Slog Entry #4 Week 4

In week 4 of this course, a few but very important concepts were introduced. First was the equivalence between implication in the form of “P->Q" and "not P or Q". This fact was made obvious through the use of a truth table but the fact itself was a powerful tool in proving the equivalence of more complex statements. I emphasize complex because the statements we had to prove were equivalent in tutorial had varying scope and it was difficult to determine where to apply the negation and disjunction. I was actually unable to prove the equivalence, but the solution written on the board by my TA made me immediately see where to apply the negation and disjunction. When it was time for the quiz, I knew precisely how to do the problem and I had confidence that I had done it perfectly.

Another notion that seemed even more shocking to me was the idea that changing the order of variable declarations in a statement changes that statement’s meaning. For example: “for all x, there exists a y” is certainly not equivalent to “there exists a y, for all x”. It took me a long time to convince myself of this but I eventually had a moment of insight and saw why this is the case. In the “there exists a y, for all x” case, a claim is being made that a single number satisfies the condition for all other numbers. In the case of delta-epsilon limit proofs, this claim is unjustified and so it is easy to see why order matters when the quantifiers are of different types.

I would like to conclude that this week has been the toughest week so far for me in terms of tutorial assignments and material presented in lecture. It is evidence that I have much more to learn in the topic of mathematical expressions and I am left desperate to gain more and more knowledge.