MATHEMATICS 300- SPRING 2000

REVIEW LIST FOR FINAL EXAM

I) LOGIC AND SET THEORY

logical connectives, truth tables

quantifiers/ negation of quantified statements

"converse" vs. "contrapositive"

simple proofs in set theory

relations/ equivalence relations/ partitions

construction of Z and Q from N using equivalence relations

2)FINITE & COUNTABLE SETS

proofs by induction (inc. inequalities)

finite sets: definition, cardinality, "pigeonhole principle"

the binomial theorem and applications

coountable sets: countability of Z, Q; uncountability of R

3) ALGEBRAIC PROPERTIES OF R

Rings, fields, integral domains: definitions, examples. The complex numbers

axioms for an ordered field; Archimedean property; well-ordered sets; partial orders.

solution of inequalities involving absolute vlalues

irrationality of square roots

4) SUPREMUM AXIOM

Definition of supremum/infimum of sets; approximation property

rational approximations to the square root of 2

Archimedean property of R; density of Q in R; dyadic rationals

Sup/inf of function on sets

5) CONSTRUCTION OF R FROM Q; CAUCHY SEQUENCES

Limits of sequences; Cauchy sequences

Subsequences; bounded sequences have convergent subsequences

Equivalence of the supremum property, convergence of bounded monotone
sequences and

convergence of Cauchy sequences

Equivalence classes of Cauchy sequences: definition of sum, product
and order

**FINAL EXAM: 8 problems, 4 from the topics in parts 1) to 3) above**
**(review exams 1,2,3!) and 4 from 4) and 5) (review class notes,
handouts)**
**CLOSED BOOK EXAM, calculators NOT allowed**