MATH 300- FALL 2012-COURSE LOG
Th 8/23 Introduction/ statements, logical connectives, truth
tables
Links to results/conjectures mentioned in class:
Goedel's
incompleteness theorems
Catalan's
conjecture (Mihailescu's theorem)
Goldbach's
conjecture
Twin primes conjecture
Collatz
conjecture
HW 1 (due Tuesday 8/28, at the beginning of class)- from text, Ch.1
4, 6, 14: choose any four items
5, 9: choose any four items except for
items where an answer is given in the book
20 (use the partial answer in the back as a hint)
For Tuesday: read Ch. 1 (all)
Tu 8/28 Discussion of hw/ Quantifiers (ch. 1)
HW
set 2 (due 8/30)
Th 8/30 Problems on quantifiers
HW3 (due Tuesday 9/4)
a) Read Chapter 2 of the text
b) Do (in writing) the following problem from CH. 2 (1 problem, listed
by student initials)
TA=3 KC=6 KD=9 EH=10 KR=14 BT=20 CZ=22
c) Now look at all the problems in Ch. 2. Choose two whose statements
look interesting to you,
and think about their proofs (if the problem has a solution in the
book, you may use that as a guide).
We'll discuss these problems in class next week, and each student will
present one proof (or more) to the class.
Tu 9/4 Proof techniques-examples
Th 9/6 Proof techniques-examples
HW (due Tuesday 9/11)
1) Prove the arithmetic-geometric mean inequality for n positive real
numbers (that's the
inequality in problem 23, p.54, but with "less than or equal to"
changed to "greater than or equal to".)
Try to show it for n=3, at least.
2) Read chapter 3 (at least through section 3.4)
3) Problem from ch.3: 1 f)g)h), 6, 12
4)For discussion: think about problem 18.
Details
on HW and participation credit (addendum to syllabus)
Tu 9/11 set theory definitions
HW (due 9/13) 14, 15, 23 (in writing)
Prepare for discussion: 18, 21, 26, 28 (you may volunteer to present
the solution, and/or turn it in in writing)
Th 9/13 Set theory: proofs (examples)
Tu 9/18 Chapter 4: read sect 4.1 to 4.4
HW: 2, 3, 10 (pick 4 items)
Prepare for discussion/ presentation: 4, 5, 8, 9, 11, 13
Th 9/20 equivalence relations: discussion of problems/ partial orders
(start)
For Tu 9/25: read section 4.5 (cardinality)
HW due Tu 9/25: Ch 4: 10 (g to j; justify your answers); 11 (show this
is an order relation, decide whether it is a
partial or total ordering; justify); also:
Problem: Given a partial order
on a set X, construct from it a strict partial order (that is, an
irreflexive, transitive relation). Conversely,
given a strict partial order on X, construct a partial order on X from
it (reflexive, antisymmetric, transitive). Hint: think of how you'd define
x<y
in terms of x less than or equal to y, and vice-versa.
For discussion: 13, 22
Tu 9/25: partial orders: graph representation, minimal/smallest
element, well-ordering/ discussion of the above problem, 11 and 22
HW (for Th 9/27): 25 b)f)h) 32, 33.
for discussion: 13, 22 (again), 31.
Th 9/27: Discussion of HW problems; cardinality (start)
HW
due 10/2
Next week we'll discuss cardinality: countable and uncountable sets.
Read Sect 4.5 in the text
and part one of the handout below:
Cardinality,
countable and uncountable sets
Tu 10/2: Comments on HW problems/ Theorems and examples on cardinality
HW (due 10/4): six problems
from the above handout: 2, 3(ii) (use the result in 3(i)), 4(i)
(directly from the definitions),
5(c): include all the details
5(d) (hint:(0,2) is the union of (0,1] and (1,2) (disjoint intervals);
use 5(c)),
6 (a)
Midterm
survey
EXAM 1: Tuesday 10/9. Included: chapters 1 through 4 (emphasis on HW
problems, or those discussed in class)
Study guide for Exam 1:
Chapter 1: Converse and contrapositive (of a conditional
statement)/ Symbolic formulation, interpretation and negation of
quantified statements
Review problems: 6, 9, 10 (also: write the negation of each
statement, symbolically and in words.)
Chapter 2: Methods of proof (direct, contradiction, induction)
Review: 3, 29, 43, 44
Chapter 3: Set operations/ union and intersection of indexed families
Review: 3, 4, 21, 22 (to "disprove": give a counterexample for specific
sets)
Chapter 4: relations (domain/range), order relations (minimal/smallest
elements, total orders, well-ordering), equivalence relations
(partitions), functions as relations, injective/surjective functions,
composition and inverse. (Sect 4.5 not included)
Review: 1, 10, 25
Practice
Test
solutions
(solutions will be posted on Monday: try to solve the problems within
75 minutes.)
Tu 10/9
Exam
1
exam
1-solutions
Th 10/11: FALL BREAK
Tu 10/16: solution of exam 1/Cantor-Bernstein-Schroeder theorem
(proof)/ problems on cardinality
HW (from cardinality handout): 5(a)(b)(c) /from text (Ch. 4): 49,
51--due Th 10/18
Th 10/18 uncountable
sets
(the handout includes a summary of the lecture and 4 problems, due
Tuesday 10/23).
Tu 10/23: Solutions of HW problems/ commutative rings and fields:
definition, examples
Th 10/25: The euclidean algorithm and multiplicative inverses in Z_p (p
prime)/ the division algorithm and decimal expansions
of rational numbers/ integral domains and quotient fields (examples)
The
euclidean algorithm
(summary of lecture; includes 5 problems, due Tuesday 10/30)
Tu 10/30: ordered rings and fields/ Archimedean property/absolute value
Th 11/1: absolute value: properties, applications/definition of limit
of a sequence/ uniqueness/
Archimedean property, density of Q in R and monotone approximation (of
reals by rationals)
HW
set: sequences, absolute values
(due 11/6, four problems)
Tu 11/6: supremum, infimum of subsets of R/ supremum property of
R/implies Archimedean property
HW
set: supremum axiom
For Thursday: from this handout, problems 1 (all items) and 2(a)(b).
Also: if lim a_n=L and lim b_n=M,
where M is not zero prove that (i) b_n is not zero for n large enough;
(ii) lim a_n/b_n=L/M.
(This was discussed in class today.)
Th 11/8: Cauchy sequences and completeness
Completeness
of the real numbers
For Tuesday: (1) read this handout (emphasis: sections 1 and 5)
HW problems from this handout: Problems 1 and 10
From the handout posted 11/6: 2(c), 3(a), 4(c)(d)(e)
Tu 11/13: discussion of HW problems/ Rational roots of polynomials/
continued fractions
HW problems (due Thursday): 6, 8, 9 from the handout. (For 9, you
may assume the limit exists.)
Also: prove that the sum and the product of two Cauchy sequences are
also Cauchy sequences.
Th, 11/15: review
Second Test: Tu 11/20 Included: Lectures from 10/16 to 11/15 (and
corresponding handouts).
Practice
test 2
solutions
Exam
2
Oral
Exam
Th 11/22: THANKSGIVING
Tu 11/27 solution of Exam 2/ review
Th 11/29 Review: sets, equivalence relations, induction, real
numbers
Tu 12/4 Review: cardinality, countable vs. uncountable sets
FINAL EXAM: comprehensive (review exams 1 and 2, practice tests,
hw sets, handouts)
Tuesday 12/11, 8:00-10:00AM
Final