UNIVERSITY OF WATERLOO TEST # 2 FALL TERM 2008
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UNIVERSITY OF WATERLOO FINAL EXAM FALL TERM 2009
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UNIVERSITY OF WATERLOO TEST # 2 FALL TERM 2009
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Math 127, Fall 2010 Assignment 1 solutions For solutions to the Part A problems, see the MapleTA website.
Page 1
1.
2.
Math 127, Fall 2010
Assignment 1 solutions
Page 2
3.
4*. Note that solutions to the challenge problem will not be posted. However, your

UNIVERSITY OF WATERLOO FINAL EXAM FALL TERM 2008
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Calculus 1, Chapter 3 Study Guide
Prepared by Dr. Robert Gardner
The following is a brief list of topics covered in Chapter 3 of Thomas
Calculus. Test questions will be chosen directly from the text. This list is
not meant to be comprehensive, but only gi

Lecture 12: Analytic Sets
Denition 12.1: A subset A of a Polish space X is analytic if it is empty or there
exists a continuous function f : X such that f ( ) = A.
We will later see that the analytic sets correspond to the sets denable by means
of 1 formu

Lecture 8: The Axiom of Choice
In the previous lectures, a number of regularity principles for sets of real
numbers emerged.
(PS) The perfect subset property,
(LM) Lebesgue measurability,
(BP) the Baire property.
We have seen that the Borel sets in have a

Lecture 6: Borel Sets as Clopen Sets
In this lecture we will learn that the Borel sets have the perfect subset property,
which we already saw holds for closed subsets of Polish spaces.
The proof changes the underlying topology so that all Borel sets becom

Lecture 17: Co-Analytic Sets
In the previous lecture we saw how to translate set theoretic denitions of sets
of reals into second order arithmetic. One can ask the converse question does
denability in second order arithmetic imply constructibility? We wil

Lecture 15: The Constructible Universe
A set X is rst-order denable in a set Y (from parameters) if there exists a rstorder formula (x 0 , x 1 , . . . , x n ) in the language of set theory (i.e. only using the
binary relation symbol ) such that for some a

Lecture 16: Constructible Reals
In this lecture we transfer the results about L to the projective hierarchy. The
main idea is to relate sets of reals that are dened by set theoretic formulas to
sets dened in second order arithmetic.
The effective projecti

Calculus 1, Chapter 4 Study Guide
Prepared by Dr. Robert Gardner
The following is a brief list of topics covered in Chapter 4 of Thomas
Calculus. Test questions will be chosen directly from the text. This list is
not meant to be comprehensive, but only gi

Problem Set 4: Additional questions
Math 125B: Spring 2013
1. Suppose that fn , f : [a, b] R are integrable functions and fn f as
n uniformly on [a, b]. Let
x
Fn (x) =
x
fn (t) dt,
a
F (x) =
f (t) dt.
a
Prove that Fn F uniformly on [a, b]. If fn f pointwi

Calculus 1, Chapter 2 Study Guide
Prepared by Dr. Robert Gardner
The following is a brief list of topics covered in Chapter 2 of Thomas
Calculus. Test questions will be chosen directly from the text. This list is
not meant to be comprehensive, but only gi

MAT 125B Final Exam
Last Name (PRINT):
First Name (PRINT):
Student ID #:
Instructions:
1. Do not open your test until you are told to begin.
2. Use a pen to print your name in the spaces above.
3. No notes, books, calculators, or any other electronic devi

Statistics 451 (Fall 2013)
Basic Set Theory
Let N denote the set of natural numbers, namely N = cfw_1, 2, 3, . . ..
Let Z denote the set of integers, namely Z = cfw_. . . , 3, 2, 1, 0, 1, 2, 3, . . ..
The non-negative integers are cfw_0, 1, 2, 3, . . . =

Statistics 451 (Fall 2013)
Some More Basic Set Theory
Suppose that f : Rn Rm is a function.The range of f is
cfw_y Rm : f (x) = y for some x Rn .
Note that the range of f is a subset of Rm . If A Rn , then the image of A under f is the
subset of the range

Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 13, 2013
Lecture #5: The Borel Sets of R
We will now begin investigating the second of the two claims made at the end of Lecture #3,
namely that there exists a -algebra B1 of subsets of [0, 1] on

Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 16, 2013
Lecture #6: Construction of a Probability (Part I)
As we showed in Lecture #4, when the sample space is = [0, 1], it is not possible to
construct a probability P : 2 [0, 1] satisfying bot

Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 18, 2013
Lecture #7: Proof of the Monotone Class Theorem
Our goal for today is to prove the monotone class theorem. We will then deduce an extremely
important corollary which we will ultimately us

Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 11, 2013
Lecture #4: There is no uniform probability on ([0, 1], 2[0,1] )
Our goal for today is to prove the rst of the claims made last lecture, namely that there
does not exist a uniform probabi

Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 20, 2013
Lecture #8: Independence and Conditional Probability
Denition. Let (, F , P) be a probability space. The events A, B F are said to be
independent if
P cfw_A B = P cfw_A P cfw_B .
A coll

Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 23, 2013
Lecture #9: Construction of a Probability (Part II)
Recall that we have been trying to construct a uniform probability on [0, 1]. As we saw in
Lecture #4, it is not possible to construct

Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 27, 2013
Lecture #11: Continuity of Probability (continued)
We will now apply the continuity of probability theorem to prove that the function F (x) =
P cfw_(, x], x R, dened last lecture is actua

1.6 Inverse Functions and Logarithms
1
Chapter 1. Functions
1.6. Inverse Functions and Logarithms
Denition. A function f (x) is one-to-one on a domain D if f (x1) =
f (x2) whenever x1 = x2 in D.
Note. A function = f (x) is one-to-one if and only if its gr

Calculus 1, Chapter 1 Study Guide
Prepared by Dr. Robert Gardner
The following is a brief list of topics covered in Chapter 1 of Thomas
Calculus. Test questions will be chosen directly from the text. This list is
not meant to be comprehensive, but only gi

Lecture 22: Hyperarithmetical Sets
Is there an effective counterpart to Souslins Theorem that Borel = 1 ? Den1
ability in second order arithmetic gives us the lightface classes 0 and 0 for
n
n
nite n, but what would a lightface 0 set be?
Instead of denabi

Lecture 7: Measure and Category
The Borel hierarchy classies subsets of the reals by their topological complexity.
Another approach is to classify them by size.
Filters and Ideals
The most common measure of size is, of course, cardinality. In the presence