Week 7

1. So we took a look into sorting algorithms and  run-time complexity. Bubble sort and Merge sort . Then we checked how quickly the two sorts run. And whether the ran linearly or quadractically. We learned that linear is faster then quadratic. Also run time means how many steps are taken rather them how many seconds it takes to run.

1. First we count the number of steps without constant factors. only the highest-order term matters . We then learned about algorithm analysis and asymptotic notation. We were introduced to this concept with an example of bubble sorting and merge sorting. We checked which of the two aforementioned methods of sorting was the most effective by analyzing their respective algorithms .Algorithms are important because they determine the running time of a function .  We also learned about how we do not really care about the number of steps withing an algorithm, but about how the number of steps increases as the size of the input increases
2. Then we started talking about Big Oh .We start off by picking a positive real number, c, and a natural number, b, that will work. Then we assume that n is a generic natural number in order to introduce ∀. We assume the antecedent, n ≥ B. We saw the definition of big Oh.                  
   
   So basically we must pick a c past the breakpoint B such that f(n) was less than or equal a constant times g(n) or n-squared.

  



Week 6

1. No class on Monday WOOT!! WOOT!!. Thanksgiving baby. With the potatoes and the turkeys and the stuffing and the rasmfrasm. So we learned more about proofs this week using floors and absolute values in proofs. Also we saw functions that didn't returned Boolean and non-Boolean functions. when using floors we must first understand the definition of floors.
   
2. Y is an integer. And Y is less than or equal to X. So basically Y becomes the immediate less than integer , I guess. Why is this so, I don't know why yet but that is what we have learnt so far. This is all so new to me. Sometimes I feel like when we do some of the proofs in class we break a lot of mathematical laws. But we also don't kind of. Alright so how do we use the floors in proofs? "Hoew waz ze exem Maximillian",
3. The 1st term test is done with, so that's good, I didn't see the practice test until 30 minutes before the test so that's bad. I don't think I did very well in general a lot of rushing and attempting to erase what I did and starting over. I almost didn't do a whole question i think.

I learned that when you are doing proofs try and manipulate expressions closer to the form of the definitions of expressions. e.g. floor so





                  Assume ....# definition of floor and antecedent of question given

                      Assume .......# Generics

                          Then .............# Try and relate expression in relation to the definition of floor

                           .                    # e.g.  For all x in R ,[x] > x+ 1, this is false so disprove. Find a  

                           .                               Contradiction like [2.5] , according to the definition of floor its 2

                           .                                etc.

                           Then .........# conclude derived conclusion

                       Conclude ............# restate the previously assumed expressions

                  Conclude ..............................

Just a simple example of a proof involving floors and how one might deal with them.



Week 5

1. Week was tough. Test 1 and assignment 1 are over and its time to start studying all over again . "Got to love UofT". Test 1 was good and fair but I was not to prepared for it considering I didn't do all too well on it but finger crossed for test two.
2. I've been working on some of the proof structures and especially the commenting on each step is bugging me out. In class we did the question on There are 5 boxes in which there are in total 51 balls. Prove that there is a box with at least 11 balls in it. I understood the premise of the question. We used negation to prove it. we assumed that there was not a a box with at least 11 and tried to prove for ten. But if every box had at most ten Then the 5  boxes would have at most 51  being a contradiction.
3. Proof example:   from handout
   

   
4. Whenever we make an assumption we indent the next line, and once we have proven our conclusion, we can unindent and start restating the initial assumptions . Also we make comments next to most lines to explain our thought process and how it relates to the net sentence. For example, real numbers are closed under addition, subtraction, and multiplication and natural numbers are closed under addition and multiplication. Cool ain't it.
5. Anyways there's a new test and assignment coming up so we'll see how that goes. Hope fully i understand how the next topics come.

 

 

Week 4

In the fourth week we are doing our first turned in assignment. it has a couple of confusing questions like in the question 5 Where we used the empty set as a precedent in the if statement. I still think it makes the antecedent true. since it can be seen as a false statement.we practiced some more operations with predicates, specifically conjunctions, disjunctions and predicates. Nowadays I don't write as much in lectures because I can hardly understand and write when Danny teaches. Anyways hopefully the assignment turns out well.