SLOG 7 November 4th 2014

The focus of this week's CSC165 was on assignment 2.  There were a total of 6 statements that were in assignment 2, which needed to be proved or disproved.  The first question involved using the floor, which was something that had been covered in class previously.  However, upon starting question one, i realized that the statements were actually quite difficult to prove fully.  For the first statement, i attempted to use part of the definition of the floor of x, which was that if a natural number z is less than or equal to x, then z is less than or equal to the floor of x.  I wanted to split the antecedent,  which was: if z ≤ x, into a disjunction of z less than x or z equal to x, and then use one part of the disjunction to prove the statement. However, my partner said that I can't just split the less than or equal to sign into a disjunction, so we ended up spending a lot of time thinking up an alternative solution. In the end, we came up with a solution which involved introducing new variables into the proof after talking to Professor Heap to make sure we can do it. The second and third statements were the most difficult of this assignment. The hardest part was to first figure out whether or not these statements are true. From calculus, i recognized 1.2 and 1.3 as definitions of limits, and so i wrote the statement in its limit format, which was: the limit as x approaches w of floor of x is equal to the floor of w. After figuring out the meaning of the statement, i drew the graph of the floor of x, and quickly realized that the statement was false. The disprove of this function involved finding the negation of the statement, which had 3 existential quantifiers. At first i wanted to pick a value for each of the existential quantifiers, but then i realized because there is also a universal quantifier before the last existential, i can only write the last existential(which is w) in terms of the universal (which was d).

The biggest lesson i learned from doing assignment 2 is that before i attempt to jump into the proof, i should first make sure i understand exactly what it is saying, and try to visualize the relationship. I realized as i were doing assignment 2 that often times i get stuck because i dont see the relationship between the numbers and variables, however after i visualize the statement graphically, i can figure out whether the statement is true of false much quicker