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.

Week 3

This week we started off by being introduced to the Conjunction (and) and Disjunction (or) which were made easy to understand by providing their set notation counterparts. I understood conjunction to occur when an element has two properties therefore two predicates hold true for that element. I also learned that we need to be careful with the and in English sentences as there could be confusion between there existing an element of each predicate and there existing an element which holds true for both predicates. I understood dis-junction to occur when at least one of  two the two predicates have to be true. Again I learnt that we need to be careful with the or in English sentences as they could be exclusive meaning that only one of two predicates may hold true for an element where as in dis-junction both predicates could hold true for an element but at least one must hold true.

The next topic we went on to discuss was negation which I understood to be the complete opposite of a logical expression.I found it very simple to negate a whole expression but it got a little tricky to place the negation sign on the smallest component of an expression. This became easier when we were introduced to the idea of negating an expression part by part. To negate a universal quantifier we use an existential quantifier. To negate an existential quantifier we use a universal quantifier. Then we just logically negate the predicates and/or implications in the expression.

We also covered the importance of placing required brackets in the right places to indicate the scope conjunctions, dis-junctions, and implications as without brackets there could be multiple meanings of an expression.

The last topic we covered this week was truth tables which should be used to represent expressions in the case that there are too many predicates which would create a very confusing Venn Diagram. i found these tables very interesting as we can easily access data from them to determine whether an expression is true or false and these tables can also be easily implemented into a program. The topic I found challenging was DeMorgans law an I plan to pay more attention to understand the exact meaning.

Week 1-2

Just from the introduction of CSC165 this course seems very interesting. I am really exited to learn how solve problems using mathematical expression and reasoning and apply these methods in programming. I also enjoyed seeing how a paragraph written in English could be translated into math making it easier to withe a program to complete whatever tasks or requirements that were in the program. In particular I enjoyed the example of how a user may ask a machine to find a phone with certain requirements which we can express mathematically and create a program to complete the tasks.

The next topic we discussed in class was precision and how precise or ambiguous to be in certain tasks. I found it very interesting how to successfully communicate with anyone we have to be the right amount of precise or ambiguous. For instance when we communicate to other humans we should be as precise as needed but when we communicate to a computer or when a computer communicates with us a high level of precision is needed.

Then we continued onto analyzing statements classifying elements of sets. Initially I found this a bit challenging as I could no picture the elements and the sets that a particular statement would refer to. But then when we were shown how ti show these elements in sets in the form of a Venn diagram, I found it a lot easier to picture the situations. The Venn diagrams helped me again when we started falsifying or verifying quantified claims as I could now easily picture what cases would have to occur for a statement to be true of false. Then i was also able identify what programs could analyze the quantified claims and also what input parameters to pass into the program to analyze the claims.

The last topics we went onto were sentences,  statements, implications, and predicates. I understood that sentences are phrases we cannot verify or falsify without further knowledge but statements can be verified or falsified.  In the predicates section we learns that if an element x has a property L then we can write L(x) to express this. When we continued onto implications, I was getting confused on the tricky phrases in identifying the assumption and the consequence. I plan on paying closer attention in class this coming week to get a better understanding of implications. Although I was able to understand the concept of Vacuous truth as even though logically an assumption or consequence is incorrect  if it is not possible to show that they are incorrect, the statement is taken to be true. That is where we ended off for the week.