Exam II will be in the usual classroom on Thursday, April 27, 2006. 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.

One of the questions will be similar to questions that appeared on Exam II. Be sure that you understand well how to do all the problems on Exam II, so that you can easily do the similar question.

The main topics will be the integers, congruence, countable and uncountable infinite sets, and proof by induction. On this exam, as on all exams in this course, “Prove” means “Give an argument”, not “Tell why”.

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

1.	divisibility in Z: the definition, proof of elementary properties such as “if a | b and a | c, then a | b+c” and so on.
2.	Euclid's proof that there are infinitely many primes
3.	countable infinite sets and Cantor's proof that the rational numbers are countable. Know the proof, and be able to adapt it to prove similar facts.
4.	uncountable infinite sets and Cantor's proof that the real numbers are uncountable. Know the proof, and be able to adapt it to prove similar facts.
5.	prime and composite numbers, the Fundamental Theorem of Arithmetic.
6.	the congruence relation for integers: definition, basic properties, cancellation properties.
7.	greatest common divisor and least common multiple
8.	induction, what it says and its use to prove assertions such as the in-class and homework examples

The following do not appear on this exam (which is not to say that they are unimportant): 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.

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 had 50-minute classes, so our exam will be somewhat longer. Different classes are different, and the exams will different, but in general it's a good technique to rehearse an exam using others from previous classes.