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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p>Read <a href="/assets/courseware/v1/96aec750e38d89a3da258929175a8532/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session11.pdf" target="[object Object]">Chapter 7 (PDF)</a> of <em>Mathematics for Computer Science</em> for 1.11 Infinite Sets.</p>
<p>View the <a href="/assets/courseware/v1/5dab1f6349109604b1497194094ca47a/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp11.pdf" target="[object Object]">Section 1.11 In-Class Questions (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Lecture Video | Cardinality</h2>
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<p>Download a copy of the slides for <a href="/assets/courseware/v1/92cf71acb4a06ea92ccc68b638ab393c/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS16_Cardinality.pdf" target="[object Object]">Cardinality (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Exercise | Cantor, Schroeder-Bernstein</h2>
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<h3 class="hd hd-3 problem-header" id="b9d6123713d64ef695c1c415675f4292-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@b9d6123713d64ef695c1c415675f4292-problem-progress" tabindex="-1">
Cantor, Schroeder-Bernstein
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@b9d6123713d64ef695c1c415675f4292-problem-progress"></div>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_b9d6123713d64ef695c1c415675f4292_2_1">
<fieldset aria-describedby="status_b9d6123713d64ef695c1c415675f4292_2_1">
<legend id="b9d6123713d64ef695c1c415675f4292_2_1-legend" class="response-fieldset-legend field-group-hd">1. Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math> be any two sets. Which of the following is true about Cantor's Idea?</legend>
<div class="field">
<input type="checkbox" name="input_b9d6123713d64ef695c1c415675f4292_2_1[]" id="input_b9d6123713d64ef695c1c415675f4292_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="b9d6123713d64ef695c1c415675f4292_2_1-choice_0-label" for="input_b9d6123713d64ef695c1c415675f4292_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_b9d6123713d64ef695c1c415675f4292_2_1"> When <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> surj <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math>, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> is as big as <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_b9d6123713d64ef695c1c415675f4292_2_1[]" id="input_b9d6123713d64ef695c1c415675f4292_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="b9d6123713d64ef695c1c415675f4292_2_1-choice_1-label" for="input_b9d6123713d64ef695c1c415675f4292_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_b9d6123713d64ef695c1c415675f4292_2_1"> When <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> inj <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math>, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> is as small as <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_b9d6123713d64ef695c1c415675f4292_2_1[]" id="input_b9d6123713d64ef695c1c415675f4292_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="b9d6123713d64ef695c1c415675f4292_2_1-choice_2-label" for="input_b9d6123713d64ef695c1c415675f4292_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_b9d6123713d64ef695c1c415675f4292_2_1"> When <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> bij <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math>, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> is the same size as <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_b9d6123713d64ef695c1c415675f4292_2_1[]" id="input_b9d6123713d64ef695c1c415675f4292_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="b9d6123713d64ef695c1c415675f4292_2_1-choice_3-label" for="input_b9d6123713d64ef695c1c415675f4292_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_b9d6123713d64ef695c1c415675f4292_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mtext>pow</mtext>
<mo stretchy="false">(</mo>
<mi mathvariant="double-struck">N</mi>
<mo stretchy="false">)</mo>
</math> bij <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<msup>
<mo fence="false" stretchy="false">}</mo>
<mi>&#969;<!-- &#969; --></mi>
</msup>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_b9d6123713d64ef695c1c415675f4292_2_1[]" id="input_b9d6123713d64ef695c1c415675f4292_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="b9d6123713d64ef695c1c415675f4292_2_1-choice_4-label" for="input_b9d6123713d64ef695c1c415675f4292_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_b9d6123713d64ef695c1c415675f4292_2_1"> Two infinite sets have the same size when there is a bijection between them.
</label>
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<span id="answer_b9d6123713d64ef695c1c415675f4292_2_1"/>
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<span class="status unanswered" id="status_b9d6123713d64ef695c1c415675f4292_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<legend id="b9d6123713d64ef695c1c415675f4292_3_1-legend" class="response-fieldset-legend field-group-hd">2. Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math> be any two sets. What does the Schroeder-Bernstein Theorem tell us?</legend>
<div class="field">
<input type="checkbox" name="input_b9d6123713d64ef695c1c415675f4292_3_1[]" id="input_b9d6123713d64ef695c1c415675f4292_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="b9d6123713d64ef695c1c415675f4292_3_1-choice_0-label" for="input_b9d6123713d64ef695c1c415675f4292_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_b9d6123713d64ef695c1c415675f4292_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> surj <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math> surj <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> IMPLIES <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> bij <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_b9d6123713d64ef695c1c415675f4292_3_1[]" id="input_b9d6123713d64ef695c1c415675f4292_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="b9d6123713d64ef695c1c415675f4292_3_1-choice_1-label" for="input_b9d6123713d64ef695c1c415675f4292_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_b9d6123713d64ef695c1c415675f4292_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo stretchy="false">|</mo>
<mi>A</mi>
<mo stretchy="false">|</mo>
<mo>&#8805;<!-- &#8805; --></mo>
<mo stretchy="false">|</mo>
<mi>B</mi>
<mo stretchy="false">|</mo>
<mo>&#8805;<!-- &#8805; --></mo>
<mo stretchy="false">|</mo>
<mi>A</mi>
<mo stretchy="false">|</mo>
</math> IMPLIES <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo stretchy="false">|</mo>
<mi>A</mi>
<mo stretchy="false">|</mo>
<mo>=</mo>
<mo stretchy="false">|</mo>
<mi>B</mi>
<mo stretchy="false">|</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_b9d6123713d64ef695c1c415675f4292_3_1[]" id="input_b9d6123713d64ef695c1c415675f4292_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="b9d6123713d64ef695c1c415675f4292_3_1-choice_2-label" for="input_b9d6123713d64ef695c1c415675f4292_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_b9d6123713d64ef695c1c415675f4292_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> surj <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math> inj <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> IMPLIES <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> bij <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_b9d6123713d64ef695c1c415675f4292_3_1[]" id="input_b9d6123713d64ef695c1c415675f4292_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="b9d6123713d64ef695c1c415675f4292_3_1-choice_3-label" for="input_b9d6123713d64ef695c1c415675f4292_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_b9d6123713d64ef695c1c415675f4292_3_1"> None of the above
</label>
</div>
<span id="answer_b9d6123713d64ef695c1c415675f4292_3_1"/>
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</span>
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<mo>&#8746;<!-- &#8746; --></mo>
<mo fence="false" stretchy="false">{</mo>
<mi>d</mi>
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<h2 class="hd hd-2 unit-title">Lecture Video | Countable Sets</h2>
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<h2 class="hd hd-2 unit-title">Lecture Video | Cantor's Theorem</h2>
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Cantor&#39;s Diagonal Argument
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input ">
<label class="problem-group-label" for="input_cf179e27cec64357965ba9fbe1c5c45b_2_1" id="label_cf179e27cec64357965ba9fbe1c5c45b_2_1">1. The Diagonal Argument was used to illustrate that the set of infinite binary sequences, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
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<mo fence="false" stretchy="false">}</mo>
<mi>&#969;<!-- &#969; --></mi>
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<option value="option_cf179e27cec64357965ba9fbe1c5c45b_2_1_dummy_default">Select an option</option>
<option value="countable"> countable</option>
<option value="uncountable"> uncountable</option>
<option value="neither countable nor uncountable"> neither countable nor uncountable</option>
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<label class="problem-group-label" for="input_cf179e27cec64357965ba9fbe1c5c45b_3_1" id="label_cf179e27cec64357965ba9fbe1c5c45b_3_1">by building a____</label>
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<option value="option_cf179e27cec64357965ba9fbe1c5c45b_3_1_dummy_default">Select an option</option>
<option value="finite binary sequence"> finite binary sequence</option>
<option value="infinite binary sequence"> infinite binary sequence</option>
<option value="Either of the above"> Either of the above</option>
<option value="Neither of the above"> Neither of the above</option>
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<option value="option_cf179e27cec64357965ba9fbe1c5c45b_4_1_dummy_default">Select an option</option>
<option value="in"> in</option>
<option value="not in"> not in</option>
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<label class="problem-group-label" for="input_cf179e27cec64357965ba9fbe1c5c45b_5_1" id="label_cf179e27cec64357965ba9fbe1c5c45b_5_1">2. The set of natural numbers is strictly smaller than the set, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> of infinite binary strings (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="double-struck">N</mi>
</math> strict<math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> ), because _____.</label>
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<option value="option_cf179e27cec64357965ba9fbe1c5c45b_5_1_dummy_default">Select an option</option>
<option value="there is no surjection from the set of natural numbers to A"> there is no surjection from the set of natural numbers to A</option>
<option value="there is a bijection from the set of natural numbers to A"> there is a bijection from the set of natural numbers to A</option>
<option value="the statement is false! The natural numbers are as large as A"> the statement is false! The natural numbers are as large as A</option>
<option value="there is no injective relation from the set of natural numbers to A"> there is no injective relation from the set of natural numbers to A</option>
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<mi>A</mi>
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<mi>A</mi>
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<option value="option_cf179e27cec64357965ba9fbe1c5c45b_6_1_dummy_default">Select an option</option>
<option value="A inj"> A inj</option>
<option value="A bij"> A bij</option>
<option value="A surj"> A surj</option>
<option value="A strict"> A strict</option>
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<h2 class="hd hd-2 unit-title">Exercise | Countable and Uncountable Sets</h2>
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Countable and Uncountable Sets
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@e50907f5dabb423dac46953a43427dea-problem-progress"></div>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p> Which of the following sets are countable?</p>
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<input type="checkbox" name="input_e50907f5dabb423dac46953a43427dea_2_1[]" id="input_e50907f5dabb423dac46953a43427dea_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="e50907f5dabb423dac46953a43427dea_2_1-choice_0-label" for="input_e50907f5dabb423dac46953a43427dea_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_e50907f5dabb423dac46953a43427dea_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="double-struck">N</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_e50907f5dabb423dac46953a43427dea_2_1[]" id="input_e50907f5dabb423dac46953a43427dea_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="e50907f5dabb423dac46953a43427dea_2_1-choice_1-label" for="input_e50907f5dabb423dac46953a43427dea_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_e50907f5dabb423dac46953a43427dea_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="double-struck">Z</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_e50907f5dabb423dac46953a43427dea_2_1[]" id="input_e50907f5dabb423dac46953a43427dea_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="e50907f5dabb423dac46953a43427dea_2_1-choice_2-label" for="input_e50907f5dabb423dac46953a43427dea_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_e50907f5dabb423dac46953a43427dea_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="double-struck">N</mi>
<mo>&#215;<!-- &#215; --></mo>
<mi mathvariant="double-struck">N</mi>
</math>
</label>
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<div class="field">
<input type="checkbox" name="input_e50907f5dabb423dac46953a43427dea_2_1[]" id="input_e50907f5dabb423dac46953a43427dea_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="e50907f5dabb423dac46953a43427dea_2_1-choice_3-label" for="input_e50907f5dabb423dac46953a43427dea_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_e50907f5dabb423dac46953a43427dea_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="double-struck">Q</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_e50907f5dabb423dac46953a43427dea_2_1[]" id="input_e50907f5dabb423dac46953a43427dea_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="e50907f5dabb423dac46953a43427dea_2_1-choice_4-label" for="input_e50907f5dabb423dac46953a43427dea_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_e50907f5dabb423dac46953a43427dea_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi mathvariant="double-struck">Z</mi>
<mo>+</mo>
</msup>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_e50907f5dabb423dac46953a43427dea_2_1[]" id="input_e50907f5dabb423dac46953a43427dea_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="e50907f5dabb423dac46953a43427dea_2_1-choice_5-label" for="input_e50907f5dabb423dac46953a43427dea_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_e50907f5dabb423dac46953a43427dea_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="double-struck">R</mi>
</math>
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<input type="checkbox" name="input_e50907f5dabb423dac46953a43427dea_2_1[]" id="input_e50907f5dabb423dac46953a43427dea_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="e50907f5dabb423dac46953a43427dea_2_1-choice_6-label" for="input_e50907f5dabb423dac46953a43427dea_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_e50907f5dabb423dac46953a43427dea_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="double-struck">C</mi>
</math>
</label>
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<div class="field">
<input type="checkbox" name="input_e50907f5dabb423dac46953a43427dea_2_1[]" id="input_e50907f5dabb423dac46953a43427dea_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="e50907f5dabb423dac46953a43427dea_2_1-choice_7-label" for="input_e50907f5dabb423dac46953a43427dea_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_e50907f5dabb423dac46953a43427dea_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
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<mn>10</mn>
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<msup>
<mn>10</mn>
<mn>10</mn>
</msup>
</math> bit-strings
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<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
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<mi>&#969;<!-- &#969; --></mi>
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<input type="checkbox" name="input_e50907f5dabb423dac46953a43427dea_2_1[]" id="input_e50907f5dabb423dac46953a43427dea_2_1_choice_9" class="field-input input-checkbox" value="choice_9"/><label id="e50907f5dabb423dac46953a43427dea_2_1-choice_9-label" for="input_e50907f5dabb423dac46953a43427dea_2_1_choice_9" class="response-label field-label label-inline" aria-describedby="status_e50907f5dabb423dac46953a43427dea_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi mathvariant="double-struck">Q</mi>
<mi>&#969;<!-- &#969; --></mi>
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</math>, the infinite sequences of rationals
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Halting Problem Basics [Optional]
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<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
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<h2 class="hd hd-2 unit-title">Exercise | Russell's Paradox and ZFC (Optional)</h2>
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Russell&#39;s Paradox [and ZFC optional]
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<legend id="e8fc93de37204f2389363f776d609d18_2_1-legend" class="response-fieldset-legend field-group-hd">1. Russell's Paradox</legend>
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<input type="checkbox" name="input_e8fc93de37204f2389363f776d609d18_2_1[]" id="input_e8fc93de37204f2389363f776d609d18_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="e8fc93de37204f2389363f776d609d18_2_1-choice_0-label" for="input_e8fc93de37204f2389363f776d609d18_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_e8fc93de37204f2389363f776d609d18_2_1"> is "buggy" - the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>W</mi>
</math> is a set is not mathematically well-defined.
</label>
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<input type="checkbox" name="input_e8fc93de37204f2389363f776d609d18_2_1[]" id="input_e8fc93de37204f2389363f776d609d18_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="e8fc93de37204f2389363f776d609d18_2_1-choice_1-label" for="input_e8fc93de37204f2389363f776d609d18_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_e8fc93de37204f2389363f776d609d18_2_1"> is "buggy" - it assumes that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>W</mi>
</math> is a set, so it only holds when <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>W</mi>
</math> is considered a set.
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<input type="checkbox" name="input_e8fc93de37204f2389363f776d609d18_2_1[]" id="input_e8fc93de37204f2389363f776d609d18_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="e8fc93de37204f2389363f776d609d18_2_1-choice_3-label" for="input_e8fc93de37204f2389363f776d609d18_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_e8fc93de37204f2389363f776d609d18_2_1"> is confirmed by Zermelo-Frankel with Choice (ZFC).
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<legend id="e8fc93de37204f2389363f776d609d18_3_1-legend" class="response-fieldset-legend field-group-hd">2. Zermelo-Frankel with Choice axioms (ZFC)</legend>
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<input type="checkbox" name="input_e8fc93de37204f2389363f776d609d18_3_1[]" id="input_e8fc93de37204f2389363f776d609d18_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="e8fc93de37204f2389363f776d609d18_3_1-choice_1-label" for="input_e8fc93de37204f2389363f776d609d18_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_e8fc93de37204f2389363f776d609d18_3_1"> are written in predicate logic.
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<input type="checkbox" name="input_e8fc93de37204f2389363f776d609d18_3_1[]" id="input_e8fc93de37204f2389363f776d609d18_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="e8fc93de37204f2389363f776d609d18_3_1-choice_2-label" for="input_e8fc93de37204f2389363f776d609d18_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_e8fc93de37204f2389363f776d609d18_3_1"> include the equality axiom and power set axiom.
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<input type="checkbox" name="input_e8fc93de37204f2389363f776d609d18_3_1[]" id="input_e8fc93de37204f2389363f776d609d18_3_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="e8fc93de37204f2389363f776d609d18_3_1-choice_4-label" for="input_e8fc93de37204f2389363f776d609d18_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_e8fc93de37204f2389363f776d609d18_3_1"> fix Russell's Paradox with the foundation axiom: a set cannot be a member of itself, directly or indirectly.
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<input type="checkbox" name="input_e8fc93de37204f2389363f776d609d18_3_1[]" id="input_e8fc93de37204f2389363f776d609d18_3_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="e8fc93de37204f2389363f776d609d18_3_1-choice_5-label" for="input_e8fc93de37204f2389363f776d609d18_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_e8fc93de37204f2389363f776d609d18_3_1"> imply, by the foundation axiom, that a collection of all sets is a set.
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<input type="checkbox" name="input_e8fc93de37204f2389363f776d609d18_3_1[]" id="input_e8fc93de37204f2389363f776d609d18_3_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="e8fc93de37204f2389363f776d609d18_3_1-choice_6-label" for="input_e8fc93de37204f2389363f776d609d18_3_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_e8fc93de37204f2389363f776d609d18_3_1"> imply, by the foundation axiom, that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>W</mi>
</math> is not a set, since it equals the collection of all sets.
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<p>Download a copy of the slides for <a href="/assets/courseware/v1/359f18cc4c914475abbc92517e37b19e/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_theroyaxioms.pdf" target="[object Object]">Set Theory Axioms (PDF)</a></p>
<p><a href="/assets/courseware/v1/2336b88813d24be3d941514611a1c62e/asset-v1:OCW+6.042J+2T2019+type@asset+block/SetTheoryAxioms_1.11_Lectrans.pdf" target="[object Object]">Lecture video transcript (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Exercise | Set Theory Axioms Problem (Optional)</h2>
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<h3 class="hd hd-3 problem-header" id="8ca2211687444148b4062a2fa7dda7dc-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@8ca2211687444148b4062a2fa7dda7dc-problem-progress" tabindex="-1">
Set Theory Axioms [Optional] Set 1
</h3>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>1. Match the name of the set theory axiom to its corresponding statement. Try not to look back at the slides!</p>
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<legend id="8ca2211687444148b4062a2fa7dda7dc_2_1-legend" class="response-fieldset-legend field-group-hd">1. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#8704;<!-- &#8704; --></mi>
<mi>x</mi>
<mi mathvariant="normal">&#8707;<!-- &#8707; --></mi>
<mi>p</mi>
<mi mathvariant="normal">&#8704;<!-- &#8704; --></mi>
<mi>s</mi>
<mo>.</mo>
<mi>s</mi>
<mo>&#8838;<!-- &#8838; --></mo>
<mi>x</mi>
</math> IFF <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>s</mi>
<mo>&#8712;<!-- &#8712; --></mo>
<mi>p</mi>
</math></legend>
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<div class="field">
<input type="radio" name="input_8ca2211687444148b4062a2fa7dda7dc_2_1" id="input_8ca2211687444148b4062a2fa7dda7dc_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="8ca2211687444148b4062a2fa7dda7dc_2_1-choice_1-label" for="input_8ca2211687444148b4062a2fa7dda7dc_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8ca2211687444148b4062a2fa7dda7dc_2_1"> Extensionality
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</label>
</div>
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<input type="radio" name="input_8ca2211687444148b4062a2fa7dda7dc_2_1" id="input_8ca2211687444148b4062a2fa7dda7dc_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="8ca2211687444148b4062a2fa7dda7dc_2_1-choice_3-label" for="input_8ca2211687444148b4062a2fa7dda7dc_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_8ca2211687444148b4062a2fa7dda7dc_2_1"> Comprehension
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<span id="answer_8ca2211687444148b4062a2fa7dda7dc_2_1"/>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8ca2211687444148b4062a2fa7dda7dc_3_1">
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<legend id="8ca2211687444148b4062a2fa7dda7dc_3_1-legend" class="response-fieldset-legend field-group-hd">2. If <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>S</mi>
</math> is a set and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math> is a predicate, then <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo fence="false" stretchy="false">{</mo>
<mi>x</mi>
<mo>&#8712;<!-- &#8712; --></mo>
<mi>S</mi>
<mo>&#8739;<!-- &#8739; --></mo>
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">}</mo>
</math> is a set.</legend>
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</label>
</div>
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<input type="radio" name="input_8ca2211687444148b4062a2fa7dda7dc_3_1" id="input_8ca2211687444148b4062a2fa7dda7dc_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="8ca2211687444148b4062a2fa7dda7dc_3_1-choice_1-label" for="input_8ca2211687444148b4062a2fa7dda7dc_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8ca2211687444148b4062a2fa7dda7dc_3_1"> Extensionality
</label>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8ca2211687444148b4062a2fa7dda7dc_4_1">
<fieldset aria-describedby="status_8ca2211687444148b4062a2fa7dda7dc_4_1">
<legend id="8ca2211687444148b4062a2fa7dda7dc_4_1-legend" class="response-fieldset-legend field-group-hd">3.<math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#8704;<!-- &#8704; --></mi>
<mi>x</mi>
<mo>.</mo>
<mo stretchy="false">[</mo>
<mi>x</mi>
<mo>&#8800;<!-- &#8800; --></mo>
<mi mathvariant="normal">&#8709;<!-- &#8709; --></mi>
</math> IMPLIES <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#8707;<!-- &#8707; --></mi>
<mi>y</mi>
<mo>.</mo>
<mi>y</mi>
</math> is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8712;<!-- &#8712; --></mo>
</math>-minimal in <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>x</mi>
</math> ]</legend>
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</label>
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<input type="radio" name="input_8ca2211687444148b4062a2fa7dda7dc_4_1" id="input_8ca2211687444148b4062a2fa7dda7dc_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="8ca2211687444148b4062a2fa7dda7dc_4_1-choice_1-label" for="input_8ca2211687444148b4062a2fa7dda7dc_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8ca2211687444148b4062a2fa7dda7dc_4_1"> Extensionality
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<span class="status unanswered" id="status_8ca2211687444148b4062a2fa7dda7dc_4_1" data-tooltip="Not yet answered.">
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8ca2211687444148b4062a2fa7dda7dc_5_1">
<fieldset aria-describedby="status_8ca2211687444148b4062a2fa7dda7dc_5_1">
<legend id="8ca2211687444148b4062a2fa7dda7dc_5_1-legend" class="response-fieldset-legend field-group-hd">4.<math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#8704;<!-- &#8704; --></mi>
<mi>x</mi>
<mo stretchy="false">[</mo>
<mi>x</mi>
<mo>&#8712;<!-- &#8712; --></mo>
<mi>y</mi>
</math> IFF <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>x</mi>
<mo>&#8712;<!-- &#8712; --></mo>
<mi>z</mi>
<mo stretchy="false">]</mo>
</math> IFF <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#8704;<!-- &#8704; --></mi>
<mi>x</mi>
<mo stretchy="false">[</mo>
<mi>y</mi>
<mo>&#8712;<!-- &#8712; --></mo>
<mi>x</mi>
</math> IFF <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>z</mi>
<mo>&#8712;<!-- &#8712; --></mo>
<mi>x</mi>
<mo stretchy="false">]</mo>
</math></legend>
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<input type="radio" name="input_8ca2211687444148b4062a2fa7dda7dc_5_1" id="input_8ca2211687444148b4062a2fa7dda7dc_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="8ca2211687444148b4062a2fa7dda7dc_5_1-choice_1-label" for="input_8ca2211687444148b4062a2fa7dda7dc_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8ca2211687444148b4062a2fa7dda7dc_5_1"> Extensionality
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aria-labelledby="8062a1f7602b4ddbbc0ca1b22f7d6c47-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@8062a1f7602b4ddbbc0ca1b22f7d6c47" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@8062a1f7602b4ddbbc0ca1b22f7d6c47/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="8062a1f7602b4ddbbc0ca1b22f7d6c47-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@8062a1f7602b4ddbbc0ca1b22f7d6c47-problem-progress" tabindex="-1">
Questions 1 - 7
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@8062a1f7602b4ddbbc0ca1b22f7d6c47-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1">
<fieldset aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1">
<legend id="8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1-legend" class="response-fieldset-legend field-group-hd">Suppose the Foundation Axiom did not hold and there are sets <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>T</mi>
<mo>,</mo>
<mi>U</mi>
<mo>,</mo>
<mi>V</mi>
</math> such that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>T</mi>
<mo>&#8712;<!-- &#8712; --></mo>
<mi>U</mi>
<mo>&#8712;<!-- &#8712; --></mo>
<mi>V</mi>
<mo>&#8712;<!-- &#8712; --></mo>
<mi>T</mi>
</math>, and there are no other membership relations among them.
<br/>
<br/>
For each of the following sets, indicate whether it has an <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8712;<!-- &#8712; --></mo>
</math>-minimal element.
<br/>
<br/>
1. The empty set <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#8709;<!-- &#8709; --></mi>
</math></legend>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1-choice_0-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1"> Yes
</label>
</div>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1-choice_1-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1"> No
</label>
</div>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1-choice_2-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1"> Maybe, but not necessarily
</label>
</div>
<span id="answer_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1">
<fieldset aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1">
<legend id="8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1-legend" class="response-fieldset-legend field-group-hd">2. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>T</mi>
</math></legend>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1-choice_0-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1"> Yes
</label>
</div>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1-choice_1-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1"> No
</label>
</div>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1-choice_2-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1"> Maybe, but not necessarily
</label>
</div>
<span id="answer_8062a1f7602b4ddbbc0ca1b22f7d6c47_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_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="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1">
<fieldset aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1">
<legend id="8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1-legend" class="response-fieldset-legend field-group-hd">3.<math xmlns="http://www.w3.org/1998/Math/MathML">
<mo fence="false" stretchy="false">{</mo>
<mi>T</mi>
<mo fence="false" stretchy="false">}</mo>
</math></legend>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1-choice_0-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1"> Yes
</label>
</div>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1-choice_1-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1"> No
</label>
</div>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1-choice_2-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1"> Maybe, but not necessarily
</label>
</div>
<span id="answer_8062a1f7602b4ddbbc0ca1b22f7d6c47_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_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="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1">
<fieldset aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1">
<legend id="8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1-legend" class="response-fieldset-legend field-group-hd">4. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo fence="false" stretchy="false">{</mo>
<mi>V</mi>
<mo>,</mo>
<mi>T</mi>
<mo fence="false" stretchy="false">}</mo>
</math></legend>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1-choice_0-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1"> Yes
</label>
</div>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1-choice_1-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1"> No
</label>
</div>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1-choice_2-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1"> Maybe, but not necessarily
</label>
</div>
<span id="answer_8062a1f7602b4ddbbc0ca1b22f7d6c47_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_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="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1">
<fieldset aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1">
<legend id="8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1-legend" class="response-fieldset-legend field-group-hd">5. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo fence="false" stretchy="false">{</mo>
<mi>U</mi>
<mo>,</mo>
<mi>V</mi>
<mo>,</mo>
<mi>T</mi>
<mo fence="false" stretchy="false">}</mo>
</math></legend>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1-choice_0-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1"> Yes
</label>
</div>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1-choice_1-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1"> No
</label>
</div>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1-choice_2-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1"> Maybe, but not necessarily
</label>
</div>
<span id="answer_8062a1f7602b4ddbbc0ca1b22f7d6c47_6_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_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="wrapper-problem-response" tabindex="-1" aria-label="Question 6" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1">
<fieldset aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1">
<legend id="8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1-legend" class="response-fieldset-legend field-group-hd">6. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo fence="false" stretchy="false">{</mo>
<mi>T</mi>
<mo>,</mo>
<mi>U</mi>
<mo>,</mo>
<mi>V</mi>
<mo fence="false" stretchy="false">}</mo>
</math></legend>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1-choice_0-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1"> Yes
</label>
</div>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1-choice_1-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1"> No
</label>
</div>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1-choice_2-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1"> Maybe, but not necessarily
</label>
</div>
<span id="answer_8062a1f7602b4ddbbc0ca1b22f7d6c47_7_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_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="wrapper-problem-response" tabindex="-1" aria-label="Question 7" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8062a1f7602b4ddbbc0ca1b22f7d6c47_8_1">
<fieldset aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_8_1">
<legend id="8062a1f7602b4ddbbc0ca1b22f7d6c47_8_1-legend" class="response-fieldset-legend field-group-hd">7. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo fence="false" stretchy="false">{</mo>
<mi mathvariant="normal">&#8709;<!-- &#8709; --></mi>
<mo>,</mo>
<mi>T</mi>
<mo>,</mo>
<mi>U</mi>
<mo>,</mo>
<mi>V</mi>
<mo fence="false" stretchy="false">}</mo>
</math></legend>
<div class="field">
<input type="radio" name="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_8_1" id="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_8_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="8062a1f7602b4ddbbc0ca1b22f7d6c47_8_1-choice_0-label" for="input_8062a1f7602b4ddbbc0ca1b22f7d6c47_8_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_8062a1f7602b4ddbbc0ca1b22f7d6c47_8_1"> Yes
</label>
</div>
<div class="field">
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<span class="icon fa fa-save" aria-hidden="true"></span>
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<span class="notification-message" aria-describedby="8062a1f7602b4ddbbc0ca1b22f7d6c47-problem-title">Answers are displayed within the problem
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<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
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"
data-graded="False">
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<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
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