Here are some general tips to help you as you navigate the homework assignments of Math 4571.
Mathematics is a creative endeavor, even though it is rarely presented that way in courses below the level of calculus. The goal of this course is to explain the foundations of number theory in scrupulous detail, and to help you develop intuition about the material that will allow you to solve a very wide variety of problems.
The homework assignments and exams will emphasize creative problem-solving, and as such, you will probably find yourself deeply confused, puzzled, or lost for a long time when you are working on the assignments. Please be aware -- this is completely normal, and to be expected! Learning and mastering abstract concepts in a rigorous way is a difficult and messy process, and being stuck, trying incorrect approaches, and making mistakes are all parts of that process. Do not expect yourself to be able to solve every problem (or even any of the problems!) as soon as you read them. The most important qualities for success in this course are perseverance, hard work, and an ability to speak up when you feel you don't understand something.
Solving mathematics problems often requires a lot of time and effort: you will need an ample supply of knowledge, patience, and hard work. Here are a few ideas that may be useful to keep in mind when you are trying to solve a problem:
Try coming up with an example, and work out the result in that special case. Find a specific example of a set (matrix, number, function, vector space, statement, equation, relation, congruence, object) that satisfies all the necessary hypotheses, and see if you can establish the result in your specific case. The ideas you use in a special case might also work in general.
Try playing around with the hypotheses. See if you can find a situation where removing a hypothesis makes the result no longer true. If you can figure out what is going wrong, it will help you see where the hypothesis is needed.
Try to establish a simpler result. If you're getting stuck, try strengthening the hypotheses and see if that allows you to prove the result. Then go back to the original problem and see if you can solve it with the weaker assumptions.
Draw a picture. If you are trying to understand a function, draw its graph, and try to use your visual intuition to understand it. Pictures do not need to be completely accurate, as long as you capture the important ideas.
Don't try to reinvent the wheel if you don't have to. If your problem looks similar to something already proven, see if you can adapt the proof to your setting: if you can, life will be much easier (someone else did all the hard work for you!).
See if you can apply a big theorem (or a little one). If you can use a known result in the context of your problem, it is good to give it a try. If it doesn't quite look like you can invoke a theorem yet (e.g., you're missing one of the necessary hypotheses), try to see if you can get to a stage where you can apply the theorem (i.e., show that the missing hypotheses are actually satisfied).
Ask dumb questions. If you don't understand something, even if you think it is very simple, don't be afraid to ask yourself (or someone else) clarifying questions. One of the best questions in mathematics is: "why did we do it that way?".
Talk to other people. If you are trying to work through a problem, talking to another person about it will help you to clarify your thoughts. Explaining your ideas out loud will force you to think through them again, and might help you gain more understanding about what is going on. And of course, the person you're talking to might have some ideas too.
Be patient. When doing real mathematics, you will spend most of your time struggling through difficult problems and being confused about the material. This is normal, even for professional mathematicians! Abstract theory is not something that is easy to absorb, but you will eventually learn the material if you work at it. If you are getting frustrated, take a break and come back to your work later.
All of the things above are good general advice. Here are some ideas that are more specific to Math 4571:
If you are having trouble getting started on a problem, you can always start by writing down the hypotheses and (separately) the conclusion. Next, make sure to translate the hypotheses and the conclusion into concrete information by applying any relevant definitions (the "definition chase"). Often, once you explicitly write out exactly what information you are given and the result you are trying to reach, a natural next step will suggest itself.
For example, instead of trying to manipulate statements like "the set A is a subset of the set B" or "the integer n is divisible by 6" you can write out what the definition means explicitly: "if x is an arbitrary element of A, then x is an element of B" or "there is some integer k with n=6k", respectively. This will give you something you can actually work with.
Examples are a useful tool to test out what is going on in a problem, but be warned: listing examples, even if they are explained, is not a solution to a problem that asks you to prove something. Most problems require you to prove that some result is true in every possible situation: just giving one or two examples where the result is true does not accomplish that task.
If you're trying to prove something directly and getting stuck, try arguing by contradiction instead, and see what can go wrong. If you want to show that all As are B, then examine what happens if you had an A that was not B: having an extra few properties (namely, "not B") lying around might give you more ideas. If you use a proof by contradiction, clearly indicate that fact at the beginning of your argument, and also note when you have reached a contradiction.
If you are trying to establish that two statements are equivalent (A if and only if B), try splitting it into two conditionals: if A then B, and if B then A. Each conditional gives you something to start with along with a goal you want to reach.
When trying to show that two numerical expressions are equal, try starting with one side of the expression and doing transformations on it until you obtain the quantity on the other side.
If a problem has multiple parts, they are often related in some way. Always be on the lookout for a way to use the results from earlier parts of the problem in the later parts. For example, if part (a) of a problem asks you to show that , and then part (b) asks you to find , you can just say "by part (a), we have ".
If you are trying to show that two sets S and T are equal, try showing that every element of S is contained in T, and that every element of T is contained in S.
If you are trying to show that an object is unique, suppose that there are two of them and then prove that they must be the same.
When a problem has a phrase like "for every positive integer", this is often a tip that induction might be useful. Induction arguments are often very simple if you organize them the right way, and can turn a single difficult direct proof into two easy pieces (namely, establishing the base case, and establishing the inductive step). When using induction (or any other special proof method), you should clearly label the base case(s) and induction step as well as what variable you are inducting on.
Solving a problem is not the same as writing up a solution. Once you have figured something out, you still have to write it up.
In this course, a full solution to a problem is a rigorous mathematical proof: a sequence of statements, written in sentences, that establish a conclusion starting from some hypotheses. This is not the same as writing up the details of a calculation (e.g., how you computed a derivative), nor is it merely a succession of equalities: it is a sequence of statements written using words and symbols (or possibly just words).
You should be able to read a proof aloud, at least in principle, and have it make perfect sense. This includes the equations you have written, as mathematical symbols and equations have verbal translations: the equation can be read aloud as "x-squared equals nine".
Standards for good writing also apply to proof-writing: you should use correct grammar, write in full sentences, organize sentences into paragraphs, etc. If the proof is hard to read (e.g., with lines erased or scribbled out, or missing parts added in separate areas of the page), you should rewrite it cleanly.
Proofs should contain no gaps. Every statement in a proof should follow logically, in an obvious way, from previous statements or known facts. The definition of "in an obvious way" is a little bit subjective, but as a general rule, if your argument would not completely convince another student in this course who hasn't seen that particular problem, it needs more detail.
A well-written proof should be easy to read and easy to follow: if there are multiple parts to an argument, they should be clearly separated from each other. If you reference other results, you should state these results by name and (if appropriate) specifically how you are using them.
You are expected to produce proofs that are of comparable rigor and accuracy as the proofs given in class and in the course materials. There are dozens of examples of properly-written proofs in the textbook and notes, and you will see many more in lecture. Some of these proofs will be much longer and more complicated than the kinds of proofs you are expected to write, but you should strive to write proofs that are similar to the textbook's and instructor's in style and form, using clear explanations and logical reasoning.
Clarity and conciseness are important. A short and crisp proof is often much easier to understand than a longer one. Of course, concise proofs are often harder to write. However, an important aspect of problem-solving is understanding which parts of an argument are necessary and which parts are unnecessary: taking the time to streamline an argument will often help you understand the important components better.