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.