Number theory I.c - Combinatorial number theory

Guidelines for preparing written homework

Guidelines for writing mathematical proofs (some parts taken from John M. Lee's "Some Remarks on Writing Mathematical Proofs")

• Write in Paragraphs: There is an overwhelming consensus that an ordinary prose narrative is much better suited to communicating the truth of a mathematical statement than formal symbolic statements. As you read increasingly complicated proofs, you’ll find that paragraph-style proofs are much easier to read and comprehend than symbolic ones or the two-column proofs of school books. Roughly speaking, one paragraph should correspond to one step in your proof.
• Grammar and Punctuation: Mathematical writing should follow the same conventions of grammar and punctuation as any other writing. Avoid sentence fragments, run-on sentences, and dangling modifiers; pay attention to subject-verb agreement and parallel structure; and use correct spelling and capitalization.
• Write with precision: In mathematical writing more than any other kind, precision is paramount. For each mathematical statement you write, ask yourself these two key questions:
  –What does it mean? Every mathematical statement must have a precise mathematical meaning. Every mathematical term you use must be well defined and used properly according to its definition (unless it’s an officially undefined term); and every symbolic name you mention must be either previously defined or quantified in some appropriate way.
  –Why is it true? Every mathematical statement in a proof must be justified in one or more of the following six ways: by an axiom; by a previously proved theorem; by a definition; by hypothesis (including as special cases an inductive hypothesis or an assumption for the sake of contradiction); by a previous step in the current proof; or by the rules of logic. Sometimes this is best accomplished by citing the reason directly: “We conclude that  by Theorem 3,” or “It follows from transitivity that .” At other times, the reason will be so obvious to the reader that it is actually more effective to leave it unstated.
• Don't start sentences with symbols: It’s bad form to begin a sentence with a mathematical symbol, because that makes it hard for the reader to recognize that a new sentence has begun. (You can’t capitalize a symbol to indicate the beginning of a sentence!) It’s usually easy to avoid this by minor rewording.
• Punctuation after formulas: If a displayed formula ends a sentence, it must be followed by a period!
• How much detail do you have to include: It might not be necessary to write down every step, but you should include just enough to give the reader the Aha! experience that makes the rest obvious (and, if you’re writing a homework assignment that will be graded, makes it clear to the grader that you’ve figured out the details yourself!).
• Proof structure: If the structure of your proof is anything other than a simple direct proof, state at the beginning what type of proof you’re using (“we will prove the contrapositive” or "we will use a proof by contradiction" or “we will prove this by induction,” for example).
• Use the first person singular sparingly: Most authors avoid using the word “I” in mathematical writing. It is standard practice to use “we” whenever it can reasonably be interpreted as referring to “the writer and the reader.” For example, write “We will prove the theorem by induction on ” instead of  “I will prove the theorem by induction on ”.
• Avoid most abbreviations: There are many abbreviations that we use frequently in informal mathematical communication: “s.t.” (such that), “w.r.t.” (with respect to), and “w.l.o.g.”(without loss of generality) are some of the most common. These are indispensable for writing on the blackboard and taking notes, but should usually be avoided in written mathematical exposition, especially in formal contexts. The only exceptions are abbreviations that would be acceptable in any formal writing, such as “i.e.” (id est, which means “that is”) or “e.g.” (exempli gratia, which means “for example”); but if you use these, be sure you know the difference between them!
• State what you are proving: If you’re writing a proof, you should always precede it with a precise statement, in one or more English sentences, of the result that you are proving. This applies even if you’re writing a proof as a homework assignment. You might do this by copying the problem statement verbatim, by summarizing the problem statement, or by paraphrasing the problem in the form of a theorem statement.
• Include more than just the logic: If you have to write a long sequence of formulas, intersperse the formulas at carefully chosen places with words about why one step follows from another, or what you are doing and why. Write with an eye to minimizing the amount of work your reader will have to do.
• Symbols and formulas in sentences: Every mathematical symbol or formula, whether in line or displayed, should have a definite grammatical function aa part of a sentence; a formula cannot stand on its own as an entire sentence. Formulas should almost always have one of the following two grammatical functions: (1) a formula representing a particular mathematical object can be used as a noun; and (2) a complete symbolic mathematical statement can be used as a clause. For example, consider the following sentence: "If , we see that must be greater than ." Here “” is a mathematical statement functioning as a clause (whose verb is “>”), while “” and “” are mathematical expressions (representing real numbers) that function as nouns.
• Relations and operators in sentences: Symbols representing mathematical relations (like =,>,∈, or ⊆) or operators (like +,−, or ∩) should be used only to connect mathematical formulas, not to connect words with symbols or with each other. For example, do not write: "If is a real number that is , then must be ". The following is much better: "If is a real number satisfying , then we must have ".