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<!--* on 2016-03-05T09:10:50-05:00 *-->
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<title>UML-MBX Definitions, referencing, and exercises</title><meta name="Keywords" content="Authored in MathBook XML"></meta><meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=0, minimum-scale=1.0, maximum-scale=1.0"></meta><script xmlns:b64="https://github.com/ilyakharlamov/xslt_base64" type="text/x-mathjax-config">
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</script><script xmlns:b64="https://github.com/ilyakharlamov/xslt_base64" type="text/javascript" src="https://code.jquery.com/jquery-latest.min.js"></script><link xmlns:b64="https://github.com/ilyakharlamov/xslt_base64" href="https://aimath.org/knowlstyle.css" rel="stylesheet" type="text/css"></link><script xmlns:b64="https://github.com/ilyakharlamov/xslt_base64" type="text/javascript" src="https://aimath.org/knowl.js"></script><script xmlns:b64="https://github.com/ilyakharlamov/xslt_base64" src="https://aimath.org/mathbook/js/lib/jquery.sticky.js"></script><script xmlns:b64="https://github.com/ilyakharlamov/xslt_base64" src="https://aimath.org/mathbook/js/lib/jquery.espy.min.js"></script><script xmlns:b64="https://github.com/ilyakharlamov/xslt_base64" src="https://aimath.org/mathbook/js/Mathbook.js"></script><link xmlns:b64="https://github.com/ilyakharlamov/xslt_base64" href="https://fonts.googleapis.com/css?family=Open+Sans:400,400italic,600,600italic|Source+Code+Pro:400" rel="stylesheet" type="text/css"></link><link xmlns:b64="https://github.com/ilyakharlamov/xslt_base64" href="https://aimath.org/mathbook/stylesheets/mathbook-3.css" rel="stylesheet" type="text/css"></link><link xmlns:b64="https://github.com/ilyakharlamov/xslt_base64" href="https://aimath.org/mathbook/mathbook-add-on.css" rel="stylesheet" type="text/css"></link></head><body class="mathbook-book has-toc has-sidebar-left"><div xmlns:b64="https://github.com/ilyakharlamov/xslt_base64" style="display:none;">\(\newcommand{\identity}{\mathrm{id}}
\newcommand{\notdivide}{{\not{\mid}}}
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\newcommand{\amp}{ & }\)</div><header id="masthead"><div class="banner"><div class="container"><a id="logo-link" href="http://faculty.uml.edu/klevasseur/uml-mbx/" target="_blank"><img src="http://faculty.uml.edu/klevasseur/images/small_umllogo.gif"></img></a><div class="title-container"><h1 class="heading"><span class="title">UMass Lowell Mathbook XML</span><span class="subtitle">a guide to creating open source STEM materials.</span></h1><p class="byline">Ken Levasseur</p></div></div></div><nav xmlns:b64="https://github.com/ilyakharlamov/xslt_base64" id="primary-navbar"><div class="container"><div class="navbar-top-buttons"><button class="sidebar-left-toggle-button button active">Contents</button><div class="tree-nav toolbar toolbar-divisor-3"><a class="previous-button toolbar-item button" href="features.html">Previous</a><a class="up-button button toolbar-item" href="features.html">Up</a><a class="next-button button toolbar-item" href="theorem.html">Next</a></div><button class="sidebar-right-toggle-button button active">Annotations</button></div><div class="navbar-bottom-buttons toolbar toolbar-divisor-4"><button class="sidebar-left-toggle-button button toolbar-item active">Contents</button><a class="previous-button toolbar-item button" href="features.html">Previous</a><a class="up-button button toolbar-item" href="features.html">Up</a><a class="next-button button toolbar-item" href="theorem.html">Next</a></div></div></nav></header><div class="page"><aside xmlns:b64="https://github.com/ilyakharlamov/xslt_base64" id="sidebar-left" class="sidebar"><div class="sidebar-content"><nav id="toc"><h2 class="link"><a href="index.html"><span class="title">Front Matter</span></a></h2><ul><li><a href="colophon-1.html">Colophon</a></li><li><a href="acknowledgement-1.html">Acknowledgements</a></li><li><a href="preface-1.html">Preface</a></li></ul><h2 class="link"><a href="overview.html"><span class="codenumber">1</span><span class="title">Overview</span></a></h2><ul><li><a href="mbx-structure.html">Structure of an MBX book</a></li><li><a href="html-conversion.html">The Conversion Process to html</a></li><li><a href="latex-conversion.html">The Conversion Process to LaTeX</a></li></ul><h2 class="link"><a href="chapter-2.html"><span class="codenumber">2</span><span class="title">LaTeX</span></a></h2><ul><li><a href="latex.html">LaTeX</a></li></ul><h2 class="link active"><a href="features.html"><span class="codenumber">3</span><span class="title">MBX Features</span></a></h2><ul><li><a href="definition-xref-exercise.html" class="active">Definitions, referencing, and exercises</a></li><li><a href="theorem.html">Theorems</a></li></ul><h2 class="link"><a href="chapter-4.html"><span class="codenumber">4</span><span class="title">Sage</span></a></h2><ul><li><a href="sage_cell.html">Sage Cell Calculations</a></li><li><a href="sageplot.html">Using sageplot to create images</a></li></ul><h2 class="link"><a href="backmatter.html"><span class="title">Reference</span></a></h2><ul><li><a href="chapter-5.html">Hints and Solutions to Selected Exercises</a></li><li><a href="chapter-6.html">Notation</a></li><li><a href="chapter-7.html">Lists of Elements</a></li></ul></nav><div class="extras"><nav><a class="mathbook-link" href="https://mathbook.pugetsound.edu">Authored in MathBook XML</a><a href="https://www.mathjax.org"><img title="Powered by MathJax" src="https://cdn.mathjax.org/mathjax/badge/badge.gif" border="0" alt="Powered by MathJax"></img></a></nav></div></div></aside><main class="main"><div id="content" class="mathbook-content"><section xmlns:b64="https://github.com/ilyakharlamov/xslt_base64" class="section" id="definition-xref-exercise"><header><h1 class="heading" alt="Section 3.1 Definitions, referencing, and exercises" title="Section 3.1 Definitions, referencing, and exercises"><span class="type">Section</span><span class="codenumber">3.1</span><span class="title">Definitions, referencing, and exercises</span><a href="definition-xref-exercise.html" class="permalink">¶ permalink</a></h1></header><article class="introduction" id="introduction-3"><h5 class="heading"></h5><p>There are three features of Mathbook XML highlighted in this chapter.
<ol style="list-style-type: decimal;"><li id="li-24">Definitions - we make a basic definition that includes an xml id for easy referencing.</li><li id="li-25">Exercises - including hints and solutions.</li><li id="li-26">Cross-referencing - making use of an xml id to create a knowl that reminds the reader of a definition that is made in a different part of the document.</li></ol>
</p></article><section class="subsection" id="subsection-1"><header><h1 class="heading hide-type"><span class="type">Subsection</span><span class="codenumber">3.1.1</span><span class="title">A definition and exercise</span></h1></header><p> This is the general form of a definition, with the notation section optional:<tt class="code-inline"><ul style="list-style-type: disc;"><li id="li-27"><definition xml:id="xxx"></li><li id="li-28"><title>xxx</title></li><li id="li-29"><statement></statement></li><li id="li-30"><notation></li><li id="li-31"><usage> actual-notation </usage><description> verbal-description</description></li><li id="li-32"></notation></li><li id="li-33"></definition></li></ul></tt></p><p>The tag that opens the following definition is <tt class="code-inline"><definition xml:id="lattice-points"></tt>. In the exercise below we reference this definition with the tag <tt class="code-inline"><xref ref="lattice-points" autoname="title" /></tt>. With the value of title, the knowl text is the title of the referenced item, in this case "lattice points." The other possible <tt class="code-inline">autoname</tt>values are
<ul style="list-style-type: disc;"><li id="li-34">yes - displays the type of item (Definition, Example, Theorem, ...) and then the number of that item.</li><li id="li-35">no - just displays the item number</li><li id="li-36">plural - Is a variation on "title" that pluralizes the title. I wasn't aware of this when I wrote the code for this example. If I had used this option, I could have used the more natural id value "lattice point" and the knowl would have added the 's' at the end.</li></ul>
You can set the default when running <tt class="code-inline">xsltproc</tt> with the argument <tt class="code-inline">--stringparam autoname 'yes'</tt>, where 'yes' can be any of the values mentioned above.
</p><article class="definition-like" id="lattice-points"><h5 class="heading"><span class="type">Definition</span><span class="codenumber">3.1.1</span><span class="title">Lattice Points</span></h5>The lattice points in \(d\) dimensions, \(d\) a positive integer, are the points in \(d\)-dimensional space with integer coordinates.</article><p>The following definition has two notation tags, each is listed in the notations section.</p><article class="definition-like" id="set_complement."><h5 class="heading"><span class="type">Definition</span><span class="codenumber">3.1.2</span><span class="title">Complement of a set</span></h5>Let \( A\) and \( B\) be sets. The complement of \( A\) relative to \( B\) (notation
\(B - A\)) is the set of elements that are in \( B\) and not in \( A\). That is, \(B-A=\{x: x\in B \textrm{ and } x\notin A\}\). If \(
U\) is the universal set, then \(U-A\) is denoted by \(A^c\) and is called simply the complement of \( A\). \(A^c=\{x\in U : x\notin A\}\).
<span id="notation-1"></span><span id="notation-2"></span></article><p>...many lines later or in a different chapter, the following exercise may appear.....</p></section><section class="exercises" id="exercises-1"><header><h1 class="heading hide-type"><span class="type">Subsection</span><span class="codenumber">3.1.2</span><span class="title">Some Exercises</span></h1></header><article class="exercise-like" id="exercise-1"><h5 class="heading"><span class="codenumber">1</span></h5>
Given nine <a knowl="./knowl/lattice-points.html" knowl-id="xref-lattice-points" alt="Definition 3.1.1 Lattice Points" title="Definition 3.1.1 Lattice Points">Lattice Points</a> in three dimensions, prove that there are at least two for which their midpoint is also a lattice point.
</article><div class="hidden-knowl-wrapper"><span class="hidden-knowl-wrapper"><a knowl="" class="id-ref" refid="hk-hint-1" id="hint-1"><span class="heading"><span class="type">Hint</span></span></a></span><span id="hk-hint-1" style="display: none;" class="tex2jax_ignore"><span class="solution">Think about the odd-even parity of the nine points</span></span><span class="hidden-knowl-wrapper"><a knowl="" class="id-ref" refid="hk-solution-1" id="solution-1"><span class="heading"><span class="type">Solution</span></span></a></span><span id="hk-solution-1" style="display: none;" class="tex2jax_ignore"><span class="solution">There are eight different odd/even parity "signature" of a point. For example \((1,4,-2)\) has the signature (odd, even, even). Since we are given nine points, the pigeon-hole principle guarantees us that there must be two different points with the same signature. The midpoint of two lattice points with the same signatures is a lattice point since \(\frac{odd + odd}{2}\) and \(\frac{even + even}{2}\) are both integers.
</span></span></div></section></section></div></main></div></body></html>