Week 2 - Jan 19, 2013

This past week we covered a number of different topics, however I feel there is one concept that stands out. We looked at universal and existential quantification; that is how to identify one from the other as well as how to prove and disprove both.  So far I feel very comfortable with this material and I believe this will set the foundation which will be used to approach proofs in the future (a foundation I had never been introduced to before this class).  

We also covered topics such as implication, equivalence and looked at some new notation which will allows us to translate English statements into mathematical statements.  The one thing I had some trouble with was predicates.  In class as well as in the online course notes I could never find an explanation as to what exactly the predicate, P(x) for example, represented.  Fortunately I was able to find a video online (https://www.youtube.com/watch?v=DmCltf8ypks) which explained in detail what a predicate was, as well as what the statement "For all x, P(x)" means.  I had first thought that when we define a predicate we indicated the situation in which that predicate was true.  For example, P(x) : x > x + 1.  Initially I thought this meant that P(x) is true when x > x + 1.  

After watching the video above I now understand that the predicate P(x) is either true or false, but only when it's input variables are bound by either a value or a quantifier thus making it a proposition.  Once the predicate P(x) is bound, it is no longer a predicate but a proposition.  This proposition is then either true or false.  In the example above with P(x) : x > x + 1, if we bind x = 1 then the proposition P(1) is false.  We could also bind P(x) with a quantifier.  For example, "For all x, P(x)".  This statement says "For all x, the proposition P(x) is true".  This statement is of course false, because for any x, x > x + 1 will always be false.