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 other scientists, since set theory has applications in all
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 lls in part of
the proof of the equivalence relation/part
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. Show < V V .
c. Show < V V .
3. Dene rank(x) = the leas
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 xn + cn1 xn1 + + c1 x + c0 = 0.
For example, every ratio
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 and < x, y > = cfw_x, cfw_y.
Prove or disprove:
a. < x,
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. Prove x +R x = 0R .
c. Prove: if x R 0R then x = |x|.
2. Pr
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 = X and one-to-one function f : X Y .
The above stateme
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(P(P(P().
b. Express x P( x) in the language of set theo
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
make from these numbers, one of which is shown to the
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, try working
through the following problems.
First, draw
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. Prove x +R x = 0R .
c. Prove: if x R 0R then x = |x|.
2
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, y and < x, y >00 = cfw_x, cfw_y.
Prove or disprove:
a
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, Y 6= X and one-to-one function f : X Y .
The above sta
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. Find P(P(P(P().
b. Express x P( x) in the language of s
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 each ON , V is transitive.
b. Show < V V .
c. Show < V V
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. (This is part of
the proof of the equivalence relation/
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 cf() = if and only if is the smallest
cardinal such that