Good writing, especially of proofs and logical arguments, is important for many reasons. It helps you think through your answer carefully, which can help you fully solve the problem. It also helps us grade solutions efficiently and fairly. We cannot grade you on what you understand, only on what you turn in. Good writing helps those things match up.
All solutions in this class, in order to pass, must meet the below requirements in addition to problem-specific rubrics.
Writing
For written responses, a solution must satisfy the following in order to pass:
Writing must be clear and easy to understand. Each statement must make sense in context of the problem and what has been written so far.
There must not be excess writing. There must not be empty or vague statements, non-sequiturs, or writing that does not contribute to the solution. There must not be excess repetition.
Even if the solution contains correct components, it must not also contain incorrect statements or other excess writing.
Terms used should be well-defined: either a term already defined in the class or the problem statement, or else clarified and defined in the solution.
Solutions should also follow the following guidelines. Solutions may not pass if they do not:
If using a fact we stated in class or in the notes, it is often a good idea to restate the fact and mention that we stated it in class.
However, do not re-write proofs, justifications, explanations, etc. from class or lecture notes. You should assume as given anything that we have done in class.
You generally do not need to restate a definition unless it is very helpful to your answer to have the definition there on the page.
The only exception to the above is when you are explicitly asked to give a statement, definition, proof, etc. from class. For example, if you are asked for the definition of big-O, you should give that definition. If you are asked to reproduce a proof you have seen in class, you should give that proof.
Proofs
For problems that ask for a proof, a solution must satisfy the following in order to pass:
The proof should be a sequence of claims and/or steps. Each claim and/or step should have a clear justification for why it is true.
For example, suppose you have an even number $n$. You might write, "By definition of even, $n = 2k$ for some integer $k$." Here, the claim/step is that $n=2k$ for some integer $k$. The justification is the definition of even.
The justification does not always need to be explicitly written if it is obvious from context. For example, if your step simplifies the expression $5n^2 + 3n^2$ to $8n^2$, you do not have to write down that it is justified by rules of arithmetic.
Generally, the last claim should be the original fact that needed to be proven.
The method of proof should be stated if it is not a direct proof and it is not obvious from context. For example, proof by induction or proof by contradiction.
A solution must also satisfy the following in order to pass:
It should be well-written and clear according to the writing guidelines above. Proofs are writing.
It should not contain false statements or irrelevant statements.
It should not contain incorrect or poor justifications, even if the claim being justified is correct.
It should not contain scratch work or an explanation of how you solved the problem. When asked for a proof, do not "show your work" or discuss how you got there. Give a concise, correct proof. We will always tell you when you are supposed to show your work, and it will usually not be in the context of a proof.
Using sequences of inequalities
When we need to prove an inequality, it is often useful to have a chain: $A \leq B$, and $B \leq C$, and $C \leq D$, so we conclude $A \leq D$.
The chain can also include some equalities as well as less-than-or-equals.
The best way to write such chains is generally as an equation array shown below.
If any of the steps need justification, it can be given in the right-hand margin, above the array, or below the array.
Example 1.
Prove that $3n^2 + 2n \leq 5n^3$ for all $n \geq 1$.