Q1. (Yue Wu and Tyler) (a) Conclude other relation (binary relation, partial order relation etc.) 0-1 point Conclude x=y without any proof 2 points (b) General: Conclude false 0-5 points Whole proof based on a wrong assumption -10 points Prove by pointer: Don’t mention the left most case or right most case (failed to explain why #a(y) and #b(z) will meet at middle) -3 points Prove by induction: Wrong base case -2 points Induction hypothesis unclear -3 points In induction, omit the case x=a or x=b -3 points Q2. (Amirhossein Kazemnajad) Reflexivity (5 point). As you instructed, I gave full mark to “obvious” or “clear”, and also everyone who used definition of R. Symmetry (6 point): It was an easy case too. Most people did it correctly. Basically, stating the “reversibility” of iff got a full mark, and also anything containing "xz iff yz” -> “yz iff xz”. Transitivity (9 points): Generally, I gave full mark if their proof contained an argument like: “xw iff yw iff zw”, either in mathematical form, or just by explaining. If their proof didn't account for the reverse direction (zw -> xw), I took 3 marks off. But, if they were proving using “iff”, they didn’t have to explicitly prove for the reverse direction so they could get a full mark on the question, if they used the argument I mentioned before. Lastly, any hand-wavy thing such as saying “obvious” or merely mentioning "property of iff” didn’t get a full mark and I took a significant portion off depending on their proof. Q3. (Maia Darmon) For the new version: In general, if they didn't actually prove anything, that is they stated things without even using the definition given or made assumptions they didn't justify, I took off quite a few marks. In showing symmetry, some people used the fact that "x div y" and "y div x" means "x \le y" and "y \le x" which by symmetry of \le implies they are equal. I took points off if this wasn't justified further, but if they showed it using the definition of "div" (that is if x=ky, since k \ge 1 then x \ge y, and vice-versa), I gave them full marks. A (thankfully not so common mistake) I saw was people using the same k as their quotient in the definition of "x div y", so for example in showing transitivity they would say y=kx and z=ky, so z=k^2x, which of course is a wrong assumption to make. This probably meant they didn't quite grasp the definition of the relation and how to construct a general proof, so I took off marks there. A minor mistake I noticed quite often was people confusing the direction of the order, that is taking "x div y" to mean "there exists an integer k such that x=ky". I wasn't too harsh there if all the rest was correct using that relation instead. For the old version: In the first part (showing it's a partial order), in showing symmetry many people were stating that since m