189-235A: Abstract Algebra 1

Assignments

The weekly assignments are an essential part of the course. You should plan to devote at least 10 hours a week or so (and quite possibly more) to the assignments. Delaying working on the assignment to later in the semester (a temptation, when you might start to be busy with other classes) is not recommended. Keeping up to date on the assignments is also a good way to keep up with the lectures, and to give yourself the best chances for success inthe written exams which are heavily based on the assignments.

If you are stuck on a problem, you may seek out the help of the professor, a TA, one of your classmates, or your favorite LLM. Just make sure that you do this intelligently, without unduly avoiding the effort without which true learning cannot really happen. Do not neglect the assignments, or allow yourself to fall behind with them: experience shows there is a strong correlation between the work you put into them and how much you learn in the course, which will of course be reflected in your exam performance.

Tips for writing proofs:

When writing up your proof, make sure that you explain your reasoning clearly and fully, using complete sentences. Mathematical notation, although admirable in its conciseness and power in many contexts, is no substitute for clearly written prose. Remember that in an exam, a wrong answer that rests on an almost correct, cogently argued justification will earn you (a lot of) extra credit. The same answer with no explanation of what led you there will earn you no points at all: the grader, unable to read your mind, will be forced to assume the worst...

It is always best to proceed linearly and in a logical order. For instance, say you want to prove the identity A=Z. If you know A=B, B=C, etc, write A=B=...=Z. This makes the proof easy to read as the reader only has to check each statement independently. In highschool some of you may have acquired the bad habit of writing the same proof by starting with A=Z, which you do not know to hold a priori (this already gives a hard time to the grader, who is left to guess whether you know what you are doing or are instead assuming what you were asked to prove), then proceed to write B=Y (since A=B, Y=Z this holds if and only if A=Z holds), C=W, ... up to L=M say. But you know L=M to be true so going up the chain this means A=Z. This is logically correct but very confusing and hard to follow as the proof is essentially written upside down. I did not penalize anyone for this on the first assignment because MATH 235 is one of the first proof based courses but make sure to write proofs where, proceeding from the top to the bottom, every statement can be deduced from the previous statements. Finally, some of you seem to be fervent advocates of the LHS, RHS school of proof writing. Note that if one can proceed linearly to show LHS=A=B=...=Z=RHS one should try to avoid showing that LHS=A=...=Z=Z', RHS=A'=B'=...=Z' which implies LHS=RHS. If lenghty computations are involved and one has difficulty using the first method of proof then one is justified in using the second.

Assignment 1: Weeks of August 31-September 11.