SLOG 2 September 27th 2014

               In this week of CSC165, I learned about conjunction, disjunction and negation.  The conjunction and disjunction were very similar to the ‘and (n)’ and ‘or (u)’ used in sets and Venn diagrams, and so it was fairly easy for me to understand what they mean.  Disjunction and conjunction are also identical to the “and” and “or” commands in Python, which I had learned in CSC108 already, so I was able to understand how they are used to evaluate statements and implications.  Basically, conjunction is used when the conditions on either side of the sign must all be true in order for the statement to be true.  For disjunction, only one of the condition being true will satisfy the requirement for the statement as a whole to be true.  However, the use of brackets can modify which requirement comes first, whether conjunction and disjunction, and during the tutorial this week I went over some example statements that gave me a better grasp of how conjunction and disjunction is used.  To not be confused by the order of the requirements, I like to underline parts of statements that need to be considered first.  For example, if the statement was (P n Q) u R, I would draw a line under (P n Q), and another under R, so I know which two objects I’m looking at and whether disjunction or conjunction is the main requirement of the statement. 

I also studied negation during in this week, and over negation wasn’t a difficult concept to grasp.  It is basically the opposite of what is being said.  That means if a statement is true, then its negation is false.  However, one problem I tend to have when dealing with negation is that I often confuse them with contrapositives.  What I mean by this is that when I try to find the negation of a statement, I end up finding the contrapositive instead.  To prevent this problem from reoccurring repeatedly in the future, I realised that the negation is the opposite of the statement, so if I put another negation sign in front of the negated statement, then I would get the contrapositive. 

Lastly, during the class on Friday I had the opportunity to work with my friend on a problem which involves finding and expressing the patterns of up and down creases of a paper strip folded n number of times.  I devised a plan to first observe the first 4 folds and discover the basic relationships between the number of folds n, the number of layers after being folded n times, and the number of creases c.  We found that the number of creases can be calculated using the formula 2^n – 1, which helps us understand how many creases there are.  However the problem of how the pattern of ups and downs go still remain unsolved.  We drew a diagram depicting the ups with arrows pointing up, and downs with arrow pointing down, and discovered that there exists a central crease which remains the same at all time.  Every time the paper is folded, new creases are generated on either side of existing creases, and the new creases have an alternating orientation of ups and downs.  The biggest issue we encountered was to express this pattern mathematically.  This was the most frustrating because although we know the pattern, we could not express it in a way that the pattern of creases for any number of folds can be calculated.  After doing some research about the problem at home, I learned that a type of function called recursive function can be used to predict the pattern.  This is because recursive functions are basically looping functions that uses the output as the input, thus if we can enter the basic pattern of how the creases are added, then the final crease pattern can be predicted with the function.  

SLOG1 September 17th 2014

In this week of class, I learned how to understand ideas that are expressed in various mathematical and linguistic forms.  During the first week of class, I was introduced to the basic if and then expressed as events P(x) and Q(x).  It was easy to memorize all 4 different combinations of truths and false of P and Q, but I had a hard time understanding why the if and then statements are true even when P is false.  Besides the if and then statements, I also had a difficult time visualising the all and any statements, and I had a lot of trouble when I had to work on some problems in class.  In the following classes during the first week, I was taught about the universal and existential quantifiers.  Unlike the if and then statements, I have never seen the two types of quantifiers before, and so I didn’t really understand what was going on in class, especially when the professor gave examples of statements that are expressed with the quantifiers, and what they meant in each scenarios.  All I could do during lectures was memorize and copy down as much material as possible, and try to think about them after class.  However, during my MAT137 class, I was taught about universal and existential quantifiers with much more detail, and I realised that the quantifiers are essentially just another way of expressing if and then statements without using P(x) and Q(x).  I also learned about vacuous truths in math, which helped me understand why if and then statements are true even if P is false. 

One of the most important thing I learned in class which really helped me understand the different ways of expressing implications was the rules to verifying and falsifying universal and existential claims.  I learned that:

-        To verify a universal claim, show that there are no counterexamples

-        To falsify a universal claim, show at least one counterexample

-        To verify an existential claim, show at least one example

-        To falsify an existential claim, show that there are no counterexamples

With these rules in mind, I began to translate if and then statements into their universal and existential forms, and by using the rules, I was able to figure out what needs to be true and what needs to be false in order for the statement to be true or false.  The tutorial exercise which involved expressing if and then statements with venn diagrams gave me a stronger visualisation of what is being described by if and then statements, and which areas must be empty and which areas must be occupied for various statements to be true.  I also found out after examining the 4 rules above that verifying a universal claim is the same as falsifying an existential claim, and falsifying a universal claim is the same as verifying an existential claim.  It is then that I saw the connection between universal and existential claims.  By the end of this week, I believe I have a good understanding of what has been covered in lectures and tutorials, and I am confident that my level of understanding increase as the weeks pass.