Math 818 - "How to" guide for proof writing
Basic rules on writing up proofs:
- (R1) Never use the words "obvious", "clearly", "trivial", "of course", or
synonyms of these.
- (R2) Write words rather than symbols - a long string of symbols
isn't as readable as an English sentence.
proofs you hand in, write out "for all",
"there is", "implies", "if and only if",
"therefore", "because". (That is,
the symbols ∀, ∃, ⇒, ⇔,
∴, ∵ are not allowed
on your graded work.)
- (R3) Write sentences and paragraphs, and use
correct grammar and punctuation.
(E.g., every sentence must end with a period.)
- (R4) Every time you apply a definition, theorem, proposition,
lemma, corollary, exercise, etc., from class or the text, in
a proof that you are writing up, reference it.
Every time you apply a definition, theorem,
proposition, lemma, corollary, exercise, etc., from
the Table of
Contents (TOC), the earlier
the text, or class, in a proof
that you are writing up, reference it. You can reference
it by its number ; for
example TOC Def 2.1, Text Thm 2.7, TOC Cor 4.5, PS 4.2, etc., or
"definition of group".
If you are using only part of a multi-part
theorem, reference the specific part(s) that
you are using; for example, TOC Thm 2.1(a).
- (R5) Do not copy the statement of a definition or
theorem inside of a proof - write them out before you
start your proof. [If the result
came from class and is in the
textbook or the TOC, then give the
full statement of the definition or theorem (or...) before
you begin your proof, so that you can reference it inside
of your proof.]
General tips on writing up proofs:
- (T1) Avoid the words "if" and "it" in proofs.
("If" expresses uncertainty over what you know is true
so far, and the object that "it" refers to is often
- (T2) Follow the standard mathematical conventions for notation:
- Notation for indices:
i for finite or countably many indices, α for arbitrarily many.
- Notation for sets, elements, and groups/rings/fields:
small letters (a,b,x,y) for elements of sets, capital letters
(A,B,X,Y) for sets and groups/rings/fields.
- (T3) Proofread your proofs! Proofs, like any other writing,
often need to go through more than one draft. Also consider exchanging papers
with a study partner, and proofreading each others papers.
- (T4) When writing a proof that is split into several
subcases, don't conclude too much within a subcase.
- (T5) Between any two clauses in a sentence
there should be a connecting word. For example,
the sentence "Since A is open in X, B is open in Y,
C is open in Z" the logical implications are not
clear: Is the fact that B is open in Y part of
the "since" that is already known or assumed, or
is it part of the "therefore" consequence of the
fact that A is open in X? Without connecting
words like "and" and "then", there is no way to
determine what the sentence's author means here.