James' Homework 2

Comments:

Thanks! Good work, see comment on 11.12 if you are courageous. Btw, do you mind if you forward the work to the group and I forward the comments to the group? There are only 2 people handing in homework (and 2 promised), and I kinda wanted to show everyone that people are doing it (to encourage them to work on it as well). You do pretty good work and by far have cleaner handwriting than paul.

12.1 (a) Good. (b) Great, lol every math guy writes the same thing. (c) Whoops, you missed the word R in the beginning of the question. It fails reflexivity since x+x=2x need not be a rational. Consider when x is pi. 2pi is irrational. (d) Spot on.

12.2. Good, I see you covered complex numbers in school. (a) Yep, and multiplication is commutative. But the transitivity and reflexivity...there is a problem in the proof. Assume x is related to y and y is related to z. Then xy is real and yz is real. Call xy = b and yz = c where b and c are the real values. This is a good step you did: y = b/x. Then plug that in to get (b/x)*z = c. This means bz=cx. But this just means the complex numbers are real valued scalings of each others, it does not mean xz is real. More importantly, squaring an imaginary number does not yield a real. For example, (1+i)^2 = 1+ 2i-1= 2i. The problem with complex numbers is things become..more complex. (b) GREAT! (c) good (not a GREAT! since problem not as hard as (b).

12.7 Whoa, why did I assign cosets when I did not talk about them. Whoops. I do not even think I talked about A_4. Let me work this out, but I think (just from reading your steps, thanks for writing them out) that you only computed gh and hg for a single h in H and g in G. You have to multiply out all g in A_4 and do it with each h in H on first the left. Then you do it separately for the ones on the right. You will realize that it gh NEED Not equal hg BUT the collection GH (all left cosets) is a reordering of the collection (HG). You can finish the question if you like, but it is super tedious and I never taught it.

11.3 And 11.4! GREAT! You're getting it. The use of this theorem is exactly what Ravi meant when he said number theory is secretly encoded.

11.12. Hhahahahahha everyone got this wrong on the midterm for the same reason (we confused it for Rp under addition so we thought 1 +1 +1... n times will generate any n.  But this is actually {1, 2, 3, ...p-1} (no 0) which has p-1 elements! ).  I was curious if anyone could get this. I would like you to think about this question, and add the question under the calendar under today's date. If you have the courage, I would like you to email [mailto:vakil@stanford.edu vakil@stanford.edu] the question (he is the one who gave the origami lecture).

Euler's theorem: Good, now remember this for the rest of your life!