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