The Final Exam will be in the usual classroom on Tuessday, May 9, 2006, from 8:00-10:15 a. m. Calculators or other mechanical assistance are not needed and are not to be used. Blank paper will be provided, so all you will need is something to write with.

Some of the questions will be similar or even identical to questions on the in-class exams, and on the homework. The final part of the course, after the Exam III material, will receive some extra weight since it has not already been tested.

On this exam, as on all exams in this course, “Prove” means “Give an argument”, not “Tell why”. Definitions are fair questions, in fact the fairest since knowing the definitions is so fundamentally important. You do not need to give them in exactly the way that they were presented in class, but the definitions you give should be precise.

The following will definitely be covered, although the exam not necessarily limited to these topics:

1.	methods of proof: ways to prove implications, how to handle quantified statements, proof by contradiction, and mathematical induction (starting at 1 or starting at some other integer).
2.	counting problems using the Product Rule and the Sum Rule
3.	countable infinite sets and Cantor's diagonal method for proving that sets are countable
4.	uncountable infinite sets and Cantor's diagonal method for proving that sets are uncountable
5.	cardinality of sets, including Cartesian products and power sets
6.	divisibility in the integers and its basic properties, prime and composite numbers, the Fundamental Theorem of Arithmetic.
7.	functions, basic definitions, composition, injective, surjective, and bijective functions
8.	greatest common divisor and relatively prime integers
9.	permutations and combinations, binomial coefficients (fairly basic questions, since this is very new material)

The following do not appear on this exam (which is not to say that they are unimportant): interpretation of quantified statements using “truth sets,”, the book's “rules of inference,”, truth tables, “vacuously true” statements, advice on good writing (but please try to use it), the concept of multiplicity, proving basic properties of union, intersection, or complementation, Venn diagrams, Rolle's theorem, tricky membership or subset relationships involving the empty set, vacuously true statements, n-tuples other than pairs, preimage of a subset, the Prime Number Theorem, the Division Algorithm, the Euclidean algorithm for finding the greatest common divisor, the proof that there is no surjective function from a set to its power set, the pseudo-proof of the Induction Theorem, well-ordering of the natural numbers and the justfication for the induction theorem, Strong Induction, the Inclusion-Exclusion principle, fine points that were mentioned only in passing.

Advice: Make a reasonable plan of study, starting by making sure you know how to do all the problems on the three in-class exams, and how to do the homework problems related to the final exam focus topics. Get started several days ahead of the exam, and most importantly, do not stay up late studying the night before the exam--- that is how disasters happen. Finally, when you take the exam, just relax, do your best, and move on. It's just an exam.

The final exams from last fall's class can be found on its course page (links to my previous course pages appear on the course pages page). That course covered more material than we did this semester, so some of it should seem unfamiliar.