Mid-term and Proofs

Mid-term Finished Confetti!

    As promised!  Mid-term went well, or at least I think it did.  The thing that caught me off-guard was how exceedingly similar it was to the provided past mid-term, so as long as you reviewed that one you were set.  That said, it's worth pointing out a single mistake in the first section could take that juicy 100% to a modest 75%, if you were to confuse any two python functions.

    But as to not dilly-dally, we also jumped right into proofs after our test was over and the various ways to do them!  And we went over arguably one of the most important methods of proofs there are:  Proof by contradiction.

    At first glance, it's difficult to prove things that are a bit more abstract in concept, or seem tedious to run through every possible combination of.  Are there an infinite number of prime numbers?  Is the square root of 2 irrational?  Indeed, by proving there is a contradiction in the opposite, you are thereby proving the original statement to be true.

    One of the simpler examples provided in class was the following statement:  There are five boxes, that all together contain a total of 51 balls.  Prove that at least one box has 11 balls.  This may seem tricky to prove at first, but becomes reasonably trivial if we merely use proof by contradiction.  By assume that every box has at most 10 balls (for the statement to be false), we can immediately then show the 5 boxes can only have a maximum 50 balls (10 * 5), thereby demonstrating a contradiction!
   

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