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Final Exam Review Outline

The final exam will be cumulative, though with more emphasis on the post-midterm material. Below is a list of topics covered by the exam.

Pre-midterm

1. Prove basic properties of the integers using only the axioms.

2. Find the negation, converse, and contrapositive of a statement.

3. Use the following techniques of proof: proof by contradiction proof by contrapositive proof by induction or strong induction

4. Prove an “if and only if” statement. Prove two sets are equal.

6. State and use the definitions of union, intersection and complements of sets Cartesian product of two sets equivalence relation equivalence class

7. Binomial Theorem definition of binomial coefficients Binomial Theorem (Thm 4.21)

Post-midterm (Note: theorems and propositions listed are useful statements that you may quote and use in your proofs. You must quote the statement, NOT the number of the theorem or proposition!)

1. Number Theory Division Algorithm (Thm 6.13) greatest common divisors Euclid’s Lemma (Prop 6.31) Unique factorization theorem (Thm 6.32) Integers mod n

2. Real numbers upper and lower bounds supremum (least upper bound) and infimum (greatest lower bound) Completeness Axiom

3. Limits Absolute values as distances, the triangle inequality (Prop 10.10) definition of convergence to a limit (be able to state the definition and use it to prove convergence) bounded sequences, increasing sequences, decreasing sequences, Thm 10.19 properties of limits (Prop 10.23)

4. Rational and Irrational Numbers Definition of rational number Thm 11.8, Prop 11.17

5. Infinite Series Definition of convergence of an infinite series Comparison test (Prop 12.3) Decimal expansions expressed as infinite series

6. Cardinality (Ch 9.1 and Ch 13) Injections, surjections, bijections Left and right inverses Definition of “same cardinality” Countable sets, Prop 13.9, Prop 13.19 Uncountable sets Power sets, ordering on cardinalities

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