after test 1

Well, I found out that the reason q3 was so troubling for me was because of an error in my math.  It was a simple arithmetical error that gave me -phi for n3/n4.  And as a result, I thought that n4 was negative.  I now see (and it makes perfect sense) that the cycle of natural numbers will proceed indefinitely, and that a contradiction can be found by trying to find a smallest number.  I guess the only thing that can be learned from this is to be more careful, and double check all my results.  Also, more time (and help sought) for the next assignment would be advisable.

I had just went through test 1 last week, and found it to be fair, yet challenging.  My last question, was where, if I remember correctly, I had to show that a set of natural numbers {1, ..., n}, which was missing either 1 or 2, or both, had 3*2^(n-2) subsets.  I initially took the wrong approach, where, for some reason, I tried to directly prove this.  I realized, with 5 minutes left, that this question was exactly like the ones seen in the PS and assignment.  I didn't need to show that it works for P(n), but only show that P(n)=>P(n+1).  I will probably get part marks for this, but again, lesson learned.  Hopefully, or eventually, I'll get quicker at seeing what kind of problem is in front of me, and then taking the right approach :).