Saturday May 10, 2008 6:15 am Lethbridge Alberta

This page last updated on: Sunday, May 11, 2008 7:43 AM

It is - 2 C with a high forecast of +14 C. Sunrise 5:53 Sunset 21:02 Hours of daylight: 15:09

A. Morning Musings

I had a good night's sleep and am looking forward to the day. Having a cup of coffee beside me is a good start.

Yesterday was not a good day. I burned a pot of barley soup while on the phone and then began snacking and gained a couple of pounds. I also had difficulty getting my head around doing some elementary mathematics. Today will be different.

I have one chore scheduled for today - to take a load of refuse out to the local dump. I will do that at about 10 this morning after it warms up a little. But my main goal is to do some serious mathematics.

Learning Category	Planned Activities for Today	Time
Literature	Begin morning with a Rumi reading
Literature	Continue reading "Late Nights on Air" by Elizabeth Hay	1 hr
Mathematics	Make notes on symmetry - 11	2 hr
Puzzles & Games	New York Times crossword puzzles	1 hr

B. Actual Learning Activities

6:30 am

Notes on Symmetry - 11

Dale Burnett

The goal today will be to review all of the books I am using and make some substantial progress with the preliminary ideas involving sets and functions.

The idea will be to review the material in the first chapter of Burn and then see what additional information I can find on the same topic in my other books. My main difficulty when I first looked at this material was the terminology: words like bijection, injection, surjection, onto, not onto, and codomain appeared to me to be unecessary and pedantic. But I now realize they each denote an important idea.

The first consequence of this strategy is to add two more books to the table below that identifies the books and the chapters that I am using. I now have 7 books, all of which I would categorize as textbooks, that focus on developing the skills to actually "do" mathematics.

Date	Mathematics	History
500 BCE	Pythagorus
399 BCE	Thaetetus classifies the 5 regular Platonic solids in 3 dimensions: tetrahedron, cube, octahedron, dodecahedron, icosahedron.
1048 - 1131	Omar Khayyam finds geometric method for solving cubic equations.
1200	Leonardo Fibonacci wtote the first original book on mathematics published in Europe. It introduced Hindu-Arabic numerals and place-value notation.
1439		Gutenberg invents the printing press
1452 - 1519		Leonardo da Vinci
1492		Columbus discovers America
early 1500s	del Ferro, Tartaglia, Cardano, Ferrari solve cubic & quartic equations
1564 - 1642	Galileo
1642 - 1727	Isaac Newton
1775 - 1783		American War of Independence
1777 - 1855	Carl Friedrich Gauss
1789 - 1799		French Revolution
1802 - 1829	Niels Henrik Abel proves that no formula exists for equations of degree 5.
1832	Evariste Galois dies at age 20.
1842 - 1899	Sophus Lie: Norwegian group theorist

Here is my chart of symmetry readings: Each cell will corresponds to a chapter. Yellow indicates the number of chapters in the book, green indicates that I have read and made notes on the chapter. Grey indicates that I used part of the chapter. The background colors of the book titles are purple for mathematics books where one actually does the mathematics and light blue indicated books that describe what others have done. In an ideal world on each day I should be able to add at least one cell to what I have read.

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
Symmetry
Fearless Symmetry
Algebra
Abstract Algebra
Creating Escher-type Drawings
Handbook of Regular Patterns
Symmetry & the Monster
The Celtic Design Book
Groups & Symmetry
Groups: A Path to Geometry
A Transition to Advanced Mathematics
Modern Abstract Algebra

I want to begin with the Burn book "Groups: A Path to Geometry". Here are the terms he uses in the first chapter:

function
cartesian product
graph
injection (one - one function)
surjection (onto)
bijection
codomain
composition
inverse function

I just realized that the chapter ends with a concise summary of both the important definitions as well as a collection of Theorems that were developed in the chapter. These theorems are important results that can then be used as "facts" in further explorations.

But one needs a firm understanding of the original definitions in order to appreciate the meaning of the Theorems.

Let me try to write definitions for each of the above terms:

function	a rule (sometimes called a mapping) from A to B that assigns a unique element in a set B to each element in a set A.
cartesian product	the set of all ordered pairs of elements from two sets, A and B.
graph	the set of all points in the cartesian product that satisfies a particular function
domain	the set of elements in the first set of a function
codomain	the set of possible elements for the second set
range	the set of elements in the second set that are mapped by the function
injection	a one-one function. If f(x) = f(y) then x = y.
surjection	an onto function. There is always an element x for every element y
bijection	an injection which is also an surjection
composition

In addition to having a very clear understanding of each of these terms (concepts) it is also necessary to have a very clear understanding of the notational conventions for expressing these ideas. The book "A Transition to Advanced Mathematics" (2006) by Douglas Smith & Maurice Eggen was particularly useful (e.g. chapter 4 on Functions).

I have now had a closer look at chapter 4 of Smith & Eggen. This provides a much more detailed and extensive treatment of the concept of function.

This leads to the question of whether I should now read the first three chapters of Smith & Eggen.

After having a close look at Smith & Eggan, I think that it makes sense to ignore the first two chapters as being too far from my area of interest, but that both chapter 3 (on Relations) and chapter 4 (on Functions) deserve very close study. Chapter 3 introduces the concept of a digraph (directed graph) for both relations and functions. This will be a new topic for me.

Now to begin chapter 3.

A Transition to Advanced Mathematics. Chap. 3 Relations

"In this chapter we will study the idea of 'is related to' by making precise the notion of a relation and concentrating on certain relations called equivalence relations. We finish the chapter with another type of relation, orderings." [p. 131]
"Definition: Let A and B be sets. The set of all ordered pairs having the first coordinate in A and second coordinate in B is called the Cartesian product of A and B and is written A x B. Thus

A x B = {(a, b): a e A and b e B}." [p. 132]
"If A has m elements and B has n elements, then A x B has mn elements." [p. 132]
"A relation is simply a set of ordered pairs." [p. 133]
"Definition: Let A and B be sets. R is a relation from A to B iff R is a subset of A x B." [p. 133]
There are several ways to present a relation in a usable form:
- list all the ordered pairs
- construct a table where the first column lists the first coordinates and the second column lists the second coordinaates
- graph the relationship by showing
  - each point on a rectangular coordinate system
  - shade an area on a rectangular coordinate system (where the areas are defined by various inequalities)
  - a line on a rectangular coordinate system (where the relation is specified by a function).
If the relation R is on a set A, then one can draw a digraph (directed graph) where each element of A is considered as a vertex and R tells us which elements are connected by edges (i.e. lines).
In general, there are many different relations from a set A to a set B." [p. 135]
"Every set of ordered pairs is a relation." [p. 136]
"A function is a relation that satisfies a special property." [p. 138]