The gathering of three brothers.

Among three basic ideas of computational logic, principle of well-ordering is my least favourite theory. The round-robin domino tournament example used in the class was somewhat puzzling and confusing for me. I still couldn’t figure it out until I visited Professor Danny Heap during his office hour. Nevertheless, principle of well-ordering proves to be of extreme importance in the golden ratio question in the assignment 1 and I loved how we don’t have to structure our proof (assume and then) when applying principle of well-ordering. Oh, I loved the golden ratio question too. It posed some manageable challenges to me and the process of resolving it was definitely satisfying!

The equivalences of three brothers (theories): principle of simple induction, principle of strong induction, and principle of well-ordering are unimaginably laborious and boring. Danny didn’t seem to be having fun proving them and neither did we. Personally, I don’t think that a perfect understanding of those equivalences will facilitate our problem-solving skills in the context of inductions.

The tricks and traps, on the other hand, are refreshing and informative. I still remember how I couldn’t figure out at all why one particular proof presented in the class because I omitted the all-important base cases. This goes to show the base cases, often neglected by me, should not be underestimated. When applying inductions, one should think seriously about the legitimacy and the number of base cases. Invalid base cases will disqualify a seemingly valid proof and unnecessary base cases will simply ruin the elegance of induction. Never ever underestimate the importance of base cases!