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<h2 class="hd hd-2 unit-title">Homework</h2>
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<p>The exercises below will count towards your grade. <strong>You have only one chance to answer these questions.</strong> Take your time, and think carefully before answering.</p>
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Problem 1
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<p>Is \(\mathcal P(\mathcal P( \mathcal P (\emptyset)))\) totally ordered by \(\in\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c9809360ded04b3bba4c5365c5c86ae7_2_1">
<fieldset aria-describedby="status_c9809360ded04b3bba4c5365c5c86ae7_2_1">
<div class="field">
<input type="radio" name="input_c9809360ded04b3bba4c5365c5c86ae7_2_1" id="input_c9809360ded04b3bba4c5365c5c86ae7_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="c9809360ded04b3bba4c5365c5c86ae7_2_1-choice_0-label" for="input_c9809360ded04b3bba4c5365c5c86ae7_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_c9809360ded04b3bba4c5365c5c86ae7_2_1"> Yes
</label>
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<input type="radio" name="input_c9809360ded04b3bba4c5365c5c86ae7_2_1" id="input_c9809360ded04b3bba4c5365c5c86ae7_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c9809360ded04b3bba4c5365c5c86ae7_2_1-choice_1-label" for="input_c9809360ded04b3bba4c5365c5c86ae7_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c9809360ded04b3bba4c5365c5c86ae7_2_1"> No
</label>
</div>
<span id="answer_c9809360ded04b3bba4c5365c5c86ae7_2_1"/>
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<div class="indicator-container">
<span class="status unanswered" id="status_c9809360ded04b3bba4c5365c5c86ae7_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
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<div class="solution-span">
<span id="solution_c9809360ded04b3bba4c5365c5c86ae7_solution_1"/>
</div><p>Is \(\mathcal P (\mathcal P (\mathcal P (\emptyset))) - \{ \{ \{ \emptyset\}\}\}\) totally ordered by \(\in\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c9809360ded04b3bba4c5365c5c86ae7_3_1">
<fieldset aria-describedby="status_c9809360ded04b3bba4c5365c5c86ae7_3_1">
<div class="field">
<input type="radio" name="input_c9809360ded04b3bba4c5365c5c86ae7_3_1" id="input_c9809360ded04b3bba4c5365c5c86ae7_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="c9809360ded04b3bba4c5365c5c86ae7_3_1-choice_0-label" for="input_c9809360ded04b3bba4c5365c5c86ae7_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_c9809360ded04b3bba4c5365c5c86ae7_3_1"> Yes
</label>
</div>
<div class="field">
<input type="radio" name="input_c9809360ded04b3bba4c5365c5c86ae7_3_1" id="input_c9809360ded04b3bba4c5365c5c86ae7_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c9809360ded04b3bba4c5365c5c86ae7_3_1-choice_1-label" for="input_c9809360ded04b3bba4c5365c5c86ae7_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c9809360ded04b3bba4c5365c5c86ae7_3_1"> No
</label>
</div>
<span id="answer_c9809360ded04b3bba4c5365c5c86ae7_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_c9809360ded04b3bba4c5365c5c86ae7_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div class="solution-span">
<span id="solution_c9809360ded04b3bba4c5365c5c86ae7_solution_2"/>
</div><p>Some definitions:
<ul><li>\(\emptyset\) is the empty set.</li><li>if \(A\) and \(B\) are sets, \(A-B\) is the set of objects in \(A\) but not in \(B\). (For instance, \(\{1,2\}-\{1\}=\{2\}\).)</li><li>\(\mathcal P\) is the powerset operation.</li></ul></p>
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data-problem-id="block-v1:MITx+24.118x+2T2020+type@problem+block@ab88fd4d59034bc7a861655496949b2c" data-url="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@problem+block@ab88fd4d59034bc7a861655496949b2c/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="ab88fd4d59034bc7a861655496949b2c-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@ab88fd4d59034bc7a861655496949b2c-problem-progress" tabindex="-1">
Problem 2
</h3>
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<div class="problem">
<div>
<p>Are the following statements true or false?</p>
<p style="padding-left:60px;">
<ul>
<li>\(\alpha + 0' = \alpha \cup \{\alpha\}\)
</li>
</ul>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ab88fd4d59034bc7a861655496949b2c_2_1">
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<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_2_1" id="input_ab88fd4d59034bc7a861655496949b2c_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ab88fd4d59034bc7a861655496949b2c_2_1-choice_0-label" for="input_ab88fd4d59034bc7a861655496949b2c_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_2_1"> True
</label>
</div>
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_2_1" id="input_ab88fd4d59034bc7a861655496949b2c_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ab88fd4d59034bc7a861655496949b2c_2_1-choice_1-label" for="input_ab88fd4d59034bc7a861655496949b2c_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_2_1"> False
</label>
</div>
<span id="answer_ab88fd4d59034bc7a861655496949b2c_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ab88fd4d59034bc7a861655496949b2c_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<span id="solution_ab88fd4d59034bc7a861655496949b2c_solution_1"/>
</div><p/>
<hr/>
<p/>
<p style="padding-left:60px;">
<ul>
<li> \(0' + \omega = \omega + 0'\)
</li>
</ul>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ab88fd4d59034bc7a861655496949b2c_3_1">
<fieldset aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_3_1">
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_3_1" id="input_ab88fd4d59034bc7a861655496949b2c_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ab88fd4d59034bc7a861655496949b2c_3_1-choice_0-label" for="input_ab88fd4d59034bc7a861655496949b2c_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_3_1"> True
</label>
</div>
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_3_1" id="input_ab88fd4d59034bc7a861655496949b2c_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ab88fd4d59034bc7a861655496949b2c_3_1-choice_1-label" for="input_ab88fd4d59034bc7a861655496949b2c_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_3_1"> False
</label>
</div>
<span id="answer_ab88fd4d59034bc7a861655496949b2c_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ab88fd4d59034bc7a861655496949b2c_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ab88fd4d59034bc7a861655496949b2c_solution_2"/>
</div><p/>
<hr/>
<p/>
<p style="padding-left:60px;">
<ul>
<li>\(0' + 0''' = 0''' + 0'\)
</li>
</ul>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ab88fd4d59034bc7a861655496949b2c_4_1">
<fieldset aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_4_1">
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_4_1" id="input_ab88fd4d59034bc7a861655496949b2c_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ab88fd4d59034bc7a861655496949b2c_4_1-choice_0-label" for="input_ab88fd4d59034bc7a861655496949b2c_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_4_1"> True
</label>
</div>
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_4_1" id="input_ab88fd4d59034bc7a861655496949b2c_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ab88fd4d59034bc7a861655496949b2c_4_1-choice_1-label" for="input_ab88fd4d59034bc7a861655496949b2c_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_4_1"> False
</label>
</div>
<span id="answer_ab88fd4d59034bc7a861655496949b2c_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ab88fd4d59034bc7a861655496949b2c_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ab88fd4d59034bc7a861655496949b2c_solution_3"/>
</div><p/>
<hr/>
<p/>
<p style="padding-left:60px;">
<ul>
<li> \((\omega + 0'') + \omega &lt;_o (\omega + \omega) + 0''\)
</li>
</ul>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ab88fd4d59034bc7a861655496949b2c_5_1">
<fieldset aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_5_1">
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_5_1" id="input_ab88fd4d59034bc7a861655496949b2c_5_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ab88fd4d59034bc7a861655496949b2c_5_1-choice_0-label" for="input_ab88fd4d59034bc7a861655496949b2c_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_5_1"> True
</label>
</div>
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_5_1" id="input_ab88fd4d59034bc7a861655496949b2c_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ab88fd4d59034bc7a861655496949b2c_5_1-choice_1-label" for="input_ab88fd4d59034bc7a861655496949b2c_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_5_1"> False
</label>
</div>
<span id="answer_ab88fd4d59034bc7a861655496949b2c_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ab88fd4d59034bc7a861655496949b2c_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ab88fd4d59034bc7a861655496949b2c_solution_4"/>
</div><p/>
<hr/>
<p/>
<p style="padding-left:60px;">
<ul>
<li> \(0' \times 0''' = 0''' \times 0'\)
</li>
</ul>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ab88fd4d59034bc7a861655496949b2c_6_1">
<fieldset aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_6_1">
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_6_1" id="input_ab88fd4d59034bc7a861655496949b2c_6_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ab88fd4d59034bc7a861655496949b2c_6_1-choice_0-label" for="input_ab88fd4d59034bc7a861655496949b2c_6_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_6_1"> True
</label>
</div>
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_6_1" id="input_ab88fd4d59034bc7a861655496949b2c_6_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ab88fd4d59034bc7a861655496949b2c_6_1-choice_1-label" for="input_ab88fd4d59034bc7a861655496949b2c_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_6_1"> False
</label>
</div>
<span id="answer_ab88fd4d59034bc7a861655496949b2c_6_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ab88fd4d59034bc7a861655496949b2c_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ab88fd4d59034bc7a861655496949b2c_solution_5"/>
</div><p/>
<hr/>
<p/>
<p style="padding-left:60px;">
<ul>
<li>\(0''' \times \omega = (\omega + \omega) + \omega\) </li>
</ul>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 6" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ab88fd4d59034bc7a861655496949b2c_7_1">
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<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_7_1" id="input_ab88fd4d59034bc7a861655496949b2c_7_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ab88fd4d59034bc7a861655496949b2c_7_1-choice_0-label" for="input_ab88fd4d59034bc7a861655496949b2c_7_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_7_1"> True
</label>
</div>
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_7_1" id="input_ab88fd4d59034bc7a861655496949b2c_7_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ab88fd4d59034bc7a861655496949b2c_7_1-choice_1-label" for="input_ab88fd4d59034bc7a861655496949b2c_7_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_7_1"> False
</label>
</div>
<span id="answer_ab88fd4d59034bc7a861655496949b2c_7_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ab88fd4d59034bc7a861655496949b2c_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ab88fd4d59034bc7a861655496949b2c_solution_6"/>
</div><p/>
<hr/>
<p/>
<p style="padding-left:60px;">
<ul>
<li> \(\omega \times 0''' = \omega + (\omega + \omega)\)
</li>
</ul>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 7" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ab88fd4d59034bc7a861655496949b2c_8_1">
<fieldset aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_8_1">
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_8_1" id="input_ab88fd4d59034bc7a861655496949b2c_8_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ab88fd4d59034bc7a861655496949b2c_8_1-choice_0-label" for="input_ab88fd4d59034bc7a861655496949b2c_8_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_8_1"> True
</label>
</div>
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_8_1" id="input_ab88fd4d59034bc7a861655496949b2c_8_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ab88fd4d59034bc7a861655496949b2c_8_1-choice_1-label" for="input_ab88fd4d59034bc7a861655496949b2c_8_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_8_1"> False
</label>
</div>
<span id="answer_ab88fd4d59034bc7a861655496949b2c_8_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ab88fd4d59034bc7a861655496949b2c_8_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ab88fd4d59034bc7a861655496949b2c_solution_7"/>
</div><p/>
<hr/>
<p/>
<p style="padding-left:60px;">
<ul>
<li> \((\omega \times 0'') + \omega &lt;_o (\omega \times \omega) + 0''\)
</li>
</ul>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 8" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ab88fd4d59034bc7a861655496949b2c_9_1">
<fieldset aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_9_1">
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_9_1" id="input_ab88fd4d59034bc7a861655496949b2c_9_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ab88fd4d59034bc7a861655496949b2c_9_1-choice_0-label" for="input_ab88fd4d59034bc7a861655496949b2c_9_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_9_1"> True
</label>
</div>
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_9_1" id="input_ab88fd4d59034bc7a861655496949b2c_9_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ab88fd4d59034bc7a861655496949b2c_9_1-choice_1-label" for="input_ab88fd4d59034bc7a861655496949b2c_9_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_9_1"> False
</label>
</div>
<span id="answer_ab88fd4d59034bc7a861655496949b2c_9_1"/>
</fieldset>
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<span class="status unanswered" id="status_ab88fd4d59034bc7a861655496949b2c_9_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ab88fd4d59034bc7a861655496949b2c_solution_8"/>
</div><p/>
<hr/>
<p/>
<p style="padding-left:60px;">
<ul>
<li> \(\omega \times \omega &lt;_o \omega \times (0'' \times \omega)\)
</li>
</ul>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 9" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ab88fd4d59034bc7a861655496949b2c_10_1">
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<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_10_1" id="input_ab88fd4d59034bc7a861655496949b2c_10_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ab88fd4d59034bc7a861655496949b2c_10_1-choice_0-label" for="input_ab88fd4d59034bc7a861655496949b2c_10_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_10_1"> True
</label>
</div>
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_10_1" id="input_ab88fd4d59034bc7a861655496949b2c_10_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ab88fd4d59034bc7a861655496949b2c_10_1-choice_1-label" for="input_ab88fd4d59034bc7a861655496949b2c_10_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_10_1"> False
</label>
</div>
<span id="answer_ab88fd4d59034bc7a861655496949b2c_10_1"/>
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<span class="status unanswered" id="status_ab88fd4d59034bc7a861655496949b2c_10_1" data-tooltip="Not yet answered.">
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</div></div>
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<span id="solution_ab88fd4d59034bc7a861655496949b2c_solution_9"/>
</div><p/>
<hr/>
<p/>
<p style="padding-left:60px;">
<ul>
<li> \(\omega \times (\omega + \omega) = (\omega \times \omega) + (\omega \times \omega)\)
</li>
</ul>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 10" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ab88fd4d59034bc7a861655496949b2c_11_1">
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<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_11_1" id="input_ab88fd4d59034bc7a861655496949b2c_11_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ab88fd4d59034bc7a861655496949b2c_11_1-choice_0-label" for="input_ab88fd4d59034bc7a861655496949b2c_11_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_11_1"> True
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</div>
<div class="field">
<input type="radio" name="input_ab88fd4d59034bc7a861655496949b2c_11_1" id="input_ab88fd4d59034bc7a861655496949b2c_11_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ab88fd4d59034bc7a861655496949b2c_11_1-choice_1-label" for="input_ab88fd4d59034bc7a861655496949b2c_11_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ab88fd4d59034bc7a861655496949b2c_11_1"> False
</label>
</div>
<span id="answer_ab88fd4d59034bc7a861655496949b2c_11_1"/>
</fieldset>
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<span class="status unanswered" id="status_ab88fd4d59034bc7a861655496949b2c_11_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
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<span id="solution_ab88fd4d59034bc7a861655496949b2c_solution_10"/>
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<h3 class="hd hd-3 problem-header" id="7062240f94d2413db13c344e36b1b2a0-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@7062240f94d2413db13c344e36b1b2a0-problem-progress" tabindex="-1">
Problem 3
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<div class="problem-progress" id="block-v1:MITx+24.118x+2T2020+type@problem+block@7062240f94d2413db13c344e36b1b2a0-problem-progress"></div>
<div class="problem">
<div>
<p>Let \(U\) be the following set:</p>
<p style="padding-left:60px;"> \(\cup \{\mathbb{N}, \mathcal P^1(\mathbb{N}), \mathcal P^2(\mathbb{N}), \dots\}\) </p>
<p>Does \(U\) contain the set \(\underbrace{\{\{\ldots\{\{}_{n \ \textrm{times}}17\underbrace{\}\}\ldots\}\}}_{n \ \textrm{times}}\) for each \(n&gt;0\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_7062240f94d2413db13c344e36b1b2a0_2_1">
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<input type="radio" name="input_7062240f94d2413db13c344e36b1b2a0_2_1" id="input_7062240f94d2413db13c344e36b1b2a0_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="7062240f94d2413db13c344e36b1b2a0_2_1-choice_0-label" for="input_7062240f94d2413db13c344e36b1b2a0_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_7062240f94d2413db13c344e36b1b2a0_2_1"> Yes
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<input type="radio" name="input_7062240f94d2413db13c344e36b1b2a0_2_1" id="input_7062240f94d2413db13c344e36b1b2a0_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="7062240f94d2413db13c344e36b1b2a0_2_1-choice_1-label" for="input_7062240f94d2413db13c344e36b1b2a0_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_7062240f94d2413db13c344e36b1b2a0_2_1"> No
</label>
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<span id="answer_7062240f94d2413db13c344e36b1b2a0_2_1"/>
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<span class="status unanswered" id="status_7062240f94d2413db13c344e36b1b2a0_2_1" data-tooltip="Not yet answered.">
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<div class="solution-span">
<span id="solution_7062240f94d2413db13c344e36b1b2a0_solution_1"/>
</div><p>Does \(U\) contain the set \(\underbrace{\{\{\ldots\{\{}_{\infty \ \textrm{times}}17\underbrace{\}\}\ldots\}\}}_{\infty \ \textrm{times}}\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_7062240f94d2413db13c344e36b1b2a0_3_1">
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<input type="radio" name="input_7062240f94d2413db13c344e36b1b2a0_3_1" id="input_7062240f94d2413db13c344e36b1b2a0_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="7062240f94d2413db13c344e36b1b2a0_3_1-choice_0-label" for="input_7062240f94d2413db13c344e36b1b2a0_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_7062240f94d2413db13c344e36b1b2a0_3_1"> Yes
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<input type="radio" name="input_7062240f94d2413db13c344e36b1b2a0_3_1" id="input_7062240f94d2413db13c344e36b1b2a0_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="7062240f94d2413db13c344e36b1b2a0_3_1-choice_1-label" for="input_7062240f94d2413db13c344e36b1b2a0_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_7062240f94d2413db13c344e36b1b2a0_3_1"> No
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Problem 4
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<p>Is there an ordinal \(\alpha\) such that \(\alpha &lt;_o \emptyset\)?</p>
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Problem 5
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Recall the Ordinal Construction Principle:</p>
<p style="padding-left:60px;"><i>Construction Principle</i>: At each stage, introduce a new ordinal: the set of all ordinals that have been introduced at previous stages.</p>
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If \(\alpha\) is an ordinal with infinitely many members, then either \(\alpha = \omega\) or \(\omega &lt;_o \alpha \).</legend>
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Problem 6
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<legend id="a5a5043e606f4b79824468393b17c065_2_1-legend" class="response-fieldset-legend field-group-hd">Which of the following sets are totally ordered by their respective relation, \(&lt;\)?</legend>
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<p>The set of natural numbers, where \( 1 &lt; 2 &lt; 3 &lt; ... &lt; 0 \)</p>
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<p>The set of rational numbers, where \( a &lt; b\) if and only if \(b = a + q \) for \(q\) a positive rational number</p>
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<p>The set of people on Earth, where \(a &lt; b\) if and only if \(a\)'s year of birth is before \(b\)'s year of birth</p>
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<p>The set \(\mathcal{P}(\{0,1,2,3\}) \), where \(a &lt; b\) if and only if \(a \subsetneq b \)</p>
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<p>The set of Olympic athletes, where \(a &lt; b\) if and only if \(a\) has won fewer medals than \(b\)</p>
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