SLOG #4

    This week's material is all about proofs: structure of proofs, actual content of proofs, and comments that help explain the proofs. The proof structures originally seemed very redundant to me, that they serve only the purpose of visual formality, and are otherwise utterly useless, but I now understand that it is important for me to familiarize myself with the train of thought following the proof structures: when I assume or assign something, I am then in the domain where something is certain to be true. I can see how that can simplify and clarify the process of drawing a conclusion as I exit the domains, one by one, all while stating that the content inside supports the assumptions, until I declare the concluding proof.

    One thing that confused me was the second question on the tutorial, which states: prove that for all x in the set of natural numbers, if (there is a y in the set of natural numbers such that x can be written as x = 3y + 1), then (there is a y in the set of natural numbers such that x^2 can be written as 3y + 1). The correct structure included assigning a y0 to be x = 3y0 + 1, and assigning a y' that depends on x and y0 (Let y' =... #some value that depends on x and y0). At first, I didn't understand why I needed two separate variables to write into the proof structures. Then, I realized that it's because they are indeed two distinct representations of different numbers when I got around doing the actual proof. (3y0 + 1)^2 gives 9y0^2 + 6y0 + 1, and then we just have to set y' = 3y0^2 + 2y0, then x^2 would be equal to 3y' + 1. 

     We left off Friday's lecture at an unfinished proof that there is an infinite amount of prime numbers. What we have is "pretend this is a finite set of prime numbers, all the numbers in the set of prime numbers are multiplied together, add 1 to the product, a new number that cannot be divided by any number in that set of prime numbers is created..." However, the fundamental theorem of arithmetic says all integers greater than 1 is either a prime number or a product of prime numbers. Since that newly created number cannot be divided by any number in the set of our finite prime numbers, then either that number is prime and not in the set, or that number can be divided by some other prime number which is not in the set. Because there is always a number to add to our finite set of prime numbers, our assumption that the set is finite is false.