Math 321 Review for Final Exam Generalities You will receive the take-home portion of the exam during the last class, on Tuesday, May 5. The in-clss portion will be held on Tuesday, May 12 from 9:45 11:45 AM, in our usual classroom. The take home exam is due when you walk in to take the in-class portion. The take-home exam require you to complete five problems (chosen from seven on the exam). It would be a good idea to complete all of the problems to help you study, and to turn in good write-ups of the ones you feel most confident about. The take-home portion of the exam is worth 40 points, and is primarily proof-based. The in-class exam has nine problems, some of them multi-part. Some require definitions, some examples, and some proofs. The proofs, for the most part, should be shorter than the ones required on the take-home exam. The in-class portion is worth 110 points. Most of the material covered on the exam comes from the last third of the class (relations, functions, and cardinality). However, the main point of the course has been learning to write proofs. The expectation is that you apply what you’ve been learning about writing good proofs to this new content. New Material Review relations, functions, and cardinality. In particular, • For any old homework problems that you missed points on, re-work the problem and be sure you understand it. • Also be sure you do problems that were assigned but not collected. If you feel unsure about your solution, stop by office hours and show it to me. Be able to define, give examples of, and decide if something is: • a function (including 1–1, onto, bijective functions); • a relation (including partial orders, total orders, and equivalence relations); • a partition (and know the correspondence between equivalence relations and partitions); • a maximal element and a minimal element for a set A; • a least element and a greatest element for a set A; • a pair of sets satisfying card A = card B, card A ≤ card B, and card A < card B; and • a set that is countable, or a set that is uncountable. Old Material You should review old exams, old review sheets, and any old homework assignments that you found particularly difficult. You should know definitions, theorems, and proofs for the big ideas in number theory, including: • Definition of a | b and theorems like: a | b and b | c ⇒ a | c. a | b and a | c ⇒ a | bx + cy for all integers x, y. • Know the definition of gcd(a, b), how to perform the Euclidean algorithm to find a gcd, and the statement of the division algorithm. • Know the statements of trichotomy and WOP on N, and how to use them in proofs. • Know about the systems Zm and how to solve equations there. • The definition of prime and theorems about primes like: If p | ab then p | a or p | b. Every integer > 1 is a product of positive primes. • The definition of units and theorems about units like: The product of two units is a unit. An element a ∈ Zm is a unit if and only if gcd(a, m) = 1. • Know how to use truth tables to prove that statements are (or are not) logically equivalent. • Review proof by induction. • Know how to construct proofs for statements of the form “For all x...,” “There exists and x...,” and “There exists a unique x...” • Know how to prove contingency (if / then) statements through direct proof, contrapositive, and proof by contradiction.