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<div id="loadlink">
<p><a>Open this file in the Lurch web app</a></p>
<script language="javascript">
const link = document.querySelector( '#loadlink > p > a' )
const thisURL = encodeURIComponent( window.location.href )
link?.setAttribute( 'href', 'http://localhost:2999/instructor.html?load=' + thisURL )
</script>
</div>
<div id="metadata" style="display: none;"><div data-category="main" data-key="header" data-value-type="html"><div class="lurch-atom" contenteditable="false" data-metadata_type=""dependency"" data-metadata_description=""none"" data-metadata_filename=""math/Prop-Rules.lurch"" data-metadata_source=""the web"" data-metadata_auto-refresh="true" style="border: 1px solid gray; padding: 0px 1em;"><div class="lurch-atom-metadata" style="display: none;"><div data-key="content">
<div id="loadlink">
<p><a>Open this file in the Lurch web app</a></p>
</div>
<div id="metadata" style="display: none;"><div data-category="main" data-key="header" data-value-type="html"><div class="lurch-atom" contenteditable="false" data-metadata_type=""dependency"" data-metadata_description=""none"" data-metadata_filename=""math/Intro.lurch"" data-metadata_source=""the web"" data-metadata_auto-refresh="true" style="border: 1px solid gray; padding: 0px 1em;"><div class="lurch-atom-metadata" style="display: none;"><div data-key="content">
<div id="loadlink">
<p><a>Open this file in the Lurch web app</a></p>
</div>
<div id="metadata" style="display: none;"></div>
<div id="document"><h1>Math 299 Rules, Definitions, and Theorems</h1>
<h3>Ken Monks - University of Scranton - Spring 2024</h3>
<p>Welcome to Math 299. Below you will find some of the rules, definitions, theorems, and declarations that we use in the course. For more detail about these rules see the Course Lecture Notes on the Math 299 Home Page accessible via the Lurch <code>Help</code> menu at the top of the page. (<em>Note that some of the definitions below differ from what is in the lecture notes as they are still in the process of being updated to be consistent with the rules below</em>.)</p>
<p>There are many ways to write the same definition, theorem, or rule in Lurch. For consistency most of the rules below use the same style that imitates the <em>Given-Conclude</em> format used in many high school geometry textbooks.</p>
<p> </p></div>
</div></div><div class="lurch-atom-body"><table><colgroup><col><col></colgroup><tbody><tr><td colspan="2"><b>Imported dependency document</b></td></tr><tr><td>Description:</td><td><tt>none</tt></td></tr><tr><td>Filename:</td><td><tt>math/Intro.lurch</tt></td></tr><tr><td>Source:</td><td>the web</td></tr><tr><td>Auto-refresh:</td><td>yes</td></tr></tbody></table></div></div></div><div data-category="settings" data-key="shell style" data-value-type="json">"boxed"</div></div>
<div id="document"><h1>Propositional Logic</h1>
<p>This is a basic library of the rules that define a natural deduction style formal system for Propositional Logic. Rule premises are denoted as 'Given' and conclusions by 'Conclude'.</p>
<hr>
<h3><strong>Constants</strong></h3>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""Declare iff, and, implies, or, not, contradiction""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Declare </span><span class="ML__cmr">⇔</span><span class="ML__text">, and, </span><span class="ML__cmr">⇒</span><span class="ML__text">, or, </span><span class="ML__ams">¬</span><span class="ML__text">, and </span><span class="ML__cmr">→←</span></span></span></span></span> to be constants.</p>
<h3><strong>Definitions</strong></h3>
<div class="lurch-atom" data-metadata_type=""rule"" data-shell_title="Rule:">
<p style="text-align: left;"><strong>and+</strong></p>
<div class="lurch-atom" data-metadata_type=""premise"" data-shell_title="">
<p><em>Given</em></p>
<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></em></p>
<p><em><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""V""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></em></p>
</div>
<p style="text-align: left;"><em>Conclude</em> <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W and V"" data-metadata_latex="" W\\text{ and } V"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""rule"" data-shell_title="Rule:">
<p><strong>and-</strong></p>
<div class="lurch-atom" data-metadata_type=""premise"" data-shell_title="">
<p><em>Given</em></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W and V""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> and </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
</div>
<p><em>Conclude </em><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W"" data-metadata_latex=""W"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""V"" data-metadata_latex=""V"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""rule"" data-shell_title="Rule:">
<p><strong>implies+</strong></p>
<div class="lurch-atom" data-metadata_type=""premise"" data-shell_title="">
<p><em>Given</em></p>
<div class="lurch-atom" data-metadata_type=""subproof"" data-shell_title="">
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""Assume W""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Assume </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""V"" data-metadata_latex=""V"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
</div>
</div>
<p><em>Conclude</em> <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W implies V"" data-metadata_latex="" W\\Rightarrow V"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""rule"" data-shell_title="Rule:">
<p><strong>implies- (modus ponens)</strong></p>
<div class="lurch-atom" data-metadata_type=""premise"" data-shell_title="">
<p><em>Given</em></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W implies V""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
</div>
<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""V"" data-metadata_latex=""V"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""rule"" data-shell_title="Rule:">
<p><strong>iff+</strong></p>
<div class="lurch-atom" data-metadata_type=""premise"" data-shell_title="">
<p><em>Given</em></p>
<div class="lurch-atom" data-metadata_type=""subproof"" data-shell_title="">
<p> <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""Assume W""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Assume </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span><br><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""V"" data-metadata_latex=""V"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""subproof"" data-shell_title="">
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""Assume V""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Assume </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span><br><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W"" data-metadata_latex=""W"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
</div>
</div>
<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W iff V"" data-metadata_latex="" W\\Leftrightarrow V"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇔</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""rule"" data-shell_title="Rule:">
<p><strong>iff- (1 of 2)</strong></p>
<div class="lurch-atom" data-metadata_type=""premise"" data-shell_title="">
<p><em>Given</em></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W iff V""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇔</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
</div>
<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""V"" data-metadata_latex=""V"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""rule"" data-shell_title="Rule:">
<p><strong>iff- (2 of 2)</strong></p>
<div class="lurch-atom" data-metadata_type=""premise"" data-shell_title="">
<p><em>Given</em></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W iff V""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇔</span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""V""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
</div>
<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""rule"" data-shell_title="Rule:">
<p><strong>or+</strong></p>
<div class="lurch-atom" data-metadata_type=""premise"" data-shell_title="">
<p><em>Given</em></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
</div>
<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W or V"" data-metadata_latex="" W\\text{ or } V"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""V or W"" data-metadata_latex="" V\\text{ or } W"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""rule"" data-shell_title="Rule:">
<p><strong>or- (proof by cases)</strong></p>
<div class="lurch-atom" data-metadata_type=""premise"" data-shell_title="">
<p><em>Given</em></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W or V""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__text"> or </span><span class="ML__mathit" style="margin-right: 0.23em;">V</span></span></span></span></span></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W implies U""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.11em;">U</span></span></span></span></span></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""V implies U""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.23em;">V</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.11em;">U</span></span></span></span></span></p>
</div>
<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""U"" data-metadata_latex=""U"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.11em;">U</span></span></span></span></span> </p>
</div>
<div class="lurch-atom" data-metadata_type=""rule"" data-shell_title="Rule:">
<p><strong>not+ (proof by contradiction)</strong></p>
<div class="lurch-atom" data-metadata_type=""premise"" data-shell_title="">
<p><em>Given</em></p>
<div class="lurch-atom" data-metadata_type=""subproof"" data-shell_title="">
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""Assume W""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Assume </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""contradiction"" data-metadata_latex=""\\rightarrow\\leftarrow"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">→←</span></span></span></span></span></p>
</div>
</div>
<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""not W"" data-metadata_latex=""\\neg W"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""rule"" data-shell_title="Rule:">
<p><strong>not- (proof by contradiction)</strong></p>
<div class="lurch-atom" data-metadata_type=""premise"" data-shell_title="">
<p><em>Given</em></p>
<div class="lurch-atom" data-metadata_type=""subproof"" data-shell_title="">
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""Assume not W""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">Assume not </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""contradiction""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">→←</span></span></span></span></span></p>
</div>
</div>
<p> Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W"" data-metadata_latex=""W"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""rule"" data-shell_title="Rule:">
<p><strong>contradiction+</strong></p>
<div class="lurch-atom" data-metadata_type=""premise"" data-shell_title="">
<p><em>Given</em></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""W""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""not W""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__mathit" style="margin-right: 0.14em;">W</span></span></span></span></span></p>
</div>
<p>Conclude <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""→←"" data-metadata_latex=""\\rightarrow\\leftarrow"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">→←</span></span></span></span></span></p>
</div>
<hr>
<p> </p></div>
</div></div><div class="lurch-atom-body"><table><colgroup><col><col></colgroup><tbody><tr><td colspan="2"><b>Imported dependency document</b></td></tr><tr><td>Description:</td><td><tt>none</tt></td></tr><tr><td>Filename:</td><td><tt>math/Prop-Rules.lurch</tt></td></tr><tr><td>Source:</td><td>the web</td></tr><tr><td>Auto-refresh:</td><td>yes</td></tr></tbody></table></div></div></div><div data-category="settings" data-key="shell style" data-value-type="json">"minimal"</div></div>
<div id="document"><p><span style="color: rgb(126, 140, 141);">Name:</span></p>
<h1 style="text-align: center;">Assignment #6</h1>
<h4 style="text-align: center;">Due: Tuesday, Feb 19</h4>
<p>Here it is, future mathematicians! Your next batch of proofs!</p>
<hr>
<p>In the following problems, <span class="lurch-atom" contenteditable="false" data-metadata_type=""expositorymath"" data-metadata_latex=""A, K, P, Q, S, T, Z"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit">A</span><span class="ML__cmr">,</span><span class="ML__mathit" style="margin-right: 0.08em;">K</span><span class="ML__cmr">,</span><span class="ML__mathit" style="margin-right: 0.14em;">P</span><span class="ML__cmr">,</span><span class="ML__mathit">Q</span><span class="ML__cmr">,</span><span class="ML__mathit" style="margin-right: 0.06em;">S</span><span class="ML__cmr">,</span><span class="ML__mathit" style="margin-right: 0.14em;">T</span><span class="ML__cmr">,</span><span class="ML__mathit" style="margin-right: 0.08em;">Z</span></span></span></span></span> all represent atomic statements.</p>
<div class="lurch-atom" data-metadata_type=""subproof"" data-shell_title="">
<p><span style="background-color: rgb(45, 194, 107); color: rgb(236, 240, 241);"><strong> Problem 1. </strong></span></p>
<div class="lurch-atom" data-metadata_type=""theorem"" data-shell_title="Theorem:">
<p><strong>Theorem (Iff is stronger): </strong><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""(P iff Q) implies (Q implies P)""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8777699999999999em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.14em;">P</span><span class="ML__cmr">⇔</span><span class="ML__mathit">Q</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">⇒</span><span class="ML__left-right" style="margin-top: -0.19444em; height: 0.8777699999999999em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit">Q</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.14em;">P</span><span class="ML__small-delim ML__close">)</span></span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""proof"" data-shell_title="Proof:">
<p><strong>Proof:</strong></p>
<p> </p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expositorymath"" data-metadata_latex=""\\square"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">□</span></span></span></span></span></p>
</div>
</div>
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<div class="lurch-atom" data-metadata_type=""subproof"" data-shell_title="">
<p><span style="background-color: rgb(45, 194, 107); color: rgb(236, 240, 241);"><strong> Problem 2. </strong></span></p>
<div class="lurch-atom" data-metadata_type=""theorem"" data-shell_title="Theorem:">
<p><strong>Theorem (the other way around):</strong> <span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""not (not A) implies A""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__text">not </span><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__text">not </span><span class="ML__mathit">A</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__cmr">⇒</span><span class="ML__mathit">A</span></span></span></span></span> </p>
</div>
<div class="lurch-atom" data-metadata_type=""proof"" data-shell_title="Proof:">
<p><strong>Proof:</strong><br><br></p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expositorymath"" data-metadata_latex=""\\square"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">□</span></span></span></span></span></p>
</div>
</div>
<hr>
<div class="lurch-atom" data-metadata_type=""subproof"" data-shell_title="">
<p><span style="background-color: rgb(45, 194, 107); color: rgb(236, 240, 241);"><strong> Problem 3. </strong></span></p>
<div class="lurch-atom" data-metadata_type=""theorem"" data-shell_title="Theorem:">
<p><strong>Theorem (Modus Tollens): </strong><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""(S implies T) and not T implies not S""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__left-right" style="margin-top: 0em; height: 0.68333em;"><span class="ML__small-delim ML__open">(</span><span class="ML__mathit" style="margin-right: 0.06em;">S</span><span class="ML__cmr">⇒</span><span class="ML__mathit" style="margin-right: 0.14em;">T</span><span class="ML__small-delim ML__close">)</span></span><span class="ML__text"> and not </span><span class="ML__mathit" style="margin-right: 0.14em;">T</span><span class="ML__cmr">⇒</span><span class="ML__text">not </span><span class="ML__mathit" style="margin-right: 0.06em;">S</span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""proof"" data-shell_title="Proof:">
<p><strong>Proof:</strong></p>
<p> </p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expositorymath"" data-metadata_latex=""\\square"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">□</span></span></span></span></span></p>
</div>
</div>
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<div class="lurch-atom" data-metadata_type=""subproof"" data-shell_title="">
<p><span style="background-color: rgb(45, 194, 107); color: rgb(236, 240, 241);"><strong> Problem 4. </strong></span></p>
<div class="lurch-atom" data-metadata_type=""theorem"" data-shell_title="Theorem:">
<p><strong>Theorem (atomic statements don't have to be variables): </strong></p>
<p style="text-align: center;"><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""0 < x or x<10 implies x<10 or 0<x""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__cmr">0</span><span class="ML__cmr"><</span><span class="ML__mathit">x</span><span class="ML__text"> or </span><span class="ML__mathit">x</span><span class="ML__cmr"><</span><span class="ML__cmr">10</span><span class="ML__cmr">⇒</span><span class="ML__mathit">x</span><span class="ML__cmr"><</span><span class="ML__cmr">10</span><span class="ML__text"> or </span><span class="ML__cmr">0</span><span class="ML__cmr"><</span><span class="ML__mathit">x</span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""proof"" data-shell_title="Proof:">
<p><strong>Proof:</strong></p>
<p> </p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expositorymath"" data-metadata_latex=""\\square"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">□</span></span></span></span></span></p>
</div>
</div>
<hr>
<div class="lurch-atom" data-metadata_type=""subproof"" data-shell_title="">
<p><span style="background-color: rgb(45, 194, 107); color: rgb(236, 240, 241);"><strong> Problem 5. </strong></span></p>
<div class="lurch-atom" data-metadata_type=""theorem"" data-shell_title="Theorem:">
<p><strong>Theorem (excluded middle): </strong><span class="lurch-atom" contenteditable="false" data-metadata_type=""expression"" data-metadata_lurch-notation=""K or not K""><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__mathit" style="margin-right: 0.08em;">K</span><span class="ML__text"> or not </span><span class="ML__mathit" style="margin-right: 0.08em;">K</span></span></span></span></span></p>
</div>
<div class="lurch-atom" data-metadata_type=""proof"" data-shell_title="Proof:">
<p><strong>Proof:</strong></p>
<p> </p>
<p><span class="lurch-atom" contenteditable="false" data-metadata_type=""expositorymath"" data-metadata_latex=""\\square"" data-metadata_given="false"><span class="lurch-atom-body"><span class="ML__latex"><span class="ML__base"><span class="ML__ams">□</span></span></span></span></span></p>
</div>
</div>
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