How to write Math Proofs

[Edgardo]

Below is a list of notes on mathematical proofs.
The notes are directed at beginners who want to learn
how to write mathematical proofs.


PROOF TECHNIQUES

1) Introduction to mathematical arguments
(by Michael Hutchings (http://math.berkeley.edu/~hutching/))
http://math.berkeley.edu/~hutching/teach/113/proofs.pdf

2) How to Write Proofs -
A short tutorial on the basics of mathematical proof writing
(by Larry W. Cusick (http://zimmer.csufresno.edu/~larryc/))
http://zimmer.csufresno.edu/~larryc/proofs/proofs.html

3) How to write proofs: a quick guide
(by Eugenia Cheng (http://math.unice.fr/~eugenia/))
Department of Mathematics, University of Chicago
http://math.unice.fr/~eugenia/proofguide/

4) Notes on Methods of Proof
by Peter Williams (http://www.math.csusb.edu/faculty/williams/index.php)
http://www.math.csusb.edu/notes/proofs/pfnot/pfnot.html

5) A brief guide to writing proofs (Polytechnic university (http://www.poly.edu/))
http://www.math.poly.edu/courses/ma2312/WritingProofs.pdf

6) A few words about proof (Berkeley Math Circle (http://mathcircle.berkeley.edu/))
http://mathcircle.berkeley.edu/proof.pdf

7) Understanding Mathematical Induction
(Idris Hsi (http://www.cc.gatech.edu/people/home/idris/))
http://www.cc.gatech.edu/people/home/idris/AlgorithmsProject/ProofMethods/index.html
http://www.cc.gatech.edu/people/home/idris/AlgorithmsProject/ProofMethods/Induction/UnderstandingInduction.html

8) Basic proof methods (David Marker (http://www.math.uic.edu/~marker/))
MATH 215, Introduction to Advanced Mathematics, Fall 2006
http://www.math.uic.edu/~marker/math215/methods.pdf

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GUIDELINES FOR MATHEMATICAL PROOFS

1) Guidelines for Writing Mathematical Proofs (Jessica K. Sklar (http://www.plu.edu/~sklarjk/))
http://www.plu.edu/~sklarjk/499f06/499proofguidelines.pdf

2) Introduction to Mathematical Reasoning (John M. Lee (http://www.math.washington.edu/~lee/))
Conventions for Writing Mathematical Proofs
(Math 310, Spring 2006 (http://www.math.washington.edu/~lee/Courses/archives.html))
http://www.math.washington.edu/~lee/Courses/310-2006/writing-proofs.pdf

3) How to do math proofs (wikiHow)
http://www.wikihow.com/Do-Math-Proofs

4) Some hints on mathematical proof by David Goss (http://www.math.ohio-state.edu/~goss/)
http://www.math.ohio-state.edu/~goss/style.html

5) Proof-Writing Tips (Ezra N. Miller (http://www.math.umn.edu/~ezra/))
Math 5707, Spring 2004
http://www.math.umn.edu/~ezra/5707/tips.html

HOW TO WRITE MATHEMATICS BADLY

6) How to write mathematics badly (Entry in the Mathematics Weblog (http://www.sixthform.info/maths/))
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.

[rocomath]

do you by chance have a list of books of proofs.

[Winzer]

Thanks Edgardo!

[Edgardo]

do you by chance have a list of books of proofs.

Hi rococophysics,

I don't have a list of books, but I found some books on amazon.com.
This book here (http://www.amazon.com/Nuts-Bolts-Proofs-Third/dp/0120885093/ref=sr_1_1/105-8711782-3552405?ie=UTF8&s=books&qid=1185432686&sr=8-1) 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.

[Edgardo]

More on proof writing:

1) Notes on Math Proof
(by Bruce Ikenaga (http://marauder.millersville.edu/~bikenaga/))
http://marauder.millersville.edu/~bikenaga/mathproof/mathproofnotes.html
Covers many topics.

2) Math 23b Proofs
(by Kiyoshi Igusa (http://people.brandeis.edu/~igusa/)
http://people.brandeis.edu/~igusa/Math23bF07/Math23b.htm
An introductory course on math proofs.

3) A Guide to Proof-Writing
(by Ron Morash, University of Michigan-Dearborn)
http://www.csd.abdn.ac.uk/~kvdeemte/teaching/CS3511/lectures/slides/proofwriting.pdf

4) Writing Proofs
(by Tim Hsu (http://www.math.sjsu.edu/~hsu/))
http://www.math.sjsu.edu/~hsu/courses/generic/proof.pdf
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 (http://community.middlebury.edu/~bremser/))
http://community.middlebury.edu/~bremser/MA091_HANDBOOK.pdf
A document written by students.

6) Proof Writing and Presentation Tips
(by Erika L.C. King (http://math.hws.edu/eking/))
http://math.hws.edu/eking/pandptips
Tips for writing good proofs.

[mathwonk]

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.

[mathwonk]

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).

[Edgardo]

Some of the links in my first post are broken, but Eugenia Cheng's (http://www.cheng.staff.shef.ac.uk/) document can still be found here (http://cheng.staff.shef.ac.uk/proofguide/proofguide.pdf).

[m00npirate]

The video link, "Serre: Writing Mathematics," (aka How to write mathematics badly) was broken as well, but can be found here:
http://modular.fas.harvard.edu/edu/basic/serre/

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.

[Edgardo]

Thanks m00npirate for the link!

More notes:

1) How to write mathematics (by Martin Erickson (http://www2.truman.edu/~erickson/)), 5 pages
http://www2.truman.edu/~erickson/manual6.pdf

2) Errors in mathematical writing (by Keith Conrad (http://www.math.uconn.edu/~kconrad/)), 5 pages
http://www.math.uconn.edu/~kconrad/math216/mathwriting.pdf
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 (http://www.math.vt.edu/people/day/)), 123 pages
http://www.math.vt.edu/people/day/ProofsBook/
This one is actually a book with 123 pages!

[psycho2499]

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 )