The Beginning to my Mathematical Expression

The Beginning to my Mathematical Expression



     Quite frankly, I wasn't sure what to expect from this class. When I enrolled for courses like calculus and biology its safe to assume I had a firm idea of what it would involve. On the contrary, I had never taken a course that placed such an emphasis on logical reasoning and expression. Not being sure what to expect, I was optimistically skeptical the first class in. That first class Professor Heap described proofs as being works of literature and this statement I believe to be fundamental. These poetic remark was thought provoking in the sense that it gave me a new insight to mathematical logic. All good works of literature have some key elements that define their success. Having little prior experience to the subject myself, I started applying this idea to not only this class but calculus as well resulting in a rather successful turnout. Giving structure to a proof gives a sense of anticipation to their mysterious nature. In other words this structure is a sort of map. If we anticipate that there will be some sort of rising action and we correlate it to some part of the proof, we know we're close to a eureka moment. That eureka moment is what leads to the falling action and the resolution. In practice, this especially helped me when I'd have moments where I'd be stuck. Taking a step back, I'd think to myself, "what's next?" Going through this logical schematic, so far I've been able to work the problems out consistently. In essence, that remark's implication is itself a puzzle which has to be deciphered using logic and reasoning.
     Furthermore, what I found really interesting was the concept of having an if statement that may not necessarily be factually correct whilst having a then statement that is correct equate the whole implication to be true. For instance if pigs can fly then the sky is blue. The reasoning to my understanding being that the sky is in fact blue to the human eye and this would remain factually correct whether or not pigs could fly. Or rather we have no basis to support that the sky wouldn't be blue if pigs in fact could fly. Progressing forward, when we reached the Venn diagrams and we began to express what the diagrams represented in notation an interesting situation came about. By not marking that a section was completely empty, the possibility that an empty set was a subset of another empty set arose. This case was then used to prove that that particular Venn diagram was done incorrectly.