Hello! So! Had my first midterm this past week for this class. The difficulty was in-between the short and simple class quizzes and the noticeably more challenging homework assignments, making it a fair midterm in my eyes. I am confident I did well, a few small mistakes here and there--but overall I understood how to 'read' math and use it properly,
Aside from this, we continued with proof structure and solving and proving examples. I must say, this is quite abstract knowledge: it can be a simple task to prove something true or false to yourself, but to prove it to completion, leaving absolutely no loose ends, can be a menial task.
Even the 'simple' idea of proving that there are an infinite number of prime natural numbers becomes complex when you have to prove it. Specifically, it is proven through contradiction--which simply assumes the negation of the thing (that you are trying to prove as true) is true, resulting in some fundamental impossibility--say, 1 = 2--which in turn means the assumption must have been false, thus stating the original statement was in fact true.
This proof by contradiction is very difficult when utilizing the English language, due to its ambiguity in communication--nonetheless, it is a logical argument if all sides/parties hold to the same definitions.
I am not at all firmly grounded in these proofs though. I am unable to complete the proof by contradiction for prime natural numbers at the moment. I have this vague idea that, since the negation of there being an infinite number of natural primes is, there exists a natural number n such that the last prime |P| <= n , I need to set this |P| to be n and then have |P| +1 become the new |P| in the form: m + 1 <= m where m is a natural number. This would be the necessary contradiction to finish off the proof, but I am probably missing a step somewhere here. Anyone have an insight?
Thank you for reading.
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