the center of a uniformly charged disk with radius R and surface charge density σ, and show that when one is very far from the disk, the field decreases with the same square...
  37. E

    Calculus Taylor Approximation Proof

    1. The question is. Show that if |nx| <1, the following is exact up to (and including) the x^2 order. The hint giving says to use the Taylor Expansion for both sides of the equation2. (1+x)^n = e^n(x-(1/2)x^2) ; the n(x-(1/2)x^2) is all an exponent3. My first attempt was to take the taylor...
  38. little neutrino

    Proof for Lorentz Transformation of Momentum: Step Explained

    Hi. In the attached proof for Lorentz transformation for momentum http://www.colorado.edu/physics/phys2170/phys2170_sp07/downloads/lorentz_transformation_E_p.pdf, there is this step that I don't understand: 1/√1-u'2/c2 = γ(1-vux/c2)/√1-u2/c2 Can someone explain how they derived this? Thanks! :)
  39. RJLiberator

    [Abstract Algebra] GCD and Relatively Prime Proof

    Homework Statement If gcd(f(x),g(x)) = 1 and m,n ∈ ℕ, show that gcd(f(x)^m, g(x)^n) = 1. Homework EquationsThe Attempt at a Solution So I had previously proved this for non-polynomials: gcd(a,b)=1 then gcd(a^n,b^n)=1 Proof: a = p1*p2*...*pn b = p1*p2*...*pm then a^n = p1^n*p2^n*...*pn^n...
  40. TeethWhitener

    Question about proof from Bishop & Goldberg

    I'm going through Bishop and Goldberg's "Tensor Analysis on Manifolds" right now and I'm stuck in Chapter 0. :H They give a proof of the statement "A compact subset of a Hausdorff space is closed" that I can't seem to wrap my head around. I'm reprinting the proof here: "Suppose that A is a...
  41. TheMathNoob

    What is Z_2^n and its role in proving the vertices of a hypercube in R^n?

    Homework Statement Attached is the problem. I didn't find anyway to apply the hamming distance to this problem, but I hope that at least this is close to show something. Homework EquationsThe Attempt at a Solution Lets consider Rn over Z 2 n, so the basis of R n under Z 2 is (0,0,………0 )all the...
  42. P

    Linear algebra : Doing a proof with a square matrix

    Homework Statement Show that all square matrix (A whatever) can be written as the sum of a symmetric matrix and a anti symmetric matrix. Homework Equations I think this relation might be relevant : $$ A=\frac{1}{2}*(A+A^{T})+\frac{1}{2}*(A-A^{T}) $$ The Attempt at a Solution I know that we...
  43. Comeback City

    Mathematical Proof that Photons have No Rest Mass

    Using these equations I am about to prove that photons have a rest mass of zero (mathematically) ________________________________________________________________________________________ E=hc/λ Photon Energy Equation E2=(pc+mc2)2 Mass-Energy Equivalence with Momentum Equation p=h/λ Momentum...
  44. A. Neumaier

    Are tracks in collision experiments proof of particles?

    I'd like to discuss the question in the title, following up on my remark quoted below. Note that I don't want to repeat the discussion in https://www.physicsforums.com/threads/tracks-in-particle-detectors-and-quantum-paths.758778 so maybe reread that one first! The traditional analysis is...
  45. RJLiberator

    Integral Domain, r^2 = r proof that r = 0 or 1

    Homework Statement Let r be an element of an integral domain R such that r^2 = r. Show that either r = 0_R or 1_R Homework Equations integral domain means no zero divisors. The Attempt at a Solution This is fundamental as 0 and 1 solve r^2 = r and are the only solutions. However, I'm not...
  46. RJLiberator

    Simple Abstract Algebra Proof: T(0_r) = 0_s

    Homework Statement Let T:R-> S be a homomorphism of rings. Show that T(0_r) = 0_s. Homework EquationsThe Attempt at a Solution First off, the terminology used is kinda confusing. I take 0_r to be the zero in R. Is this correct? For some reason I recall my teacher quickly saying that it was...
  47. C

    Clarification: proof that perfect subsets of R^k uncountable

    From Baby Rudin "Thm: Let P be a non-empty, perfect subset of R^k. Then P is uncountable. Pf: Since P has limit points, P must be infinite. Suppose P is countable, list the point of P {x1 ...xn }. Construct a sequence of nbhds. as follows. Let V1 be any nbhd of x1 . Suppose Vn has been...
  48. P

    Proof involving group of permutations of {1,2,3,4}.

    Homework Statement Let ##\sigma_4## denote the group of permutations of ##\{1,2,3,4\}## and consider the following elements in ##\sigma_4##: ##x=\bigg(\begin{matrix}1&&2&&3&&4\\2&&1&&4&&3\end{matrix}\bigg);~~~~~~~~~y=\bigg(\begin{matrix}1&&2&&3&&4\\3&&4&&1&&2\end{matrix}\bigg)##...
  49. C

    Proof: integral of continuous function is 0 so function is 0

    I've just encountered this somewhere and I need some sort of formal proof for why a continuous function ##f(x)## can equal zero because its integral is zero. Are there any out there? I've seen similar forum posts on places like Stack Exchange and one here, but I can't exactly follow the logic...
  50. Math Amateur

    MHB Vector Spaces .... Linear Dependence and Indepence .... Basic Proof Required

    In Andrew McInerney's book: First Steps in Differential Geometry, Theorem 2.4.3 reads as follows:https://www.physicsforums.com/attachments/5252McInerney leaves the proofs for the Theorem to the reader ... I am having trouble formulating a proof for Part (3) of the theorem ... Can someone help...
Back
Top