Mathematics 404A        Spring 2005

• Use complete sentences.
• Explain what you're doing.
• Avoid abbreviations and some mathematical shorthand; for example, don't use the symbols for "there exists", "for all", "implies", "if and only if", or "therefore" – write the words instead. If you're not sure whether a symbol is okay, look at our textbook: if Artin uses it, it's okay, and if he doesn't, it's probably not. I've provided a list of some symbols to avoid to my Math 310 class, and you might want to look at that.
• In general, the proofs and explanations in Artin (or in most textbooks for material at this level) provide good models for your writing. You can also look at the proofs I've posted on the web site (although some of those are missing details). If you write something that looks completely different, stylistically, from what is in our textbook or in my solutions, you might want to reconsider how you wrote it.
• Let me expand on the concept "explain what you're doing". A good model for a proof is this: first you should say what you are going to do, then you should do it, and then you should say what you did. For example, the start of my solution to problem 4 in section 2.2 says

  Suppose that H is a subgroup of G, and fix elements x and y of H. I want to show that xy^-1 is in H. Since y is in H, H contains its inverse y^-1. Now by closure, the product xy^-1 is in H.

  The sentence "I want to show..." is saying what I'm going to do. Then I do it. At the end I could have added "This finishes the first part of the proof," to say what I had just done; when I was writing this, I thought that this part was short enough that I didn't need to do that.

  The solution continues

  Conversely, suppose that H is a nonempty subset of G such that xy^-1 is in H for all x,y in H.

  I probably should have added "I want to show that H is a subgroup of G."

  Then there were a bunch of details verifying the subgroup axioms for H, which I'll skip here. My solution ended like this:

  Thus H is closed, so H is a subgroup.

  In the clause "so H is a subgroup," I'm reminding you of the point of this part of the solution, and thus pointing out that the proof is over.
• Use italics for letters used as variables or representing mathematical symbols. For example, this is standard: "Let a be an element of the group G", while "Let a be an element of the group G" is not standard. (Using italics also helps to distinguish the word "a" from the variable or element "a".)
• If you have a proof by contradiction, check it: are you really using contradiction in a crucial way, or can the proof be modified (and simplified) to be a direct proof or a proof using the contrapositive?

       John Palmieri    Padelford C-538    (206) 543-1785    email    web page