Week 5 and 6

In the fifth and sixth week, my professor covered how to write proofs for various mathematical statements. This includes formatting them by logically indenting them, commenting them and concluding them with the original statement. Proofs take a lot of time to structure and write because you must consider all possible situations for where the mathematical equation is true or false. Some proofs though, are quick to prove. For example:        $xeR, x ** 3 + 3x ** 2 - 4x = 12
This is an existential, so it is not hard to prove. But you must find the right term to prove it true. This is where it sometimes gets tricky, as you may have to conduct a trial and error method before finding the right term.
So, in this, when x = 2, the existential proves to be true. This is written as:
                                    Pick x = 2. Then xeR    # well known
                                    Then x ** 3 + 3x ** 2 - 4x = 8 + 12 - 8
                                                                                = 12              # sub 2 for x
                                    Then $xeR, x ** 3 + 3x 
This is an example of how a proof is written.

My professor continued teaching us how to write more proofs after this, especially those with floor division of x (ëxû). But, he also reemphasized a limit and its graph and taught us how to write a proof based on limits.
At first, I had difficulty understanding how to write proofs, what conditions to include and how to indent them, but now I have started to understand it. Writing proofs seems simpler for me than it used to because I have learnt that you just have to follow a certain logical order by looking at the mathematical expression.

Week 4

In the fourth week, my professor delved further into the complicated aspects of logic and reasoning. By far, this course has been relatively tough for me. I am constantly introduced to new ways of understanding logic, but I struggle to understand the concept that the professor is outlining and the description in the slides. We learnt about concepts ranging from limits and their graphs to outlining proofs and finding them. One thing I struggle to do is find the difference between two seemingly identical expressions. For example, there are two expressions:     "xeS1, $yeS2, x + y = 5     &                                                                                                                   $yeS2, "xeS1, x + y = 5.
with S1 = S2 = {1, 2, 3, 4}. I could not really see the difference between these 2 expressions except for the fact that their conditions switched places.

One thing that I understood was the definition of a graph of x squared as x approaches infinity. In this case, y approaches infinity. My professor explained that y approaches infinity means that y is moving farther and farther away from zero or becoming larger than zero. He also demonstrated the concept of the graph of x squared for x more than zero. Double quantifiers was another concept that my professor explained. He said that they prove that a certain subset of the Cartesian product N X N (N squared) is not empty and has some properties, in 3 ways. I didn't fully understand this concept either as he covered the material quite quickly and I didn't know what to ask him to clarify my doubt. All I knew was that I didn't understand it. This is why I try and review my notes after class to make sense of the content. As well, I aim to further strengthen my knowledge through the help of teaching assistants and also through the detailed course notes.

I also just completed an assignment that combined whatever was taught until last week. It was very challenging, although I did try my best to understand what I wrote and checked my answers thoroughly. Right now, I am quite nervous about the results of this assignment and look forward to the upcoming material in the course.