Here are the preliminary instructions for the TAs (the instructions sometimes evolve as we mark the assignment.) Problem 1 =============== There are separate marks for "ideas" (i.e., the actual proof) and "details" (i.e., not saying things that don't make sense, justifying *every* step, etc.) They obviously will be correlated, but that's okay. For Q1a, the ideas are *Pick an example *Go from the example to existence *Combine two existence statements into a conjunction. For the counterexamples, the ideas would be the counterexamples themselves, and the details are justification of every statement. Do not deduct marks for not negating the statement and then proving the negation. I did it for uniformity and because that's what you do when using the detailed proof format, but that wasn't required. [NOTE: I *am* requiring that for Assignment 4] Non-structured proofs are fine in principle, but every statement there has to be justified, like it says in the assignment. Problem 2 ============== For Q1a and Q1b, the instructions are the same as for Q1. Q1a: Ideas: x+1 = (p+q)/q p+q is whole since the integers are closed under addition Q2a: Ideas: x-1 = (p-q)/q Taking the contrapositive For Q2c, I separated out the ideas so that they're easier to mark. Ideas: Prove that x irrational => (x+n) irrational for whole n Prove that x irrational => x/n irrational (or they could prove x*r is irrational for any rational r) Prove that -sqrt(2) is irrational Prove things separately for the two roots The rest of the marks are assigned to details + overall structure The components above should be there somewhere, even if maybe some of them are hidden in the justification. Potential common mistakes that shouldn't be given marks: * "irrational numbers are closed under addition" (they're not) * Just assuming irrational + rational = irrational without proof * A mistake that happens is to say something like (-6+sqrt(2))/2 = p/q for whole p and q and sqrt(2)-6 is not whole. That's not actually enough since 2*sqrt(2)/sqrt(2) is actually rational, even though 2*sqrt(2) is not whole. Correct alternative solutions (like the one given in the sample solutions), should also get marks for ideas, even if the ideas are different. Problem 3 =============== *Explanations are not required from the students *True and false cases are given equal weight *50% for reasonable attempts that didn't succeed *Leave brief explanations for what went wrong Problem 4 ================ Hopefully this is easy to mark. 50% for reasonable attempts. Problem 5 ================ "Good" for just the truth table. "Excellent" for truth table + some kind of correct statement that concludes that the statement is true from the truth table. 50% for reasonable attempts that don't succeed.