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. C

    Proving Set Subsets and the Cauchy-Schwarz Inequality: Insights and Techniques

    Lets say you are given a bunch of statements and you need to ask some questions to prove them: (a) How do you show that a set is a subset of another set. I said to show that x\in A and x\in B [/tex]. What else can you do to show what A\subset B ? Could you assume from the following: If...
  2. W

    Valid Methods for Proving Statements: Assumption and Contradiction

    I have read that a valid method for prooving a statement is to assume the opposite and show a contradiction. This tells me the assumption is an "either or". If this is not true, then that must be. Is this always valid?
  3. dextercioby

    Visual Proofs in Mathematics: Does Pictures Tell More than 1000 Words?

    People usually say that pictures tell more than 1000 words. Is that still true in mathematics...? I think so. Let me first say what i mean by 'visual proof'. Let's say we have an identity. To prove it's true one may write from a line to more than one page. But what if one was able to write only...
  4. J

    Some momentum proofs needed

    A jet of liquid of cross-sectional area A and density p moves with speed vj in the positive x-direction and impinges against a perfectly smooth blade B, which deflects the stream at right angles but does not slow it down. (a) If the blade is stationary, prove that the rate of arrival of mass at...
  5. S

    What books should I read to prepare for advanced math classes in university?

    I'll be starting university in just over a month. The school I'm going to has advanced section classes that basically cover the first year math classes (Algebra and Calculus) in a more rigorous fashion than what is usually offered to first year math students. I am interested in taking part of...
  6. P

    Experimental Proofs of Special Relativity: Myons, Bertozzi & More

    Hi Im new in this Forum. I am from Switzerland, in the first year of Physics at University. Please forgive some mistakes I might will make in english I read a little bit in advance for the next years about the SRT and its relation to other fields of study. Basically I wanted to know...
  7. A

    Are these proofs correct(bounded and finite variation).

    First of all if you read this and the latex is all messed upp I am probably working on getting it right so please be patient till I get it right. No need to post a comment that it doesn't work. Thanks :wink: I haven't taken a pure maths class in over 2,5 years so I can hardly remember how to...
  8. E

    Proofs of Basic Linear Algebra Concepts - A Guide for Beginners

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  9. E

    Proving Linear Dependence in Vector Spaces

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  10. C

    Proving Polynomial Proofs: Using the Expansion Method

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  11. C

    How to Prove the Induction Rule for Sum of Cubes?

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  12. C

    Measurable Sets/ Proofs: Apostol

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  13. G

    Smallest Set Proofs: How to Construct?

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  14. B

    Geometry (circles and triangles) proofs

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  15. H

    Prove: Every Int Ending in 5 to Square End in 25

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  16. T

    Mathematica Mathematical Proofs: Ideas Beyond Grade 12

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  17. JasonJo

    Nightmares with formal proofs in set theory

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  18. N

    Big Bang : 'Proofs' and observations

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  19. M

    Proving Evenness of n(n+1): Etiquette for Proofs

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  20. M

    Proving Triangle and Angle Theorems: Tips and Examples | Get Help Here!

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  21. Apost8

    Proving the Pythagorean Identity: A Brief Analysis

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  22. M

    Solving Trigonometric Proofs: Struggling with Two Challenging Examples

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  23. S

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  24. dextercioby

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  25. H

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  26. S

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  27. J

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  28. F

    Vector Proofs: A Quadrilateral thing #2

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  29. F

    Vector Proofs: A Quadrilateral thing

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  30. J

    Can You Solve These Prime Number Proofs?

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  31. N

    Why do we use |f(x)-L|< E instead of |f(x)-L|≤ E in delta-epsilon proofs?

    I am currently having a lot of troublw with delta-epsilon proofs--can someone please help explain how they work and how to do them Thanks
  32. F

    How to Learn Math and Write Proofs

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  33. R

    How to learn how to do math proofs?

    This year I'm a freshman at university - physics - and we are just starting with mathematical analysis. I don't find it that difficult, but my problem are proofs. They are not hard, but I sometimes can't prove even the easiest things (I know why it is so, but can't put it down on the paper). Can...
  34. S

    God, I hate Number Theory Proofs

    I really really hate proofs! I've done 3 of my 5 problems, which took me 2 days and over 30-50 pieces of scrap paper. I'm serious, I didn't even eat dinner today because I spent straight hours just staring at questions, thinking I was coming close to solutions, then only to find out I've...
  35. R

    Mathematica Proving Mathematical Statements: a Real-Life Example

    Hi all, If I have these two statements given to me, and I have to determine whether they are true or not. a) \forall x \epsilon R \exists y \epsilon R (y^2 = x^2 + 1) b) \exists y \epsilon R \forall x \epsilon R (y^2 = x^2 + 1) Now, to me, they both mean exactly the same thing...
  36. phoenixthoth

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  37. B

    Quantum Proofs Made Easy: Get Help from Experts Today!

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  38. S

    Epsilon-Delta Proofs: A Comprehensive Guide for Calc1

    Hi, right now I am struggeling with this (calc1). To be honest, I nearly don't understand a thing. What's going on, and when am I done with the proof? I can plug in the limits and the approached value into the formal definiton of a limit, but that's as far as I get. (I semi-get the easy...
  39. K

    Determinants and Matrix Inverses Proofs

    Can anyone help me start this out? I got absolutely no clue. Q: If A and B are n x n matrices, AB = -BA, and n is odd, show that either A or B has no inverse. I know that we have to show that either det A is 0 or det B is 0, but I have no clue how to show it with the given information...
  40. murshid_islam

    Proofs for 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\ldots=\frac{\pi^2}{6}

    can anyone help me with the proofs: 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\ldots=\frac{\pi^2}{6} if F_i is the ith Fibonacci number, then F_1+F_2+F_3+\ldots+F_n=F_{n+2}-1 F_2+F_4+F_6+\ldots+F_{2n}=F_{2n+1}-1 F_1+F_3+F_5+\ldots+F_{2n-1}=F_{2n}...
  41. E

    Proofs for recursive sets using induction

    T* is a recursive definition, a subset of the family of ternary strings. Let T* be the smallest set such that: BASIS: 0 is in T* INDUCTION STEP: If x,y in T*, then so are x0y, 1x2 and 2 x1. a) Prove that if k in N, then there is no string in T* with exactly 3^k +1 zeros. b) Prove that if...
  42. mattmns

    Geometry Proofs - An online archive?

    Are there any sites that have most of the popular geometry proofs? Thanks.
  43. H

    Can I Be a Physicist Without Mastering Proofs?

    I want to be a physicist when I'm out of college, but I see a humongous obstacle in proofs. See, I had taken Calc III and Diff Eq earlier this year, and aced them, but when I got to linear algebra, my first proof-based course, my grade dropped to a C+. (I have my final tomorrow, btw, and I'm...
  44. H

    Some linear algebra proofs I couldn't figure out: Help

    I am "scared" (to put it mildly) of these problems, which I need to review before my final tomorrow. Just to let all of you know, this is not homework. There are 25 or so problems, and I only understand around 10 of them. :frown: Help me! I need an A on the final to get a B in the class. :cry...
  45. honestrosewater

    Favorite definitions, theorems, proofs, etc.?

    After seeing infinite sets defined negatively, I liked seeing them defined as sets that are equivalent to one of their proper subsets. I always thought diagonal argument[/url] was cool. Do you have a favorite definition, theorem, proof, bit of knowledge you found especially insightful or...
  46. L

    Understanding Proofs Involving Subsets

    Hi, I am having trouble with these proofs; I don't know if I am doing these right. I'd appreciate some help. Thank you. If X---> y is a map, then let B1, B2, B \subseteq X. i. f(B1 U B2) = f(B1) U f(B2) To prove this I have: f(B1 U B2)=f(B1) U f(B2) Since B1 U B2 \subseteq B1, we...
  47. B

    Proofs in Linear Algebra: Countable Sets, Algebraic Numbers, and Fields

    Linear Algebra proof I would appreciate any help with any of the foolowing: 1. Let C be a countable set. Prove that any linear well-ordered on C with the property that whatever c in C there are only finitely elements c` with c`<c, is unduced from the canonical order on N via a bijection N->...
  48. C

    Help with Proving Trigonometric Identities

    I need help solving these 2 proofs: (sec^2x-1)/(sec^2x) = sin^2x I am not sure what direction to go in. I know the top of the left side could be changed into: tan^2x/sec^2x, but I don't know what to do after that. The second one I need help with is: cos^2x/(1+tan^2x) = cot^2x I...
  49. C

    Help Solving Two Proofs - Tan X + Cot X = (Sec X)(Csc X)

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  50. P

    Physics Major Struggling with Math Proofs

    hello everyone. i was once a math major until i got a bit taste of what upper div math is like. proofs! can't say that i like it at all. i guess I am the type of guys who can't do proofs whatsoever. currently taking linear algebra, struggling a bit. so sometime ago, i changed my major to...
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