Week 6

This week we were given our tests back and I feel I did pretty well however so did the rest of the class which is fine at least it shows that we are all learning. We started off the lecture this week with a definition of boolean and non boolean functions which is very easy to understand. We were then shown the floor function which is just the largest integer which is less than the number we are taking the floor of. We were then shown a definition for the floor function and a simple proof by contradiction which helped in understanding the function.We were also shown how to disprove a statement by just proving the negation of the statement which was pretty easy to understand. The next part was where the material got tricky. We were shown a delta epsilon proof about limits. We were to prove that as x approaches 2 then x^2 approaches 4. The key was to finding a delta in terms of epsilon such that the statement is true. this was a little tricky at first but once we were shown the complete proof it became easier to understand.

Week 5

This week we had our first test which I feel I did really well on as I had practiced a lot before hand doing the previous years test. I think the assignment really helped on this test as a lot of the questions were similar.. This week went into more detail with proofs and proof structures.  We were taught how to prove a statement using the contrapositive of the statement.  If we know the statement is true and we know how to prove the statement the we  can simply add a line to the proof to state that the contrapositive is also true since it is equivalent to the statement. We were advised to use contrapositive to prove a statement if proving the reverse direction is easier than proving the original direction. We then went through the steps of proving by contradiction. For a statement which is true if we assume it is false then we should reach a contradiction which results in the statement is true. We were shown the proof by Euclid showing that there are infinitely many prime numbers. This proof seemed difficult at first but now its seems easier to understand after a while of looking at it. We were then shown a proof for an existential statement which was very easy to understand as all we have to do is find one example making our argument true to prove it of false to disprove it. 

Week 4

Here we are going into our fifth week of class. At the end of this week we had our first assignment due which I feel pretty comfortable with as the questions on the assignment did not deviate from what we had covered in class so I felt prepared and capable while doing Assignment 1. We started off the week by learning how to express implications as conjunctions and dis-junctions which I felt was pretty intuitive. I also personally found it easier to identify conditions which would make a statement false so when it came to expressing implications as conjunctions and dis-junctions I would just negate the condition making a statement false which of course would give us the condition for a statement to be true. This was just a personal thing as for some of the implications I could not figure out all the possibilities for a statement to be true but I was able to identify what conditions made a statement false. The next topic we covered was transitivity which again seemed intuitive as it says if something1 implies something2 and something2 implies something3 then something1 has to imply something3 which is easy to picture and we can reason it out using some simple logic. We then proved transitivity by negating the implications which had a contradictions which meant that transitivity was always true. The challenging part of this week I found was the first mathematical example. I am still having trouble understanding the graph a little. From what I understand this statement says that for all d there exists a d such that if the distance between any x and 2 is less than d implies the distance between the line x^2 and 4 is less than e. Actually now that I write it out it is becoming easier to understand but it was still a challenging problem.

We were then introduced to how to structure proofs which I believe will be very helpful for this course and the other math related courses I am taking which require proofs too be written frequently. We were taught to find a chain between what we are given an what we want to prove.