## Anki Spaced Repetition

Posted in Uncategorized by pgadey on 2016/11/03

If you’re looking for spaced repetition software, check out Anki. It is a great program.

There are other good programs that do spaced repetition.

## Extra Office Hours

Posted in Uncategorized by pgadey on 2016/10/13

Happy Thanksgiving!

There will be additional office hours from MAT A29 and MAT A31 on Friday Oct 14th from 11am to 2pm in IC 481.
Feel free to drop by for a cookie and some math!

## Running Late

Posted in Uncategorized by pgadey on 2016/09/27

Hi — MAT A29,

I am running late this morning. Please expect a short delay.

## The Epsilon-Delta Story and Examples

Posted in 2016 -- MATA31H3F, Lecture Notes, Uncategorized by pgadey on 2016/09/21

Epsilon-delta proofs are the cornerstone of rigorous mathematical analysis.

• Examples of epsilon-delta proofs: contains some proofs with epsilons and deltas.
• The Delta-Epsilon Story: This is extra material to illustrate the purpose of epsilon-delta proofs. It’s a fictitious dialogue between Alice and Bob, two students in MAT A31, who disagree about the value of $\lim_{x \rightarrow 3} (2x + 3)$.

[Graphic by User:HiTeOwn work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=2314281]

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## Office Hours!

This week we had great attendance at office hours. We solved some questions on the white board in my office; the photos are available below the cut.

To make things more convenient:

Office hours are now 11:30-12:30 on Tuesday and Thursday, and 11-1pm on Wednesday.

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## Trigonometry Review

Posted in 2016 -- MATA29H3F, 2016 -- MATA31H3F, Uncategorized by pgadey on 2016/09/15

Some notes that I prepared about trigonometry are available here.

The notes sketch things like $y = 2 \sin\left( \frac{1}{3}x \right) + 4$ and solve equations like $\sqrt{2} \cos( 3\theta) = 1$.

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## MAT 237 — Week 3a

Posted in 2016 -- MAT 237, Lecture Notes, Uncategorized by pgadey on 2016/06/01

This blog post contains my notes for MAT 237. It includes links to the peer instruction slides, practice problems, and some additional references for the curious. Blackboard photos are below the cut. Peer Instruction questions and problem solving work sheets are available two days after being shown in class.

• The peer instructions questions for this lecture are available here.
• The problem solving worksheets for this lecture are available here.

## Question 4 and Question 6

Posted in Uncategorized by pgadey on 2016/02/12

These questions kept coming up during my MAT A33 office hours. The notes below are an extended set of hints for these questions.

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## The Hardware and Software of Theoretical Machines

Posted in Uncategorized by pgadey on 2015/08/01

The Hardware and Software of Theoretical Machines

(This blog entry is a written summary of an pair of talks given at Canada Math Camp on 2015-07-28 and 2015-07-30. The official notes are on pgadey.ca.)

Computers are all around us. This fact is so self-evident that writing it down, on a computer no less, is slightly embarassing. And yet, very few people ever talk about exactly what a computer is. Whenever it comes up one feels like St. Augustine, who said: “What is a computer? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know.”

This talk is an informal introduction to the formalizations of computation adopted by Alonzo Church and Alan Turing, the two Great Definers of Computation. We’ll also say something about Charles Babbage because he too was awesome.

## Basic Combinatorics.

Posted in Uncategorized by pgadey on 2014/07/27

First we recall a little bit of terminology:

1.1. Sets and functions

A set is a collection of elements . We write a set by surrounding its list elements with curly braces. For example: ${X = \{1,2,3\}}$, ${Y = \{\heartsuit, \clubsuit, \star\}}$. We also use set constructor notation ${Y = \{x : P(x)\}}$ where ${P(x)}$ is some statement about ${x}$ that can be true or false. For example: ${X = \{n : n\text{ is even}\}}$, ${Z = \{n : n\text{ is prime}\}}$. We write: ${\{\} = \emptyset}$, ${{\mathbb N}}$ for the set of natural numbers, ${{\mathbb Q}}$ for the set of rational numbers, ${{\mathbb Z}}$ for the set of integers.

We write ${x \in X}$ to mean that ${x}$ is in the set ${X}$. We write ${X \cup Y = \{x : x \in X \text{ or } x \in Y\}}$. We write ${X \cap Y = \{x : x \in X \text{ and } x \in Y\}}$. We write ${X \sqcup Y}$ for ${X \cup Y}$ if ${X \cap Y = \emptyset}$. If ${X \cap Y = \emptyset}$ then we say that ${X}$ and ${Y}$ are disjoint sets.

We write ${X \times Y = \{(x,y) : x \in X,\ y \in Y\}}$ for the set of ordered pairs of elements.

Definition 1 A function ${f : X \rightarrow Y}$ is injective (one to one) if: ${x \neq y}$ implies ${f(x) \neq f(y)}$. A function is surjective (onto) if: for all ${y \in Y}$ there is ${x \in X}$ such that ${f(x) = y}$. A function is bijective if: for all ${y \in Y}$ there is ${x \in X}$ such that ${f(x) = y}$. The number of elements in a set ${X}$ is written ${|X|}$.

1.2. Basic formulae

The basic facts of combinatorics are very simple.

1. If ${k < n}$ then there is no injective function from a set with ${n}$ elements to a set with ${k}$ elements. (This is called pigeon hole principle.)
2. If there is a bijective function from ${X}$ to ${Y}$ then ${|X| = |Y|}$.
3. If ${X}$ and ${Y}$ are disjoint then ${|X \cup Y| = |X| + |Y| - |X \cap Y|}$.
4. If ${X}$ and ${Y}$ are disjoint then ${|X \sqcup Y| = |X| + |Y|}$.
5. ${|X \times Y| = |X| \cdot |Y|}$.
6. The number of ${k}$ element subsets of a set with ${n}$ elements is: ${\binom{n}{k} = \frac{ n! }{ (n-k)! k! }}$. (Why is this an integer? Prove it.)
7. The number of functions from an ${n}$-element set to a ${k}$ element set is ${k^n}$.
8. If ${|X| = n}$ then the number of bijective functions from ${X}$ to ${X}$ (permutations of ${X}$) is ${n!}$.

There are a couple formulae that are handy to remember:

1. There are ${2^n}$ subset of ${\{1, \dots, n\}}$.
2. Suppose you have ${n_1, \dots, n_k}$ objects of types 1, 2, ${\dots}$, ${k}$ respectively. The number of ways of arranging all the objects is: ${{ (n_1 + \dots + n_k)! }/{ n_1! n_2! \dots n_k! }}$
3. Suppose you have ${n}$ identical objects that you want to distribute among ${k}$ containers. The number of ways to do this is: ${\binom{n+k-1}{k-1}}$. (Why?)

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