On proving functional incompleteness in symbolic logic classes

/ Introduction: Functional completeness A set of truth-functional connectives is said to be functionally complete if every truth function can be represented by some formula which uses connectives only from that set. In the first semester of a sequence of introductory symbolic logic courses, one normally remarks that the usual connectives {-, v, -•, Λ, <->} form a functionally complete set. Typically, one does not rigorously prove this since such proof requires use of mathematical induction—a concept usually reserved for the second semester. However, a method of constructing disjunctive (and conjunctive) normal forms is often given, and the claim is made that every formula of the propositional logic can be treated by this method. The method (for disjunctive normal form) is this: given an arbitrary formula A with n distinct sentence letters in it, represent v4's truth table in the usual way. For example consider (the three displayed T rows are the only T rows):