TUTORIAL 9: THEOREM STYLES

Transcription

1 TUTORIAL 9: THEOREM STYLES CHRISTOPHER RARIDAN Abstract. Upon completion of this tutorial, the author should be able to construct the numbered theorem styles that are typically necessary to write a mathematical paper and will be able to label and reference these constructs. In addition, the author will learn how to change the equation numbering so that the equations are numbered relative to the section in which they occur and how to change whether these numbers appear on the left- or right-hand side of the page. 1. Welcome Hello. My name is Dr. Christopher Raridan (Dr. R). I want to welcome you to the L A TEX Tutorial Series. In this tutorial, we will introduce a few of the common theorem style environments. By the end of this tutorial, the author should be able to construct the theorem styles that are typically necessary to write a mathematical paper: theorem, lemma, corollary, proposition, conjecture, definition, and example. The author also will learn how to label and reference these constructs. Part of Math 3006, Communication in Mathematics, Clayton State University. 1

2 2 C. RARIDAN In addition, the author will learn how to change the equation numbering so that the equations are numbered relative to the section in which they occur and how to change whether these numbers appear on the left- or right-hand side of the page. This is the ninth in a series of tutorials meant introduce you to L A TEX. We assume that you have successfully completed Tutorial 8 and we will use the results of this tutorial in subsequent tutorials. Hopefully, Windows and Mac users will have a similar experience. 2. Getting Started (1) Navigate to the LaTeX Tutorials folder on your computer. (2) Inside this folder, create a new folder entitled Math and Theorem Styles. (3) Open the Typesetting Mathematical Expressions folder. (4) Double-click on the Typesetting-Mathematical-Expressions.tex file. (5) In the TeXworks editor, click [Window] > Show > Tags if the Tags window is not already displayed. (6) Change the title of this article to Mathematics and the Theorem Styles with a short title of Math and Theorems. (7) Verify that your author information is correct.

3 TUTORIAL 9: THEOREM STYLES 3 (8) Click [File] > Save As... (9) Navigate to your Math and Theorem Styles folder. (10) Inside that folder, name your file Math-Theorem-Styles.tex. (11) Click [Save] to save your file. 3. Theorem Style Environments There are several environments within L A TEX that are automatically numbered. In previous tutorials, we discussed the \section{} numbered environment and how to label and reference them. We continue in that vein by introducing now the numbered environments for theorem, lemma, corollary, proposition, conjecture, definition, and example Updating the Preamble. Before we can use the L A TEX numbered environments, we must tell L A TEX how to behave with respect to them. This will require us to add some information to the preamble of our L A TEX document. (1) After \renewcommand{\baselinestretch}{1.5} hit [Enter] twice. (2) Type % THEOREM STYLES DEFINED and hit [Enter] twice. (3) Type % Each of these will numbered relative to each other and the text will be italicized. and hit [Enter]. (4) Type \theoremstyle{plain}.

4 4 C. RARIDAN (5) The command \theoremstyle{plain} tells L A TEX that the constructs that follow it will be plain, which in the amsart document class produces italicized font. (6) After \theoremstyle{plain} hit [Enter]. (7) We are now going to tell L A TEX how to format our first theorem style, which will be Theorem. (8) Type \newtheorem{thm}{theorem}[section]. (9) This first \newtheorem command has the form \newtheorem{key}{name}[relto] and tells L A TEX: The key is thm, so we type \begin{thm} to begin the theorem environment and \end{thm} to end it. If you prefer, you could set the key as theorem instead we are just abbreviating. The Name is Theorem, so L A TEX will display Theorem. The relto is section, so L A TEX will number each theorem relative to the section that it is in, i.e., Theorem 2.4 will be the fourth of the numbered theorem styles in Section 2. (10) Now, scroll to the end of the document. (11) After the displayed equation \[ \lim_{x \rightarrow a} f(x) = f(a). \], hit [Enter] twice. (12) Type \section{theorem Styles} and hit [Enter]. (13) Type \label{sec:thm-styles} and hit [Enter] twice.

5 TUTORIAL 9: THEOREM STYLES 5 (14) Type In this section, we give examples of the numbered theorem styles. and hit [Enter] twice. (15) Type \begin{thm} and hit [Enter] twice. (16) Type \end{thm}. (17) Remember, it is always a good idea to type \begin{} and \end{} in pairs. (18) Place your cursor between these two commands and type The number $\sqrt{2}$ is irrational.. (19) [Ctrl]+[T] to typeset. (20) We should see that we now have our first theorem of section 3, named and numbered appropriately as Theorem 3.1. (21) Note that the \theoremstyle{plain} command that we entered in the preamble actually typesets the text of the theorem in italics, as it should be. (22) Let s add a label: place your cursor after \begin{thm}, hit [Enter], and type \label{thm:sqrt-2-irrational}. (23) Now, let s add a few more theorem styles: Lemma, Corollary, Proposition, and Conjecture. (24) Scroll back up to the preamble and after \newtheorem{thm}{theorem}[section] hit [Enter]. (25) Type the following:

6 6 C. RARIDAN \newtheorem{lem}[thm]{lemma} \newtheorem{cor}[thm]{corollary} \newtheorem{prop}[thm]{proposition} \newtheorem{conj}[thm]{conjecture} (26) Each of these \newtheorem commands has the form: \newtheorem{key}[relto]{name}. Notice that the relto portion is in a different location and that it is thm. This tells L A TEX to follow the the convention established for the theorem environment created previously. (27) Scroll back down to the bottom of your document. (28) Place your cursor at the end of the paragraph that ends with numbered theorem styles. and hit [Enter] twice. (29) Type \begin{lem} and hit [Enter] twice. (30) Type \end{lem}. (31) Place your cursor between these two commands and type \label{lem:even-squares} and hit [Enter]. (32) Type For all integers $x$, $x^2$ is even if and only if $x$ is even.. (33) [Ctrl]+[T] to typeset. (34) We should now see that L A TEX has added this lemma before the theorem and renumbered appropriately.

7 TUTORIAL 9: THEOREM STYLES 7 (35) Note that the \theoremstyle{plain} command that we entered in the preamble actually typesets the text of this lemma in italics, as it should be. (36) In fact, each of the constructs Theorem, Lemma, Corollary, Proposition, and Conjecture will be typeset in italics. (37) We now need to prove both of these statements! (38) After \end{lem} hit [Enter] twice. (39) Type \begin{proof} and hit [Enter] twice. (40) Type \end{proof}. (41) The amsart documentclass has a built in construction for proofs, which is why we did not have to create one ourselves. (42) Place your cursor on the blank line between \begin{proof} and \end{proof}. (43) Type the following proof for the lemma: Let $x$ be an integer. First, we show that if $x$ is even, then $x^2$ is even. Let $x = 2k$, where $k$ is any integer. Then $x^2 = (2k)^2 = 4k^2 = 2(2k^2)$, which is even. Now, we show that if $x^2$ is even, then $x$ is even by proving the contrapositive. If $x$ is odd, then $x = 2j + 1$ for some integer $j$. Then $x^2 = (2j + 1)^2 = 4j^2 + 4j + 1 = 2(2j^2 + 2j) + 1$, which is odd. (44) [Ctrl]+[T] to typeset. (45) L A TEX has added the proof environment.

8 8 C. RARIDAN (46) In the amsart documentclass, proofs start with Proof. and end with an open box,. (47) Note that other documentclasses may not predefine the proof environment! (48) Notice that the open box has been right-justified (automatically), just as it should be. (49) Let s add a proof for the theorem. (50) After \end{thm} hit [Enter] twice. (51) Type \begin{proof} and hit [Enter] twice. (52) Type \end{proof}. (53) On the blank line between these two commands, type The proof relies on the result of Lemma~\ref{lem:even-squares}.. (54) [Ctrl]+[T] to typeset. (55) Of course, this is not a real proof of the theorem, as there are some details that need to be added. (56) This is just an example of how we often use a lemma, and a reminder of how to use internal labels and references. (57) Typically, a lemma is used as a helping theorem in a proof of another larger result. (58) Let s enter our last few numbered theorem styles, Definitions and Examples.

9 TUTORIAL 9: THEOREM STYLES 9 (59) Although we certainly could follow the same mold prescribed for the theorem styles we have already developed, we would like to provide an alternative. (60) Typically, we want numbered definitions and examples to appear in regular Roman font; that is, we want the text of these constructs to use the same font style as the rest of the paper instead of the italic font used for theorems, lemmas, etc. (61) Scroll back up to the preamble. (62) Place your cursor on the line that contains \newtheorem{conj}[thm]{conjecture} and hit [Enter]. (63) Type % Definitions and examples are numbered relative to theorems but the font is not italicized. and hit [Enter]. (64) Type \theoremstyle{definition}. (65) The command \theoremstyle{definition} tells L A TEX that the constructs that follow it will be typeset using the standard Roman font (i.e., not italics) of the amsart document class. (66) After \theoremstyle{definition} hit [Enter]. (67) Type \newtheorem{defn}[thm]{definition} and hit [Enter]. (68) Type \newtheorem{ex}[thm]{example}.

10 10 C. RARIDAN (69) Please note that we cannot use the abbreviation def for the definition theorem style because that command is a reserved L A TEX command. (70) Just to see what these theorem styles will look like and how they are numbered, scroll to the end of your L A TEX document. (71) Place your cursor at the end of the paragraph that ends with numbered theorem styles. and hit [Enter] twice. (72) Type \begin{ex} and hit [Enter] twice. (73) Type \end{ex}. (74) on the blank line between these two commands, type \label{ex:even-squares and hit [Enter]. (75) Type If $x$ is 2, 4, or 6, then $x^2$ is 4, 16, or 36, respectively.. (76) [Ctrl]+[T] to typeset. (77) Notice that this example is typeset in the usual upright Roman font rather than in italics, which was used for the lemma and theorem. (78) In addition, L A TEX has updated the numbering of these theorem styles appropriately. 4. Changing the way L A TEX numbers equations In this section, we examine how to change the numbering of equations so that they are numbered relative to the section in which they occur

11 TUTORIAL 9: THEOREM STYLES 11 and how to move the equation number from the left-hand side of the page to the right-hand side of the page. (1) Scroll back up to the preamble. (2) Place your cursor on the line that contains \newtheorem{ex}[thm]{example} and hit [Enter] twice. (3) Type % Equations will be numbered by section. and hit [Enter]. (4) Type \numberwithin{equation}{section}. (5) There should be only one blank line between this last L A TEX command and % END PREAMBLE. (6) The command \numberwithin{equation}{section} tells L A TEX that equations will be numbered within sections, but relative only to each other and not any other numbered construct. (7) [Ctrl]+[T] to typeset. (8) We should notice that the integral expression (on page 2) is now numbered (2.1) ; it is the first numbered equation in Section 2. (9) To move the equation numbering from the left-hand side of the page to the right-hand side, we need to add to the optional argument in the \documentclass command. (10) Scroll back to the top of your L A TEX document and locate \documentclass[12pt]{amsart}.

12 12 C. RARIDAN (11) Place your cursor after 12pt but before the right closing bracket. (12) Type,reqno. (13) That is, type a comma then reqno. (14) This optional argument tells L A TEX to place the equation numbering on the right-hand side of the page. (15) [Ctrl]+[T] to typeset. (16) You should see that equation number (2.1) has been moved accordingly. (17) This is not a required argument, but an optional one! (18) Although we have seen equations numbered on both the leftand right-hand side of the page within journal articles, we believe it looks better when they are numbered on the right. 5. Updating the Abstract Please update your previous abstract (from Tutorial 8) to reflect what you learned in this tutorial as well. Please do not add more than 2 or 3 sentences or about 50 words. Department of Mathematics, Clayton State University, Morrow, Georgia address: ChristopherRaridan@clayton.edu