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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p>Read <a href="/assets/courseware/v1/4402c4fa2ba3fecab92f41a6f43005bf/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session3.pdf" target="[object Object]">Chapter 2.1–2.3 (PDF)</a> of <em>Mathematics for Computer Science</em> for 1.3 Well Ordering Principle.</p>
<p>View the <a href="/assets/courseware/v1/b9289f9f419268bed259c034100f4f63/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp3.pdf" target="[object Object]">Section 1.3 In-Class Questions (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Lecture Video | Well Ordering Principle I</h2>
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<p>Download a copy of the slides for <a href="/assets/courseware/v1/fd4c27877380edac9f603bfa52ea4d7c/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS16_Well_Ordering_1_Slides.pdf" target="[object Object]">Well Ordering Principle I (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Exercise | Domain for Well Ordering Principle</h2>
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Domain for Well Ordering Principle
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<legend id="b6cddb38a83a44fdb05998e30afbec78_2_1-legend" class="response-fieldset-legend field-group-hd">The Well Ordering Principle says that every nonempty set of ______ ______has a least/smallest element.</legend>
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<h2 class="hd hd-2 unit-title">Lecture Video | Well Ordering Principle II</h2>
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<h2 class="hd hd-2 unit-title">Exercise | Well Ordering Proofs and Counterexamples</h2>
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Well Ordering Proofs and Counterexamples
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<h2 class="hd hd-2 unit-title">Lecture Video | Well Ordering Principle III</h2>
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<h2 class="hd hd-2 unit-title">Exercise | WOP Proof for Geometric Sum</h2>
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Well Ordering Proof for Geometric Sum
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<label class="problem-group-label" for="input_c10f5f73fb1e4e60a830f444b02d463a_2_1" id="label_c10f5f73fb1e4e60a830f444b02d463a_2_1">In Well Ordering Proofs, first we assume that there is a nonempty set <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>C</mi> </math> of counterexamples and that <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>m</mi> </math> is the smallest element of <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>C</mi> </math>. Then we reach a contradiction.
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What was the contradiction in the proof for the closed expression of the sum of a geometric series?</label>
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<option value="it holds for m"> it holds for m</option>
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<h2 class="hd hd-2 unit-title">Exercise | Well Ordering Principle Examples</h2>
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Well Ordering Principle - Examples
</h3>
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<legend id="1cb9a9f4038840139b2dc248925b6afa_2_1-legend" class="response-fieldset-legend field-group-hd">A set of numbers is <em>well ordered</em> when each of its nonempty subsets has a minimum element. The Well Ordering Principle says that the set of nonnegative integers is well ordered, but so are lots of other sets. For example, the set <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
</math> of numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>r</mi>
<mi>n</mi>
</math>, where <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>r</mi>
</math> is a positive real number and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
<mo>&#8712;<!-- &#8712; --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
</math>.
<br/>
<br/>
Indicate which of the following sets of numbers are well ordered.</legend>
<div class="field">
<input type="checkbox" name="input_1cb9a9f4038840139b2dc248925b6afa_2_1[]" id="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="1cb9a9f4038840139b2dc248925b6afa_2_1-choice_0-label" for="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_1cb9a9f4038840139b2dc248925b6afa_2_1"> The integers <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8805;<!-- &#8805; --></mo>
<mo>&#8722;<!-- &#8722; --></mo>
<msqrt>
<mn>2</mn>
</msqrt>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_1cb9a9f4038840139b2dc248925b6afa_2_1[]" id="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="1cb9a9f4038840139b2dc248925b6afa_2_1-choice_1-label" for="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_1cb9a9f4038840139b2dc248925b6afa_2_1"> The integers <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&gt;</mo>
<msqrt>
<mn>2</mn>
</msqrt>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_1cb9a9f4038840139b2dc248925b6afa_2_1[]" id="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="1cb9a9f4038840139b2dc248925b6afa_2_1-choice_2-label" for="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_1cb9a9f4038840139b2dc248925b6afa_2_1"> The rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8805;<!-- &#8805; --></mo>
<msqrt>
<mn>2</mn>
</msqrt>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_1cb9a9f4038840139b2dc248925b6afa_2_1[]" id="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="1cb9a9f4038840139b2dc248925b6afa_2_1-choice_3-label" for="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_1cb9a9f4038840139b2dc248925b6afa_2_1"> The set of rationals of the form <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</math> where <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
</math> is a positive integer.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_1cb9a9f4038840139b2dc248925b6afa_2_1[]" id="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="1cb9a9f4038840139b2dc248925b6afa_2_1-choice_4-label" for="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_1cb9a9f4038840139b2dc248925b6afa_2_1"> The set of rationals of the form <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</math> where <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
</math> is a positive integer <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8804;<!-- &#8804; --></mo>
<msup>
<mn>10</mn>
<mrow class="MJX-TeXAtom-ORD">
<mn>100</mn>
</mrow>
</msup>
</math> <i>(a googol)</i>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_1cb9a9f4038840139b2dc248925b6afa_2_1[]" id="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="1cb9a9f4038840139b2dc248925b6afa_2_1-choice_5-label" for="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_1cb9a9f4038840139b2dc248925b6afa_2_1"> The set <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>G</mi>
</math> of rationals of the form <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mi>m</mi>
<mi>n</mi>
</mfrac>
</math> where <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>m</mi>
<mo>,</mo>
<mi>n</mi>
</math> are positive integers and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
<mo>&#8804;<!-- &#8804; --></mo>
<msup>
<mn>10</mn>
<mrow class="MJX-TeXAtom-ORD">
<mn>100</mn>
</mrow>
</msup>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_1cb9a9f4038840139b2dc248925b6afa_2_1[]" id="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="1cb9a9f4038840139b2dc248925b6afa_2_1-choice_6-label" for="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_1cb9a9f4038840139b2dc248925b6afa_2_1"> The set <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
</math>, of fractions of the form <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mi>n</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mn>0</mn>
<mn>1</mn>
</mfrac>
<mo>,</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mo>,</mo>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
<mo>,</mo>
<mo>&#8230;<!-- &#8230; --></mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_1cb9a9f4038840139b2dc248925b6afa_2_1[]" id="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="1cb9a9f4038840139b2dc248925b6afa_2_1-choice_7-label" for="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_1cb9a9f4038840139b2dc248925b6afa_2_1"> The set <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>W</mi>
<mo>::=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
<mo>&#8746;<!-- &#8746; --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
</math> consisting of the nonnegative integers along with all the fractions of the form <math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac>
<mi>n</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_1cb9a9f4038840139b2dc248925b6afa_2_1[]" id="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_8" class="field-input input-checkbox" value="choice_8"/><label id="1cb9a9f4038840139b2dc248925b6afa_2_1-choice_8-label" for="input_1cb9a9f4038840139b2dc248925b6afa_2_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_1cb9a9f4038840139b2dc248925b6afa_2_1"> The set, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math>, of real numbers with the property that every element of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> has only finitely many elements of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> below it. All the sets <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
</math> for real <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>r</mi>
<mo>&gt;</mo>
<mn>0</mn>
</math> have this property.
</label>
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<h2 class="hd hd-2 unit-title">Exercise | A Bogus Well Ordering Principle Proof</h2>
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<h3 class="hd hd-3 problem-header" id="c9428f6c501b4a57bf3a87da60d68468-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@c9428f6c501b4a57bf3a87da60d68468-problem-progress" tabindex="-1">
A Bogus Well Ordering Principle Proof
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c9428f6c501b4a57bf3a87da60d68468_2_1">
<fieldset aria-describedby="status_c9428f6c501b4a57bf3a87da60d68468_2_1">
<legend id="c9428f6c501b4a57bf3a87da60d68468_2_1-legend" class="response-fieldset-legend field-group-hd">The Fibonacci numbers <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mn>3</mn>
<mo>,</mo>
<mn>5</mn>
<mo>,</mo>
<mn>8</mn>
<mo>,</mo>
<mn>13</mn>
<mo>,</mo>
<mo>&#8230;<!-- &#8230; --></mo>
</math> are defined as follows. Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math> be the <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mi>h</mi>
</mrow>
</msup>
</math> Fibonacci number. Then
<center><math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>F</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>::=</mo> <mn>0</mn> </math></center>
<center><math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>F</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>::=</mo> <mn>1</mn> </math></center>
<center><math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>::=</mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>&#8722;<!-- &#8722; --></mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mspace width="thickmathspace"/>
<mspace width="thickmathspace"/>
<mtext>&#160;for&#160;</mtext>
<mi>n</mi>
<mo>&#8805;<!-- &#8805; --></mo>
<mn>2</mn>
<mtext>&#160;</mtext>
<mo stretchy="false">(</mo>
<mo>&#8902;<!-- &#8902; --></mo>
<mo stretchy="false">)</mo>
</math></center>
<br/>
<br/>
<div align="center">
<strong> Identify which step(s) contain the logical error!</strong>
</div>
<br/>
<br/>
<strong>Bogus Claim</strong> Every Fibonacci number is even.</legend>
<div class="field">
<input type="checkbox" name="input_c9428f6c501b4a57bf3a87da60d68468_2_1[]" id="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="c9428f6c501b4a57bf3a87da60d68468_2_1-choice_0-label" for="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_c9428f6c501b4a57bf3a87da60d68468_2_1"> The proof is by the WOP.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_c9428f6c501b4a57bf3a87da60d68468_2_1[]" id="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="c9428f6c501b4a57bf3a87da60d68468_2_1-choice_1-label" for="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c9428f6c501b4a57bf3a87da60d68468_2_1"> Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mi>v</mi>
<mi>e</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math> mean that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math> is even.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_c9428f6c501b4a57bf3a87da60d68468_2_1[]" id="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="c9428f6c501b4a57bf3a87da60d68468_2_1-choice_2-label" for="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c9428f6c501b4a57bf3a87da60d68468_2_1"> Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>C</mi>
</math> be the set of counterexamples to the assertion that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mi>v</mi>
<mi>e</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math> holds for all <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
<mo>&#8805;<!-- &#8805; --></mo>
<mn>0</mn>
</math>. That is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>C</mi>
<mo>::=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>n</mi>
<mo>&#8805;<!-- &#8805; --></mo>
<mn>0</mn>
<mspace width="thickmathspace"/>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mspace width="thickmathspace"/>
<mtext>NOT</mtext>
<mo stretchy="false">(</mo>
<mi>E</mi>
<mi>v</mi>
<mi>e</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">}</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_c9428f6c501b4a57bf3a87da60d68468_2_1[]" id="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="c9428f6c501b4a57bf3a87da60d68468_2_1-choice_3-label" for="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_c9428f6c501b4a57bf3a87da60d68468_2_1"> We prove by contradiction that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>C</mi>
</math> is empty.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_c9428f6c501b4a57bf3a87da60d68468_2_1[]" id="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="c9428f6c501b4a57bf3a87da60d68468_2_1-choice_4-label" for="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_c9428f6c501b4a57bf3a87da60d68468_2_1"> Assume that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>C</mi>
</math> is not empty.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_c9428f6c501b4a57bf3a87da60d68468_2_1[]" id="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="c9428f6c501b4a57bf3a87da60d68468_2_1-choice_5-label" for="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_c9428f6c501b4a57bf3a87da60d68468_2_1"> By the WOP, there is a least nonnegative integer, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>m</mi>
<mo>&#8712;<!-- &#8712; --></mo>
<mi>C</mi>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_c9428f6c501b4a57bf3a87da60d68468_2_1[]" id="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="c9428f6c501b4a57bf3a87da60d68468_2_1-choice_6-label" for="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_c9428f6c501b4a57bf3a87da60d68468_2_1"> Then <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>m</mi>
<mo>&#8805;<!-- &#8805; --></mo>
<mn>0</mn>
</math>, since <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math> is an even number.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_c9428f6c501b4a57bf3a87da60d68468_2_1[]" id="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="c9428f6c501b4a57bf3a87da60d68468_2_1-choice_7-label" for="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_c9428f6c501b4a57bf3a87da60d68468_2_1"> Now, suppose <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>m</mi>
<mo>&#8805;<!-- &#8805; --></mo>
<mn>2</mn>
</math> so the definition <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo stretchy="false">(</mo>
<mo>&#8902;<!-- &#8902; --></mo>
<mo stretchy="false">)</mo>
</math> of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>m</mi>
<mo stretchy="false">)</mo>
</math> applies.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_c9428f6c501b4a57bf3a87da60d68468_2_1[]" id="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_8" class="field-input input-checkbox" value="choice_8"/><label id="c9428f6c501b4a57bf3a87da60d68468_2_1-choice_8-label" for="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_c9428f6c501b4a57bf3a87da60d68468_2_1"> In this case, both <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>m</mi>
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>m</mi>
<mo>&#8722;<!-- &#8722; --></mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math> are both even, since <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>m</mi>
</math> is the minimum counterexample such that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>m</mi>
<mo stretchy="false">)</mo>
</math> is not even.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_c9428f6c501b4a57bf3a87da60d68468_2_1[]" id="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_9" class="field-input input-checkbox" value="choice_9"/><label id="c9428f6c501b4a57bf3a87da60d68468_2_1-choice_9-label" for="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_9" class="response-label field-label label-inline" aria-describedby="status_c9428f6c501b4a57bf3a87da60d68468_2_1"> But by <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo stretchy="false">(</mo>
<mo>&#8902;<!-- &#8902; --></mo>
<mo stretchy="false">)</mo>
</math> in the case that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
<mo>=</mo>
<mi>m</mi>
</math>, we see that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>m</mi>
<mo stretchy="false">)</mo>
</math> is the sum of two even numbers, so it is also even; thus <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>E</mi>
<mi>v</mi>
<mi>e</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>m</mi>
<mo stretchy="false">)</mo>
</math> is true.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_c9428f6c501b4a57bf3a87da60d68468_2_1[]" id="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_10" class="field-input input-checkbox" value="choice_10"/><label id="c9428f6c501b4a57bf3a87da60d68468_2_1-choice_10-label" for="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_10" class="response-label field-label label-inline" aria-describedby="status_c9428f6c501b4a57bf3a87da60d68468_2_1"> This deduction contradicts the condition in the definition of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>m</mi>
</math> that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mtext>NOT</mtext>
<mo stretchy="false">(</mo>
<mi>E</mi>
<mi>v</mi>
<mi>e</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>m</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math> is true.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_c9428f6c501b4a57bf3a87da60d68468_2_1[]" id="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_11" class="field-input input-checkbox" value="choice_11"/><label id="c9428f6c501b4a57bf3a87da60d68468_2_1-choice_11-label" for="input_c9428f6c501b4a57bf3a87da60d68468_2_1_choice_11" class="response-label field-label label-inline" aria-describedby="status_c9428f6c501b4a57bf3a87da60d68468_2_1"> This contradiction implies that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>C</mi>
</math> must be empty. Hence, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math> is even for all <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
<mo>&#8805;<!-- &#8805; --></mo>
<mn>0.</mn>
<mspace width="thickmathspace"/>
<mspace width="thickmathspace"/>
<mi>&#9724;<!-- &#9724; --></mi>
</math>
</label>
</div>
<span id="answer_c9428f6c501b4a57bf3a87da60d68468_2_1"/>
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