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How to write Math Proofs

  1. Apr 22, 2007 #1
    Below is a list of notes on mathematical proofs.
    The notes are directed at beginners who want to learn
    how to write mathematical proofs.


    1) Introduction to mathematical arguments
    (by Michael Hutchings)

    2) How to Write Proofs -
    A short tutorial on the basics of mathematical proof writing
    (by Larry W. Cusick)

    3) How to write proofs: a quick guide
    (by Eugenia Cheng)
    Department of Mathematics, University of Chicago

    4) Notes on Methods of Proof
    by Peter Williams

    5) A brief guide to writing proofs (Polytechnic university)

    6) A few words about proof (Berkeley Math Circle)

    7) Understanding Mathematical Induction
    (Idris Hsi)

    8) Basic proof methods (David Marker)
    MATH 215, Introduction to Advanced Mathematics, Fall 2006



    1) Guidelines for Writing Mathematical Proofs (Jessica K. Sklar)

    2) Introduction to Mathematical Reasoning (John M. Lee)
    Conventions for Writing Mathematical Proofs
    (Math 310, Spring 2006)

    3) How to do math proofs (wikiHow)

    4) Some hints on mathematical proof by David Goss

    5) Proof-Writing Tips (Ezra N. Miller)
    Math 5707, Spring 2004


    6) How to write mathematics badly (Entry in the Mathematics Weblog)
    Part 1: http://www.sixthform.info/maths/?p=147
    Part 2: http://www.sixthform.info/maths/?p=148
    Part 3: http://www.sixthform.info/maths/?p=149

    If you know more links, feel free to post them here.
    Last edited: Apr 22, 2007
  2. jcsd
  3. Jul 25, 2007 #2
    do you by chance have a list of books of proofs.
  4. Jul 26, 2007 #3
    Thanks Edgardo!
  5. Jul 26, 2007 #4
    Hi rococophysics,

    I don't have a list of books, but I found some books on amazon.com.
    This book here looks good, though I haven't read it. At the bottom of the page you will find
    other books ("Customers Who Bought This Item Also Bought").

    If you click on the image of the book "The Nuts and Bolts of Proofs" ("Search Inside" function) and go to the page after page 13 you will find a list of books.
    Last edited: Jul 26, 2007
  6. Dec 16, 2008 #5
    More on proof writing:

    1) Notes on Math Proof
    (by Bruce Ikenaga)
    Covers many topics.

    2) Math 23b Proofs
    (by Kiyoshi Igusa
    An introductory course on math proofs.

    3) A Guide to Proof-Writing
    (by Ron Morash, University of Michigan-Dearborn)

    4) Writing Proofs
    (by Tim Hsu)
    44 pages long

    5) ∀ Proof Writing ∃ This Reference Book
    A Student’s Guide to Intermediate Mathematical Proofs
    (by Kiddo Kidolezi, David Molk, Maurice Opara, Dan Shea and Priscilla S. Bremser)
    A document written by students.

    6) Proof Writing and Presentation Tips
    (by Erika L.C. King)
    Tips for writing good proofs.
  7. Dec 16, 2008 #6


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    to write a proof: begin by stating what you want to prove, precisely.

    then make sure you know what all the words in that statement mean.

    then identify the hypotheses of your statement, and start using them to

    head towards your desired conclusion.

    look at the desired conclusion and see if you know any other theorems that have that as a conclusion, then try to see if their hypotheses can be verified in your setting.

    another approach is to negate the desired statement, and try to deduce a known false statement.
  8. Dec 21, 2008 #7


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    learn how to use basic language correctly. e.g. learn the distinction between the converse of a statement (whose truth is unrelated to that of the original statement), and the contrapositive, whose truth is equivalent.

    E.g. if the statement has form "P implies Q", the converse is "Q implies P", and the contrapositive is "notQ implies notP".

    e.g. Every good boy does fine is equivalent to "if X is a good boy, the X does fine".

    the converse is: "If X does fine, then X is a good boy".

    the contrapositive is: "If X does not do fine, then X is not a good boy".

    Or, if f is continuous on [0,1] then f is bounded on [0,1] (true).

    converse: if f is bounded on [0,1] then f is continuous on [0,1] (false).

    contrapositive: If f is not bounded on [0,1] then f is not continuous on [0,1] (true).
  9. Dec 24, 2008 #8
    Some of the links in my first post are broken, but Eugenia Cheng's document can still be found here.
  10. Jan 9, 2009 #9
    The video link, "Serre: Writing Mathematics," (aka How to write mathematics badly) was broken as well, but can be found here:

    I also recommend the book "An Introduction to Mathematical Reasoning" by Peter J. Eccles
    Its pretty basic, but useful for someone like me who had never been exposed to formal math or proofs.
  11. Jan 10, 2009 #10
    Thanks m00npirate for the link!

    More notes:

    1) How to write mathematics (by Martin Erickson), 5 pages

    2) Errors in mathematical writing (by Keith Conrad), 5 pages
    This one is good because Keith shows an example for good and bad mathematical writing.

    3) An Introduction to Proofs and the Mathematical Vernacular
    (by Martin V. Day), 123 pages
    This one is actually a book with 123 pages!
  12. Apr 3, 2009 #11
    Just to add my two cents on this I just started this whole proof writing business and learning proof writing from the wrong book can just confuse you more. So I'd recommend if you want to take up proof writing to learn it from more then one source especially if you don't have a teacher.

    I'd Recommend:
    How To Prove It: A Structured Approach by Daniel J. Velleman
    (It is great at explaining some basic concepts and the majority of the common proof writing concepts however it isn't as heavy in math as most books.)
    For that I'd recommend
    The Nuts and Bolts of Proof Writing ( I forget Who its by) but its heavy in example.)
    If you're really adventurous and want a challenge with almost no explanation my college uses
    "A Transition To Advanced Mathematics" ( Its a hard book for beginners if you don't have teacher since it has few examples and little to no explanation )
  13. Dec 20, 2011 #12
    More material:

    1) Why do we have to learn proofs
    Joshua Cooper, Associate Professor Department of Mathematics, University of South Carolina

    2) Book of Proof
    A complete book on mathematical proof writing by Richard Hammack, Associate Professor, Virginia Commonwealth University, Department of Mathematics and Applied Mathematics
    You can download the book as PDF for free. Or you buy it for 13$ or 11€ on amazon which I'd say is a very good investment.

    3) Writing Mathematics
    Ethan Bloch, Professor of Mathematics, Bard College

    4) Video lectures by Dr Joel Feinstein
    How and why we do mathematical proofs
    Definitions, Proofs and Examples
  14. Dec 20, 2011 #13
  15. Oct 14, 2012 #14
    Thanks a lot. Very useful. I shouldn't have bought Velleman or Solow. These notes are just enough, no need to buy a book...
  16. Oct 14, 2012 #15
    Your first link is dead. Here's what is should be:
  17. Feb 15, 2014 #16
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