Solutions to Elements of Set Theory
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January 14, 2011
Contents
1 Introduction
1
2 Axioms and Operations
3
3 Relations and Functions
3.1 Ordered Pairs . . . . . . . .
3.2 Relations
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 th
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 ma
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 nat
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
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.
Measuremen
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 relatio
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 b
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
(E
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 .
D
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 n
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 Ded
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
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 Can
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
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
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)
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,
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
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
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 subse
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 meanneithe
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 ove
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
Uni
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
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
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
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
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 car