Teaching info.

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Spring 2010 Teaching.

Math 4730: Set Theory

The first exam has been moved to Friday February 26.

Office hours	M 8:00-8:50, W 12:00-12:50, F 10:00-10:50
Class times	MWF 9:00-9:50
Class location	ECCR 118
Book	Notes handed out in class

Lectures

Date
Monday January 11	Introduction. Basic ideas about logic: truth tables and standard arguments (section introduction)
Wednesday January 13	The first axioms: meaning and some use. (section 1) the key idea about axioms is that they don't describe what a set is, but what a universe of sets is.
Friday January 15	Axioms, defined properties, and how to write a proof (section 1) the key idea in writing a proof is often using the standard methods to get down to the hart of the matter, and then reasoning very carefully.
Monday January 18	No class: MLK day.
Wednesday January 20	Ordered pairs and relations (section 2) The key point was the idea that notions that a priori are not about sets, can be "implemented" in sets. They should then have the main properties of the original notions, but will often also "accidentally" have more properties.
Friday January 22	Functions (section 3, a small start) We discussed the reason for developing the axioms the way we are, not too strong, not too weak. We also discussed what the axioms are describing.
Monday January 25	Functions (section 3) Worked some of the theorems in the section emphasizing the role of definitions, that they are not randomly chosen, but need to fit our intuitions. Side effects are to be expected though.
Wednesday January 27	Functions (section 3) The axiom of choice was introduced and explained.
Friday January 29	replacement procedure (section 3) Equivalence relations and partitions (section 4)
Monday February 1	Finishing equivalence relations (section 4) Pictures of relations: easy way of obtaining examples. Definition and (non-)examples of wellorders (section 5)
Wednesday February 3	Induction: for any well order (section 5) and on N defining N (section 6)
Friday February 5	Properties of N (section 6): Induction and some basic properties of our implementation.
Monday February 8	Doing inductive proofs The Recursion Theorem
Wednesday February 10	Proof of the recursion theorem
Friday February 12	Defining addition. Properties of addition.
Mon Feb 15-Feb19	Essentially finish Chapter 6
Monday February 22	On the proof of the closure theorem.
Wednesday February 24	On proving existence choice functions from AC.
Friday February 26	First midterm
Monday March 1	The ideas surrounding ordinals and cardinals.
Wednesday March 2	Discussed the exercise on the recursion theorem in great detail.
Friday March 5	Definition cardinals and some properties (Chapter 7)
Monday March 8	Properties of cardinals (Chapter 7)

Homeworks

Due

Assignment

Friday January 22

Chapter 1: 1, 2, 5, 6, 7, 11

Friday January 29

Chapter 1: 13

Chapter 2: 1, 2, 6, 7

Chapter 3: 1

Friday February 5

Chapter 1: 12

Chapter 2: 14

Chapter 3: 6, 15, 16, 21

Friday February 12

Chapter 3: 22, 26

Chapter 4: 1, 2

Chapter 5: 1

Friday February 19

Chapter 4: 4, 8

Chapter 5: 2, 6

Chapter 6: 2 (only do (i), (ii), and (v))

Friday March 5

Chapter 5: 5, 8

Chapter 6: 7, 13, 19

Friday March 12

Chapter 3: 24

Chapter 4: 7

Chapter 6: 12, 14, 21, 23

Subjects for Basic Notions Quiz Mar 15

Truth tables for basic connectives (and/or/if X, then Y/not/iff).
Form of argument by contradiction.
Form of argument by contrapositive.
Form of standard argument to prove there exists x. P(x) .
Form of standard argument to prove for all x. P(x) .
Form of standard argument to prove A => B .
The axioms (up to and including the 8th).
Definition of union of sets (pairs and families).
Definition of intersection of sets (pairs and families).
Definition of {x in A | P(x)} .
Definition of {a,b} .
Definition of {a} .
Definition of the empty set .
Definition of disjoint.
Standard argument for use of disjunction.
Standard argument for equality of sets.
Definition of (a,b) , and its key property.
Definition of Domain.
Definition of Range.
Definition of (binary) relation.
Definition of inverse relation.
Definition of image of a set under a relation.
Definition of function.
Definition of surjection (onto function).
Definition of injection (one-to-one function).
Definition of cartesian product of a system of sets.
Definition of equivalence relation.
Definition of partition.
Definitions of the different properties of orders.
Definition of the principle of complete induction.
Definition of the natural numbers .
Definition of addition, mutiplication, and exponentiation of
Definition of equipotent.
Definition of finite.
Definition of F -closed set.
Definition of {Cl}_F(X) .
Definition of ordinal and cardinal.
The second form of the axiom of choice.
Definition of denumerable and countable.

pdf file is also available.