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. For all 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 problem sets, 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 unclear.)

• (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.