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