
{\bf\centerline{   Math 223 02, Spring 2004 syllabus}}
\smallskip
{\sl Text} Mathematical thinking, problem-solving and proofs,
   John D'Angelo and Douglas West, Prentice-Hall, 2000.
  \medskip
  {\bf  Introduction to Introduction to Mathematical Reasoning}
absolute value,  elementary inequalities such as  triangle inequality and
arithmetic-geometric mean inequality as introduction to the notion of a
proof
   (preface and pp4-10)

   \smallskip

{\bf   Sets, functions, and numbers (Chapter 1 of text)}  sets,
equality of sets, standard sets (integers, natural numbers, reals,
rationals, empty set), equality of sets, subsets, operations on sets
(union, intersection, difference,  Cartesian product,power set), Venn
diagrams, definition of functions, sum and product of real-valued
functions,  real polynomial functions, identity function, constant
functions, bounded real valued functions,  monotone real-valued
functions, discussion of  axioms for real numbers  as complete
ordered field

   \smallskip

{\bf   Language  and Proofs (Chapter 2)} mathematical statements,
negation and quantification of such, compound statements (conjunction,
disjunction, implication), elementary proof techniques (direct
proof, contrapositive, proof by contradiction, counterexamples),  iff,
converse of a statement
\smallskip
{\bf   Induction (Chapter 3)} natural numbers, principle of induction,
   strong induction, well-ordering of the natural
numbers. Many examples of inductive proofs  including binomial theorem
and  existence of at most $n$ zeros of a degree $n$ real polynomial (using
factor theorem) \smallskip

{\bf Bijections and cardinality  (Chapter 4)} Bijections,
inverse functions, injections and surjections, composition of functions,
associativity of composition, preservation of injections, surjections,
bijections under composition, permutations of a set,
two-line and cycle notation for permutation of a finite set,
  cardinality of a finite set and elementary
properties thereof,  cardinality of sets, countable and uncountable sets,
countability of ${\bf N}\times {\bf N}$, statement (no proof) of
Schroder-Bernstein theorem, proof of uncountablity of set of
sequences from $\{0,1\}$\smallskip

{\bf   Divisibility (Chapter 6)}  Divisibility of integers, division
algorithm for integers, integer linear combinations, ideals (of
integers), theorem that an ideal (in $\bf Z$) is principal,
  greatest common divisors and least common multiples, relation between
gcd (of two integers) as generator for the ideal of their integer
combinations, prime numbers, fundamental theorem of
arithmetic, Euclidean algorithm  (expression of a gcd as an integer
linear combination), infinitude of the set of primes.\smallskip

{\bf Modular Arithmetic (Chapter 7)} Relations, reflexive, symmetric,
transitive, anti-symmetric properties of a relation, partial and total
orders, equivalence relations, fundamental theorem on equivalence
relations (equivalence relations and  partitions of a set), congruence
modulo $n$, binary operations, addition and multiplication modulo $n$,
the set (commutative ring) of integers modulo $n$,  discussion of
existence and computation of a multiplicative inverses modulo $n$, fact
${\bf Z}_p$ is a field for $p$ prime and not an integral domain if $p$ is
composite, Chinese remainder theorem,  Fermat's little theorem, Wilson's
theorem, definition of group, basic examples of groups (integers, additive
and multiplicative groups of  fields, permutations of a set), very
basic properties of group (uniqueness of identity element and
inverses, formula for inverse of a product)
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{\bf  The rational numbers (Chapter 8 and Appendix A1-A20)}
Discussion (without proof) of constructions of $\bf N$ from set theory
and ${\bf Z}$ from ${\bf N}$, detailed construction of $\bf Q$ as field of
quotients of
$\bf Z$,  rational functions (field of quotients of polynomial rings over
a field), rational zero's theorem (on zeros of a polynomial with integer
coefficients), parameterization of rational points on the unit circle,
determination of all Pythagorean triples\smallskip



{\bf   The real numbers (Chapter 13)} Sups and infs, completeness axiom
for
$\bf R$,  application  to existence of $\sqrt 2$, Archimeadean property
of $\bf R$, limit of a real sequence, uniquencess of limit,
characterization of sups by sequences, monotone convergence theorem,
existence of canonical decimal (or $k$-ary) expansion of a real
number, fact every decimal (or $k$-ary) expansion represents a real
number, uncountability of ${\bf R}$\smallskip

{\bf Sequences (Chapter 14, 14.1--14.19)} Limit of sum, product, quotient
sequences, squeeze theorem, Cauchy sequences, subsequence of a
sequence, Bolzano-Weierstrass theorem, Cauchy convergence criterion,
brief discussion (without proof) of construction of ${\bf R}$ from
${\bf Q}$ using Cauchy sequences

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