Problem 3 Rubric (17 pts.) Grading Notes. This is a general idea of how points are distributed. We give partial credit where we can, so it is worth submitting any work you've done even if you know it is not fully correct. (a) (2) You must provide the correct negation of the claim. (b) (11) Correctness. If your proof is not correct, this is where you'll get docked. (7) Regardless of how you formulate your proof, somewhere you'll need certain facts without which the proof wouldn't work. E.g. if it weren't true that the sum of two integers is integer, would your proof fail? If so, then that is a fact I need to see. (1) The order of the facts must make sense, so that you're not inferring something before you have all the facts to inferit. E.g. you cannot use hat the sum of two integers is integer if you don't already know that you have two integers to begin with. (3) You must use a proof by contradiction, which clearly states it is a proof by contra- diction, states what the negated assumption is, finds a contradiction, and clearly states what and where that contradiction is. (4) Communication. We need a mix of notation and intuition in the "column" format, with precise mathematical statements on the left, and approved reasons on the right. You should have between 6 and 10 non-trivial steps for this particular proof. (1) Use the "column format": each row is a precise mathematical statement on the left, and the reason that statement is true on the right (or wherever it fits.) (1) The mathematical statements must be precise and mostly (if not all) algebraic. (1) The reasons must be taken from the approved list and should be written out in full. (1) Provide enough rows / steps so that we can follow your proof, but do not have redundant or useless steps that might make the proof unnecessarily confusing. If you use any reason not on the approved list, you will lose 4 points automatically. Our problem sets encourage a very particular kind of practice; pulling from these particular reasons requires proofs with a certain level of detail that is appropriate for this course.