Course Notes for Math 161
Set Theory
Winter 2017
Rick Sommer
0
Introduction
What is Set Theory?
Four viewpoints:
1. Set Theory is a tool for mathematicians, computer scientists, statisticians, and oth
Math 161
Fall 2014
Problem Set # 3 (due October 16, 2014)
1. Suppose is an equivalence relation on A and F : A A.
a. Prove [x] = [y] x y.
b. Use a to show distinct elements of A/ are disjoint. This ll
Math 161
Fall 2014
Problem Set # 6 (due November 13, 2014)
1. Prove:
R well orders A ON (< A, R > , >)
=<
2. Recall: V0 = , V + = P(V ), and V =
V if LIM .
<
a. Show for each ON , V is transitive.
b.
Math 161
Fall 2014
Problem Set # 7 (due November 20, 2014)
1. Prove:
a. |< 2| = .
b. |< | = .
c. | R| = |R|.
d. |R R| = |R|.
2. A real number, x, is algebraic if for some n and c0 , . . . , cn Q,
cn x
Math 161
Fall 2014
Problem Set # 2 (due October 9, 2014)
1.
a. Find < , , , >.
b. Find A, B, C such that (A B) C = A (B C).
c. Prove: (A B) C = (A C) (B C).
def
def
2. Let < x, y > = cfw_x, cfw_x, y a
Math 161
Fall 2014
Problem Set # 5 (due November 7, 2014)
1. For x R (i.e., x is a Dedekind cut), let
x = cfw_r Q : s Q(s x r < s)
/
and let
|x| = x x
a. Prove x R (i.e., x is a Dedekind cut).
b. Prov
Math 161
Fall 2014
Problem Set # 4 (due October 23, 2014)
1. Prove that the following are equivalent (i.e. a implies b, and b implies a).
a. There is a one-to-one function f : X.
b. There is a Y X, Y
Math 161
Fall 2014
Problem Set # 1 (due October 2, 2014)
1. Express the following in the language of set theory (without dened symbols).
a. w [(x y) (y x)] [x y].
b. w x y z.
c. = cfw_x.
2.
a. Find P(
CS106B
Winter 2017
Handout #20S
February 27, 2017
Section Solutions 7
_
Problem One: Binary Search Tree Warmup!
There are several trees that are tied for the tallest possible binary search tree we can
CS106B
Winter 2017
Handout #20
February 27, 2017
Section Handout 7
_
Problem One: Binary Search Tree Warmup!
Binary search trees have a ton of uses and fun properties. To get you warmed up with them,
Math 161
Winter 2017
Problem Set # 5 (due February 24, 2017)
1. For x R (i.e., x is a Dedekind cut), let
x = cfw_r Q : s Q(r < s s
/ x)
and let
|x| = x x
a. Prove x R (i.e., x is a Dedekind cut).
b.
Math 161
Winter 2017
Problem Set # 2 (due January 27, 2017)
1.
a. Find < , , , >.
b. Find A, B, C such that (A B) C 6= A (B C).
c. Prove: (A B) C = (A C) (B C).
def
def
2. Let < x, y >0 = cfw_x, cfw_x
Math 161
Winter 2017
Problem Set # 4 (due February 10, 2017)
1. Prove that the following are equivalent (i.e. a implies b, and b implies a).
a. There is a one-to-one function f : X.
b. There is a Y X,
Math 161
Winter 2017
Problem Set # 1 (due January 20, 2017)
1. Express the following in the language of set theory (without defined symbols).
a. w [(x y) (y x)] [x y].
b. w x y z.
c. =
6 cfw_x.
2.
a.
Math 161
Winter 2017
Problem Set # 6 (due Tuesday, March 7)
Please note the due date!
1. Prove:
R well orders A ON (< A, R >
=< , >)
S
V if LIM .
2. Recall: V0 = , V + = P(V ), and V =
<
a. Show for e
Math 161
Winter 2017
Problem Set # 3 (due February 3, 2017)
1. Suppose is an equivalence relation on A and F : A A.
a. Prove [x] = [y] x y.
b. Use part a to show distinct elements of A/ are disjoint.
Math 161
Fall 2014
Problem Set # 8 (due December 5, 2014)
1. Prove that if is an innite singular cardinal then is the union of < many
sets of cardinality < .
2. Prove that if is a limit ordinal then c