10. CONSTRUCTION OF Q
10
Math 361, Course Notes
Construction of Q
Most proofs are left as reading exercises.
10.1
The set Q
Denition 10.1.
(a) Z = Z \ cfw_0.
(b) Let be the binary relation dened on the set Z Z by
a, b c, d i ad = bc
Proposition 10.2. The
9. THE INTEGERS
9
Math 361, Course Notes
The Integers
Since we have succeeded in building together with its successor operation n+ , its order < , and
its arithmetical operations a, m (or + , ), we make the following convention.
Notation 9.1. N is another
8. ARITHMETIC
8
Math 361, Course Notes
Arithmetic
Now that we understand the order on , we will write < rather than < . One should bear in
mind that each natural number N is the set of all smaller natural numbers:
N = cfw_m : m < N
A familiar (informal)
7. NATURAL NUMBERS: SUCCESSOR AND ORDER
7
Math 361, Course Notes
Natural Numbers: Successor and Order
The natural numbers play three distinct mathematical roles.
Ordinal numbers are used for counting.
These are not really 1, 2, 3, . . . ; they are 1st,2n
6. ORDER RELATIONS
6
Math 361, Course Notes
Order Relations
The natural numbers have three personalities, corresponding to three stages of education:
Counting: going from one to the next.
Measurement: Comparison of size. This refers to the order (earlie
5. EQUIVALENCE RELATIONS
5
Math 361, Course Notes
Equivalence Relations
In this section and the next we discuss two kinds of relation: equivalence relations and order
relations. We rst discuss relations in general.
5.1
Binary Relations
Functions
Operation
4. THE THEORY OF FUNCTIONS
4
Math 361, Course Notes
The Theory of Functions
(Enderton: Chapter 3, pp. 4454.)
4.1
Composition
Denition 4.1. If f : A B and g : B C , the composition g f : A C is dened by
(g f )(a) = g (f (a)
Question 1. How do we prove exis
3. FUNCTIONS AND THE POWER SET AXIOM
Math 361, Course Notes
(Enderton: Chapter 3, pp. 3538 and 4244, with some stray items from Chaps. 1 and 2)
3
3.1
Functions and the Power Set Axiom
The Power set
(Enderton, Chap. 2)
The following very powerful axiom is
2. BASIC SET THEORY
2
1
Basic Set Theory
(Enderton, Chapter 2.)
2.1
Axioms of Extensionality and the Empty Set
Axiom (I: Extensionality). If the sets A, B have exactly the same members, then A = B .
Denition 2.1. The set A is a subset of the set B (writte
Introduction
Math 361, Notes
1
Introduction
Text: Enderton, Elements of Set Theory, or these notes. The treatment in the text is much fuller.
We give the essentials here.
Note: Theexercises in these notes are suggestions for the reader, but are not assign
Math 361Solutions to Review problems for Part II
Problem 1. [R]
(a) Give the denition of a Dedekind cut, <R , and +R .
(b) Show that if a set of Dedekind cuts is bounded above, then its union is a Dedekind cut.
(c) Show that the set cfw_E (q ) : q Q is de
Math 361Review problems for Part I
Problem 1. Find a pair of sets A, B which refutes all of the following statements, simultaneously.
(a) (A A) (B B ) = (A B ) (A B )
(b) P (A) P (B ) = P (A B )
(c) A B = (A B )
Solution. We take A and B disjoint and none
Math 361Final Review Problems
Checklist
Denitions and proofs in Basic Set Theory
Dedekind cuts, the order on R, and completeness
Proof of Cantors Theorem (A
P (A) and applications
Statement of Cantor-Bernstein Theorem and applications
General theory
Math 361Review problems for Part II
Problem 1. [R]
(a) Give the denition of a Dedekind cut, <R , and +R .
(b) Show that if a set of Dedekind cuts is bounded above, then its union is a Dedekind cut.
(c) Show that the set cfw_E (q ) : q Q is dense in R.
(d)
Math 361Review problems for Part I
Problem 1. Find a pair of sets A, B which refutes all of the following statements, simultaneously.
(a) (A A) (B B ) = (A B ) (A B )
(b) P (A) P (B ) = P (A B )
(c) A B = (A B )
Problem 2. For each of the statements in Pr
Math 361Final Review Problems
Review of Basic Set Theory and R
1. Denitions. Let A, B be sets.
(a) A B = cfw_x | x A or x B ; A B = cfw_x A | x B , A \ B = cfw_x A | x B ,
/
cfw_A = cfw_x | x = A
(b) A, B = cfw_A, cfw_A, B
A B = cfw_x P (A B ) | x = a, b
11. THE ORDERED SET R
11
11.1
Math 361, Course Notes
The Ordered Set R
2
Theorem 11.1. There is no q Q such that q 2 = 2.
Proof. Suppose q Q and q 2 = 2. We may suppose q > 0. Express q as a/b with a, b N
relatively prime. Our assumption is
a2
= 2;
b2
a2
12. ARITHMETIC IN R
12
12.1
Math 361, Course Notes
Arithmetic in R
Addition
Denition 12.1. We dene +R on R by
x + y = cfw_q + r| q x, r y
Question 1. Is it clear that E (q ) +R E (r) = E (q +Q r)?
We have more basic issues to deal with rst.
Lemma 12.2. I
13. DECIMALS
13
Math 361, Course Notes
Decimals
13.1
The goals
We wish to prove that real numbers may be represented by decimals. Its unlikely that well go
into this very deeply in class, but we write out the ideas fairly carefully, to serve as a referenc
Math 361Homework Set 7Solutions
End of Part 1
Exercise 1. Let r R. Show that there is a unique pair n, s with n Z and s [0, 1)
such that
r =n+s
Proof. Let Z(r) = (, r) Z. Then Z(r) is a nonempty subset of R which is bounded
above by r. By an earlier resul
Part II
Higher Set TheoryOrdinals,
Cardinals, and the
Cumulative Hierarchy
14
Cardinality
When I use a word, Humpty Dumpty said, in rather a scornful tone, it
means just what I choose it to meanneither more nor less. (Lewis Carroll,
Through the Looking Gl
25.1 Hartogs Theorem
Math 361Lecture Last p. 1
Another look at the end
This is another presentation of the material covered at the end of the course. The presentation is
more condensed, to give an overivew of the main ideas coming in at the end, in a sing
Math 361, Course Notes
Appendix: Axioms
A listing of all axioms, with the section of the course notes where they are introduced.
#
Name
Sec.
Axiom I
Extensionality
2
Axiom II
Empty Set
2
Axiom III
Union
2
Axiom IV
Subset
2
Axiom V
Pairing.
2
Axiom VI
Powe
25. THE CUMULATIVE HIERARCHY AND ORDINAL ARITHMETIC
Math 361, Course Notes
25
25.1
The Cumulative Hierarchy and Ordinal Arithmetic
Transnite Induction
In the previous section we dened a function F by transnite induction, using a choice function c:
c(A \ F
24. ZERMELOS COMPARISON THEOREM
24
Math 361, Course Notes
Zermelos Comparison Theorem
We will pay o two debts in this section:
The proof of Zermelos Comparison Theorem and the Well Ordering Principle
The precise denition of the cardinality of a set
24.1
23. ORDINALS AND THE AXIOM OF REGULARITY
23
Math 361, Course Notes
Ordinals and the Axiom of Regularity
In set theory we know that each natural number n is a particular set, that serves as a standard
representative of all nite sets of size n. Thus we writ
22. COMPARISON OF WELL ORDERED SETS
22
22.1
Math 361, Course Notes
Comparison of Well Ordered Sets
Initial Segments
Denition 22.1. Let L; < be an ordered set.
(a) A subset S L is an initial segment if for all a, b L with a < b and b S , we have a S .
(b)
21. WELL ORDERED SETS
21
Math 361, Course Notes
Well Ordered Sets
We have the following unfullled promises to deal with.
1. Dene what is meant by the cardinality of a set A (card A).
2. Prove that cardinalities are comparable: if and are cardinals, then e