Commit ea5244bb authored by Jim Hefferon's avatar Jim Hefferon

revision of first chapter after spring 2018 teaching

parent 21ef9f74
No preview for this file type
No preview for this file type
No preview for this file type
No preview for this file type
No preview for this file type
No preview for this file type
No preview for this file type
No preview for this file type
No preview for this file type
No preview for this file type
No preview for this file type
No preview for this file type
No preview for this file type
No preview for this file type
......@@ -38,7 +38,7 @@
\end{center}
\vspace*{\fill}
\begin{center}
{\large\textsc{Greek letters with pronounciation}}
{\large\textsc{Greek letters, with pronounciation}}
\\[3ex]
\newcommand{\pronounced}[1]{\hspace*{.2em}\small\textit{#1}}
\begin{tabular}{cl@{\hspace*{3em}}cl}
......
This diff is collapsed.
No preview for this file type
\chapter*{Preface}
This is a course in mathematical proof.
It is for math majors, typically sophomores in the US, although since
its only prerequisite is high school mathematics
it can be used with first year students.
It is for math majors, typically sophomores.
Its only prerequisite is high school mathematics.
......@@ -11,19 +10,19 @@ it can be used with first year students.
\noindent\textsc{Approach.}
This course is inquiry-based (sometimes called Moore method
or discovery method).
This text is a sequence of exercises,
This text is a sequence of exercises
along with definitions and a few remarks.
Students work through the material together by
proving statements or by providing examples or counterexamples.
proving statements or providing examples.
This makes each person grapple directly with the
mathematics\Dash the instructor only
lightly guides, while the students pledge not to use outside sources\Dash
lightly guides and the students pledge not to use outside sources\Dash
talking out misunderstandings,
sometimes stumbling in the dark, and sometimes
having beautiful flashes of insight.
For these students, with this material,
this is the best way to develop mathematical maturity.
Besides, it is a lot of fun.
Besides, it is a great deal of fun.
\medskip
......@@ -32,8 +31,8 @@ We cover sets, functions and relations, and elementary number theory.
We start with number theory instead of sets
for the same reason
that the baseball team's annual practice starts with tossing the ball and
not with reading the rulebook.
that the baseball team's annual practice starts by tossing the ball and
not by reading the rulebook.
Math majors take readily to proving things about
divisibility and primes,
whereas weeks of preliminary material is less of a lure.
......@@ -50,17 +49,17 @@ intellectual habits that we established at the start.
\noindent\textsc{Exercises.}
As much as the material allows,
nearby exercises have about the same difficulty.
This standard gradually rises.
This gradually rises.
Some exercises have multiple items; these come in two types.
If the items are labeled \textsc{A}, \textsc{B}, etc.,
then each one is hard enough to be a separate assignment.
If the labels are (i), (ii), etc., then they together make
a single assignment.
I have students put proposed solutions on the board
for the group to discuss and
I have students put proposed answers on the board
for discussion, and
if the items are labelled alphabetically then I ask a different student
to do each one, while for the others I ask a single student to do them all.
to do each one while for the others I ask a single student to do them all.
% This text comes in versions that differ in the number of exercises,
% so it is adoptable for courses with different needs.
......@@ -76,12 +75,41 @@ This book is Free; see \url{http://joshua.smcvt.edu/proofs}.
That site has other material related to this text, including
its \LaTeX{} source.
\ifbool{jiblm}{%
\medskip
\noindent\textsc{For students.}
This course asks you to write
proofs.
We start with simple statements, things that you already
know.
You will ask yourself:~if I can't assume this, what can I
assume?
For one thing, you can always use a prior result.
Also,
you can use the rules of high school algebra such as associativity
of addition $x+(y+z)=(x+y)+z$, or distributivity of multiplication
over addtion $x\cdot(y+z)=xy+xz$, or that a positive times a positive
equals a positive.
Finally, you can use elementary
logic, such as that a statement like ``$P$ and~$Q$'' is true if and only if
both halves are true.
(For the logic, perhaps your instructor will
go over some of this with you.
Certainly you'll get better at it as
you move through the course.
But in any event, the proofs at the
start require only logic that is obvious to people with a
mathematical turn of mind.)
}{}
\vspace*{.1in}
\vspace{\fill}
\noindent\parbox{.95\textwidth}{\textit{The most important thing [is that] proving things in math [i]s a skill like any other that you get good at through practice.}\hspace{1.5em}---Cathy O'Neil} % mathbabe blog
\vspace{.1in}
\noindent\parbox{.95\textwidth}{\raggedright\textit{At the first meeting of the class Moore would define the basic terms and either challenge the class to discover the relations among them, or, depending on the subject, the level, and the students, explicitly state a theorem, or two, or three. Class dismissed. Next meeting: "Mr Smith, please prove Theorem 1. Oh, you can't? Very well, Mr Jones, you? No? Mr Robinson? No? Well, let's skip Theorem 1 and come back to it later. How about Theorem 2, Mr Smith?" Someone almost always could do something. If not, class dismissed. It didn't take the class long to discover that Moore really meant it, and presently the students would be proving theorems and watching the proofs of others with the eyes of eagles.}\hspace{1.5em}---Paul Halmos}
%% \vspace{.1in}
%% \noindent\parbox{.95\textwidth}{\raggedright\textit{At the first meeting of the class Moore would define the basic terms and either challenge the class to discover the relations among them, or, depending on the subject, the level, and the students, explicitly state a theorem, or two, or three. Class dismissed. Next meeting: "Mr Smith, please prove Theorem 1. Oh, you can't? Very well, Mr Jones, you? No? Mr Robinson? No? Well, let's skip Theorem 1 and come back to it later. How about Theorem 2, Mr Smith?" Someone almost always could do something. If not, class dismissed. It didn't take the class long to discover that Moore really meant it, and presently the students would be proving theorems and watching the proofs of others with the eyes of eagles.}\hspace{1.5em}---Paul Halmos}
\vspace{.1in}
\noindent\parbox{.95\textwidth}{\textit{It's a kind of art that may change lives.}\hspace{1.5em}---Peter Schjeldahl} % , \textit{New Yorker}
......
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment