Introductory Logic: Universal Quantification

On a normal circumstance, I am not easily fascinated by crude and factual topics I might encounter in a course, unless I can somehow manage to find an application for such concepts in my everyday life.  Thus far, I have found the introductory logic laws and rules covered in CSC165 to be incredibly engaging; you can apply the concepts to anything from a problem set in your homework to the logical reasoning behind the validity of cosmic laws that govern our universe. You can logically account for every situation and come up with a beautiful argument that an intellectual is unlikely to reject.

To start simply and cover one very basic example of a logical statement, one must mention the concept of a universal claim.  Things in logic are looked at from two perspectives: "all" or  "some" elements in a set are claimed to have a certain property.  For now, I will go over the universal quantification; a method that accounts for and studies all items in a set.  Let's take on example: For all elements in the set of Animal Kingdom denoted as AK, each element is an Animal, with the set of "Animal" being denoted as A. To write that using symbols, ∀ x ∈ AK,  A(x), or "Every element in the set of  Animal Kingdom is an Animal."

To prove this statement, one must go through each and every item in the set of Animal Kingdom and make sure that each of those elements is indeed an Animal; prove that there are no counter-examples.  To disprove this statement, one needs to find find only one element in the set of Animal Kingdom which is not also an element in the set Animal; provide one counter-example.