On Proofs

      When a student is first introduced to proofs in a math course, they are always presented in as cryptic a way as possible and dressed up in all the fancy math terms the textbook writers can think of. Thankfully, someone with a compassionate heart created this course and so that us poor students get a real introduction to such a fundamental can actually digest proofs and be able to write our own.

      In my experience, students never understand a proof fully if it is just given to them without all the details explained. The "eureka" moment when you get to that connects the antecedent to the consequent is usually glossed over in many proofs, and that leaves me confused when I suddenly see something surprising, but the usage of it does indeed affirm the implication. It's always a good idea to explain why something is used in a proof because not everyone is smart enough to understand that extraordinary algebra in a rearrangement, or why and how the function with a and b in it is necessary in showing a certain relationship relationship. And it goes without saying that the lectures in class also tell us where we should elaborate in the proofs that we construct in the future.

      Now that I am taking this course along with second year calculus, I find that I can really dig into the proofs and understand the thinking process of the person who came up with them. I can fully appreciate just how ingenious some of them are, like the proof for the Bolzano-Weierstrass theorem is a bit long written out but I can see the basic idea is sort of like searching for an element in a list and that's the point in the set that the sequence is approaching...sort of. Well,  before I couldn't even find analogies to help me understand proofs so I would say that this course is definitely putting me on the right path.
 

Translating logical statements in a way YOU understand

       As a person who is absolutely terrible at explaining what is on their mind, naturally I love the idea of formal language. There aren't any double meanings that arise from subtle contextual differences, it stubbornly adheres to only one layer of context, which is that it means what it means. Novelists and poets, go eat your "lavished tomes of nuance and texture" (wow, count those double meanings!). So understanding formal language is the easy part most of the time, it's just like looking at an equation. The hard part is translating our imprecise natural language into formal language symbols. I'm still having difficulties with this, but generally when an English sentence uses "and", it means conjunction. "Or" could be inclusive or exclusive though (the "or" in this sentence is exclusive), but it is easy to tell the different between them. Inclusive or is disjunction, but exclusive or is more analogous to A "and not" B "or" B "and not" A. "And not" is also usually represented in English as "except", but the word "except" could translate into other symbols or operators depending on the context. One could say "and not" implies "except" but "except" does not imply "and not"!
       Logical statements might have precise diction that I appreciate a lot, but their meaning and equivalent statements are a bit harder to internalize. For me, I don't really see why "antecedent implies consequent" could also be written as "(not antecedent) or (consequent)". de Morgan's law was more straightforward though, and I was able to see how that works with Venn diagrams, so maybe if I tried to draw a Venn diagram for any statement or rule I don't understand I would see the connection. Actually Venn diagrams help with vague English statements as well, so I might as well just draw them for everything. Actually truth tables would probably be better for most things. The point is to draw a diagram or use whatever tools you have at your disposal to help you understand.