Commit ea5244bb by Jim Hefferon

### revision of first chapter after spring 2018 teaching

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 ... ... @@ -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} ... ...
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 \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} ... ...
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