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