What is Proofs: Definition and 698 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. Dembadon

    Intro to Proofs: Properties of Relations

    Hello, I would like to check my arguments for this problem. Homework Statement Consider the relation R = \{(x,y) \in \mathbb{R} \times \mathbb{R}: x-y \in \mathbb{Z}\} on \mathbb{R} . Prove that this relation is symmetric, reflexive, and transitive. Homework Equations Supposing a relation...
  2. 1

    Hrm Congruence Proofs. Don't remember the rules

    Hrm... Congruence Proofs. Don't remember the "rules" Homework Statement Take equals sign as congruence or equals based on context, please. Itex does not work in Opera. Prove that for all n, 8^n = 1 (mod 7) Homework Equations The Attempt at a Solution This will be a proof by...
  3. T

    5 Vector Field Proofs - apparently easy

    Homework Statement http://gyazo.com/94783c14f2d2d05e62e479ab33c73830 Homework Equations I know the dot product and cross product, but even for the first one I don't see how either helps. The Attempt at a Solution 1. the gradient of the 2 scalars multiplied together (not crossed...
  4. A

    Intro to Logic: Constructing Proofs

    Hey, I'm new to the forum so I'm not sure if I posted this in the right section. I'm taking Intro to Logic and I'm having some problems. Proofs: Construct proofs for each of the following symbolic arguments. Commas are used to mark the breaks between premises. (Each proof can be completed in...
  5. K

    Studying Textbook or online website with Epsilon-Delta Proofs?

    I don't do well by just reading a proof and internalizing it. I need problems to solve and would LOVE to internalize epsilon delta proofs by practicing 100s of them. It's how I got decent at integrals. It's how anybody gets good at math and music and in general your craft right? I Don't know a...
  6. K

    Various Proofs Regarding Divisors and Properties of Divisors

    Hello there. I have been reading G.H. Hardy's book "A Course of Pure Mathematics". It is a fantastic introduction to Analysis. I have no problems with the book so far, however, it does assume some knowledge in number theory. I just want to make sure that the following proofs for properties of...
  7. M

    Thermodynamics: Proofs of work done on/by gas during adiabatic process

    Hello I'm really confused with this and would appreciate any help. Homework Statement a) Show that the work done on a gas during a quasistatic adiabatic compression is given by: W = \frac{P_f V_f - P_i V_i}{\gamma - 1} b) Show that the work done by a gas during a quasistatic...
  8. Shackleford

    Discrete Math Exam Proofs: Senioritis & Graduation

    These are potential proofs for the discrete math exam on Tuesday. I haven't been able to find proofs online. I have senioritis, and I'm graduating in a few weeks. Is a proof by contraposition the best way to prove this? If you assume h is not a function or g is not a function, then that would...
  9. G

    Do professional mathematicians remember all the proofs they come across?

    Hello, I am a college freshman currently taking Real Analysis. Calculus was fairly mechanical, and dare I say it trivial, the concepts were easy to grasp and it required little memorisation. As I have began to study more abstract areas of mathematics, I have found my speed and confidence have...
  10. K

    Proofs, Exercises & Mathematica - Training the same skills?

    First post on this forum, that IMO is amazing! I was reading the introduction of the book “A gentle introduction to the art of Mathematics” and I was wondering about what the authors wrote on whom the book is for. In particular he stated that the book is in particular for people who can...
  11. M

    Derivative product rule and other rule proofs.

    Homework Statement Prove that the functions: (u+v)'(x0) and αu and u*v are derivable. Homework Equations in other words prove that : (u+v)'(x_{0})=u'(x_{0})+v'(x_{0}) (\alpha u)'(x_{0})=\alpha u'(x_{0}) (u\cdot v)'(x_{0})=u'(x_{0})\cdot v(x_{0})+u(x_{0})\cdot v'(x_{0}) The...
  12. M

    Can Mediocrity and Proofs Co-Exist? Physics Major Asks

    I'm a physics major a bit of inclination to mathematics. The semester just ended, and I didn't particularly have a bad one. It's just I had a really mediocre grade after the semester, I'm a bit disappointed since while I'm busy reading through the proofs it seems it didn't really do me much good...
  13. M

    How Do You Differentiate Between Pure and Applied Mathematics and Start a Proof?

    Hello, all :) I was just wondering a few things: 1) what is the difference between pure mathematics and applied mathematics, and which classes do you need to take in order to get your Ph.D. in either subject? 2) I know this is a really large branch off, but I was wondering how do you start...
  14. D

    MHB Linear Algebra Proofs and Problems

    We used to have a bunch of problems and proofs that were in a pdf could be downloaded by anyone. Since we aren't able to upload pdf files of a certain size, I provided a link to google docs. If there is an error, typo, or something is just drastic wrong let me know. Undgraduate Final Review...
  15. B

    Direct Proofs: Are They Just Introductions?

    Hey guys, I'm in a proof class right now. We've covered direct proofs and moved on, but I'm still curious about them. Is there any important theorem that has even been derived using a direct proof (assume p to show q) or are they mainly just used to introduce proofs? In class, we only ever...
  16. I

    Proofs and puzzles for beginning mathematicians

    I am a freshman in High school, however I've been working quite a lot in the field of number theory for quite some time. However, I've been beginning to feel slightly bad that I haven't actually proven anything. It's not like I want to make a brand new theorem, no; but I would like to start to...
  17. S

    Why is there uncertainty in combinatorial proofs?

    There's something I can not understand about proofs in combinatorics. Whenever I solve a counting problem, there's a non-negligible amount of uncertainty about the solution which I really don't feel when I solve problems in other fields, say in analysis or abstract algebra. It happens too often...
  18. H

    How can extrema points be used to prove mathematical inequalities?

    I'm reading a math book and found a couple of proofs I can't do. 1. Given x \in R^n, a \in R, \sum\limits_{i=1}^n{x_i}=na, prove that \sum\limits_{i \in A}\prod\limits_{j = 1}^k {x_{i_j}} \leq \binom{k}{n}a^k where A = \{i \in \{1, 2, ... n\}^k : i_1 < i_2 < ... < i_k\} which essentially...
  19. Saladsamurai

    The Triangle Inequality: How to Prove It?

    Homework Statement Prove the following: (i) ##|x|-|y| \le |x-y|## and (ii) ##|(|x|-|y|)| \le |x-y|\qquad## (Why does this immediately follow from (i) ?) Homework Equations ##|z| = \sqrt{z^2}## The Attempt at a Solution (i) ##(|x|-|y|)^2 = |x|^2 - 2|x||y| + |y|^2 = x^2 - 2|x||y| + y^2...
  20. C

    Is the Set C Nonempty and Unbounded for Given Linear Programming Constraints?

    Homework Statement Let C be the set of all points (x,y) in the plane satisfying x≥0, y≥0, -x-2y≤-8. a. Show that C is nonempty and unbounded. b. Prove that the LP problem: Max M=2x+3y subject to the constraint that (x,y) lie in C has no feasible, optimal solution. c. Show that the LP...
  21. Saladsamurai

    Proofs: Absolute Values and Inequalities

    Homework Statement I am wondering if the general approach to these proofs involving absolute values and inequalities is to do them case-wise? Is that the typical approach (unless pf course you see some 'trick')? For example, I have: Prove that if |x-xo| < ε/2 and Prove that if |y-yo| <...
  22. L

    Proving \exists x \in (1, \infty): xy\geq1

    Hey! I tried to make the title as descriptive as possible, but ran out of characters. I am trying to prove that.. Homework Statement "There exists x \in (1, \infty) such that for all y \in (0,1), xy\geq1. \exists x \in (1, \infty) s.t. \forall y \in (0,1), xy\geq1. Homework...
  23. S

    Eigen Vector Proofs: Proving Real Symmetric Matrix M is Positive Definite

    Homework Statement Let M be a symmetric matrix. The eigenvalues of M are real and further M can be diagonalized using an orthogonal matrix S; that is M can be written as M = S^-1*D*S where D is a diagonal matrix. (a) Prove that the diagonal elements of D are the eigenvalues of M...
  24. K

    Proving Sigma Field Properties on Set S

    I have the following to prove: Given a sigma field/sigma algebra B on a set S. Prove: i) 0 E B ii) if B1,..,Bk E B then UBi E B for i = 1 to n and nBi E B for i = 1 to n iii) if B1,B2... E B then nBi for i = 1 to infinity E B so this is what I have so far. i) A sigma algebra is...
  25. J

    Writing mathematical proofs has greatly improved my life

    After 1-2 years of writing formal math proofs in undergraduate school, I now speak and write much more eloquently than I used to. Now, before uttering or writing a statement, I take a quick pause to ask myself whether it's logically valid; it's unambiguous; it's relevant and sequentially...
  26. J

    Math proofs vs physics proofs

    Math proofs vs physics "proofs" I'm a senior level physics major interested in taking a 400-level class in the math department for which I do not meet a prereq for (Graph Theory requires Intro to Abstract Math). I emailed the professor, and he stressed to me that a very important part of the...
  27. K

    Can someone explain these 2 linear algebra proofs

    Homework Statement The proofs: show (A')^-1 = (A^-1)' and (AB)^-1 = B^-1A^-1Homework Equations The Attempt at a Solution for the first one: (A^-1*A) = I (A^-1*A)' = I' = I A'(A^-1)' = I but I am not sure how this proves that a transpose inverse = a inverse transpose... the second i have...
  28. T

    Can Real Number Properties Simplify Complex Analysis Proofs?

    Homework Statement Let x, y, and z be real numbers. Prove the following: 1. If x * z = y * z, then x = y. 2. If x is not equal to 0, then x^2 > 0. (consider the two cases x > 0 and x < 0 ). 3. 0 < 1 4. For each n ∈ N, if 0 < x < y, then x^n < y^n 5. If x > 1, then x^2 > x...
  29. S

    Inquiry about proofs involving families of sets

    Homework Statement This post does not concern a particular problem or exercise, but instead a peculiarity (for me) in one genre: proofs involving families of sets (that is, sets containing sets as elements). I have noticed that in some statements of theorems which involve families of sets...
  30. M

    I am realizing these proofs are meaningless

    I've been looking at this proof thinking that if I read it over and over that what I am reading that seems so obvious that something else will actually pop out that I am not realizing, but what I realized the proofs that I am reading seem meaningless and pointless. I added a paint doc with...
  31. P

    Help w/ Math Proofs: cos(n∏+θ), ln|sec x|=-ln|cos x|

    I'm having trouble with these two proofs. cos(n∏+θ)=(-1)^n cos θ ln|sec x|= -ln|cos x| I know for the first one that I have to incorporate log somehow but that's about all I got from it.
  32. L

    Good Proofs for Math Prep: High School Senior

    I am a high school senior who is planning to major in math in college. I am currently in a break until the second semester of calculus at a local college starts at the end of January. I took the first half as an AP class at my school last year. I have been going back and reviewing topics from...
  33. M

    Location of proofs of trigonometric identities

    Where would be the best place to find every trigonometric identity, from sin[2] + cos[2] = 1, to the matrix identities (and Euler's equation would be helpful, also) Also the location of mathematical analysis symbols would be helpful, also. Thank you very much in advance :)
  34. L

    What is the best way to prove basic set theory statements?

    Homework Statement I'm working on some set theory stuff to prepare for Topology next semester. I'm actually working out of a Topology book from Dover Publications. I could really use some direction/correction. 1. If S ⊂ T, then T - (T - S) = S. 2. If S is any set, then ∅ ⊂ S. The...
  35. 3

    Intuition and Proofs: A Scientist's Perspective

    Homework Statement This isn't a homework question so I apologize if I'm in the wrong section, but I'm wondering if proofs are 'easy' or 'intuitive' to you. I recently took a linear algebra course in which I was sometimes able to get through the proofs without any trouble but was completely...
  36. H

    Binomial Theorem related proofs

    Homework Statement Let a be a fixed positive rational number. Choose (and fix) a natural number M>a. Use (a^n)/(n!)\leq(a^M/(M!))(a/M)^(n-M) to show that, given e>0, there exists an N\inN such that for all n\geqN, (a^n)/n! < e. Homework Equations The Attempt at a Solution In a...
  37. H

    Binomial Theorem related proofs

    Homework Statement Let a be a fixed positive rational number. Choose(and fix) a naural number M > a. a) For any n\inN with n\geqM, show that (a^n)/(n!)\leq((a/M)^(n-M))*(a^M)/(M!) b)Use the previous prblem to show that, given e > 0, there exists an N\inN such that for all n\geqN, (a^n)/(n!)...
  38. S

    Do All Uncountable Sets Share the Same Cardinality?

    Mathmatical proofs help please! [b]1. Must two uncountable sets have the same cardinality? a countable union of countable sets is countable. Is a finite set necessarily countable? If the union of A and B is infinite, then A or B must be inifinte [b]2. Just use definitions of...
  39. B

    I'm glad I could help! Let me know if you have any other questions.

    If lim x_n=x n to infinity and lim y_n=y n to infinity prove rigorously lim n to infinity (x_n/5+10y_n)=x/5+10y. My attempt let ε>0. Must find n_0 \in \mathbb{N} such that ||(x_n/5+10y_n)-(x/5+10y)||<ε for all n>n_0 ||(x_n/5+10y_n)-(x/5+10y)||=||(x_n/5-x/5)||+||10y_n-10y|| \le...
  40. I

    Trying to understand proofs, help me solve this one

    Suppose that f, g : \mathbb{R} \rightarrow \mathbb{R} are surjective (ie onto functions with domain \mathbb{R} and allowable output values \mathbb{R}). Prove that f \circ g is also surjective (ie, prove f \circ g is also onto). First of all, I have absolutely no math theory experience, so I...
  41. J

    Was plagiarizing a math proof unethical?

    Suppose there's a difficult proof on one of my homework problems in an undergrad course, and suppose I find on the internet a clever, elegant proof whose basic framework I use to construct a slightly modified proof, perhaps with some added explanation (for example, add a "because" or "since"...
  42. V

    Theoretical Biologist + Proofs

    How do you learn how to use applied math not only as a modeling tool/statistics tool/numerical analysis tool, etc., but as a theoretical tool? How do you know when a proof holds true in applied math/physics/biology? For example...
  43. R

    Intro to Proofs: Greatest Common Divisors

    Homework Statement (a) Let a and b be integers with gcd(a,b)=d, and assume that ma+nb=d for integers m and n. Show that the solutions in Z to xa+yb=d are exactly x=m+k(b/d), y=n-k(a/d) where k∈Z. (b) Let a and b be integers with gcd(a,b)=d. Show that the equation xa+yb=c...
  44. F

    What's the purpose of Epsilon proofs for limits?

    In all the problems I have done so far, the limit was already given. So the goal is to utilize the theorem to see whether the limit really holds. But what's the point? If we already know how to find the limit, why must we go through a process of ingenuity algebra to tell ourselves, "okay it...
  45. R

    Proofs Question: Equivalence Relation and Classes

    Homework Statement Define a relation ~ on ℝ by a~b if and only if a-b∈Q. i) Show that ~ is an equivalence relation. ii) Show that [a]+=[a+b] is a well-defined addition on the set of equivalence classes. Homework Equations Q represents the set of rational numbers. An Equivalence...
  46. R

    Introduction to Proofs: One-to-One and Onto Problem

    1. Given: Let f: X → Y be a function. Then we have an associated function f-1: P(Y) → P(X), where f-1 (B)⊂X is the inverse image of B⊂Y. Question: Show that f^(-1) is one-to-one if and only if f is onto. [Notes: ⊂ represents subspace, I just couldn’t find a way to put the line under the...
  47. T

    Polar Coordinates Improper Integral Proofs

    Homework Statement (a) we define the improper integral (over the entire plane R2) I=\int\int_{R^2}e^{-(x^2+y^2)}dA=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dy dx=\lim_{a\rightarrow\infty}\int\int_{D_{a}} e^{-(x^2+y^2)} dA where Da is the disk with radius a and center the...
  48. S

    How to Translate and Prove a Complex Predicate Logic Statement?

    Homework Statement No matter what positive real number x we choose, there exists some positive real number y such that yz2 > xz + 10 for every positive integer z. Translate the above statement to predicate logic and prove it using a direct approach. Homework Equations I don't...
  49. I

    Introduction to Proofs texts/resources?

    Homework Statement My intro to Proofs class uses How to Prove It, 2nd edition by Velleman. I would like a couple other references on introduction to proofs. What do you recommend? I don't mind spending hours agonizing over proofs, but I'd like to be able to check my work with answers...
  50. E

    Logical Proofs Regarding Sets and Subsets

    Homework Statement The following is all the information needed: Homework Equations There are, of course, all the basic rules of logic and set identities to be considered. The Attempt at a Solution Not really sure how to attempt this one, to be honest. I know that (A ⊆ B) can...
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