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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>
</div>
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<h3 class="hd hd-3 problem-header" id="ca3732ea2be3483a8317cdf0dd3e578b-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@ca3732ea2be3483a8317cdf0dd3e578b-problem-progress" tabindex="-1">
Problem 1
</h3>
<div class="problem-progress" id="block-v1:MITx+24.118x+2T2020+type@problem+block@ca3732ea2be3483a8317cdf0dd3e578b-problem-progress"></div>
<div class="problem">
<div>
<p>Or each of the follow sets of real numbers, determine whether it is a Borel Set.</p>
<p>If it <i>is</i> a Borel Set, specify its Lebesgue Measure.</p>
<p>And if it is <i>not</i> a Borel Set, enter '\(0\)' into the numeric entry field.</p>
<p>(You may avail yourself of the fact that every Borel Set has a Lebesgue Measure.)</p>
<p>As usual, \([a,b] = \{x : a \leq x \leq b\}\) and \([a,b) = \{x : a \leq x &lt; b\}\).</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p><br/><br/>\( [\frac{1}{4}, \frac{5}{6}] \)</p>
<div class="choicegroup capa_inputtype" id="inputtype_ca3732ea2be3483a8317cdf0dd3e578b_2_1">
<fieldset aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_2_1">
<div class="field">
<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_2_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_2_1-choice_0-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_2_1"> Borel Set
</label>
</div>
<div class="field">
<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_2_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_2_1-choice_1-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_2_1"> Not a Borel Set
</label>
</div>
<span id="answer_ca3732ea2be3483a8317cdf0dd3e578b_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ca3732ea2be3483a8317cdf0dd3e578b_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 id="formulaequationinput_ca3732ea2be3483a8317cdf0dd3e578b_3_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_ca3732ea2be3483a8317cdf0dd3e578b_3_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_3_1" data-input-id="ca3732ea2be3483a8317cdf0dd3e578b_3_1" value="" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_ca3732ea2be3483a8317cdf0dd3e578b_3_1"/>
<span class="status unanswered" id="status_ca3732ea2be3483a8317cdf0dd3e578b_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ca3732ea2be3483a8317cdf0dd3e578b_3_1" class="answer"/>
<div id="input_ca3732ea2be3483a8317cdf0dd3e578b_3_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<div class="solution-span">
<span id="solution_ca3732ea2be3483a8317cdf0dd3e578b_solution_1"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><p><br/><br/>\( \{\frac{5}{6}\} \)</p>
<div class="choicegroup capa_inputtype" id="inputtype_ca3732ea2be3483a8317cdf0dd3e578b_4_1">
<fieldset aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_4_1">
<div class="field">
<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_4_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_4_1-choice_0-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_4_1"> Borel Set
</label>
</div>
<div class="field">
<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_4_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_4_1-choice_1-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_4_1"> Not a Borel Set
</label>
</div>
<span id="answer_ca3732ea2be3483a8317cdf0dd3e578b_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ca3732ea2be3483a8317cdf0dd3e578b_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 id="formulaequationinput_ca3732ea2be3483a8317cdf0dd3e578b_5_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_ca3732ea2be3483a8317cdf0dd3e578b_5_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_5_1" data-input-id="ca3732ea2be3483a8317cdf0dd3e578b_5_1" value="" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_5_1" size="20"/>
<span class="trailing_text" id="trailing_text_ca3732ea2be3483a8317cdf0dd3e578b_5_1"/>
<span class="status unanswered" id="status_ca3732ea2be3483a8317cdf0dd3e578b_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ca3732ea2be3483a8317cdf0dd3e578b_5_1" class="answer"/>
<div id="input_ca3732ea2be3483a8317cdf0dd3e578b_5_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<div class="solution-span">
<span id="solution_ca3732ea2be3483a8317cdf0dd3e578b_solution_2"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><p><br/><br/>\( [\frac{1}{4},\frac{5}{6}) \)</p>
<div class="choicegroup capa_inputtype" id="inputtype_ca3732ea2be3483a8317cdf0dd3e578b_6_1">
<fieldset aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_6_1">
<div class="field">
<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_6_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_6_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_6_1-choice_0-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_6_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_6_1"> Borel Set
</label>
</div>
<div class="field">
<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_6_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_6_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_6_1-choice_1-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_6_1"> Not a Borel Set
</label>
</div>
<span id="answer_ca3732ea2be3483a8317cdf0dd3e578b_6_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ca3732ea2be3483a8317cdf0dd3e578b_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 id="formulaequationinput_ca3732ea2be3483a8317cdf0dd3e578b_7_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_ca3732ea2be3483a8317cdf0dd3e578b_7_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_7_1" data-input-id="ca3732ea2be3483a8317cdf0dd3e578b_7_1" value="" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_7_1" size="20"/>
<span class="trailing_text" id="trailing_text_ca3732ea2be3483a8317cdf0dd3e578b_7_1"/>
<span class="status unanswered" id="status_ca3732ea2be3483a8317cdf0dd3e578b_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ca3732ea2be3483a8317cdf0dd3e578b_7_1" class="answer"/>
<div id="input_ca3732ea2be3483a8317cdf0dd3e578b_7_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<div class="solution-span">
<span id="solution_ca3732ea2be3483a8317cdf0dd3e578b_solution_3"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 7" role="group"><p><br/><br/>\( [\frac{1}{4},\frac{5}{6}] - [\frac{1}{3},\frac{1}{2}] \)</p>
<div class="choicegroup capa_inputtype" id="inputtype_ca3732ea2be3483a8317cdf0dd3e578b_8_1">
<fieldset aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_8_1">
<div class="field">
<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_8_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_8_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_8_1-choice_0-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_8_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_8_1"> Borel Set
</label>
</div>
<div class="field">
<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_8_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_8_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_8_1-choice_1-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_8_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_8_1"> Not a Borel Set
</label>
</div>
<span id="answer_ca3732ea2be3483a8317cdf0dd3e578b_8_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ca3732ea2be3483a8317cdf0dd3e578b_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="wrapper-problem-response" tabindex="-1" aria-label="Question 8" role="group"><div id="formulaequationinput_ca3732ea2be3483a8317cdf0dd3e578b_9_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_ca3732ea2be3483a8317cdf0dd3e578b_9_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_9_1" data-input-id="ca3732ea2be3483a8317cdf0dd3e578b_9_1" value="" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_9_1" size="20"/>
<span class="trailing_text" id="trailing_text_ca3732ea2be3483a8317cdf0dd3e578b_9_1"/>
<span class="status unanswered" id="status_ca3732ea2be3483a8317cdf0dd3e578b_9_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ca3732ea2be3483a8317cdf0dd3e578b_9_1" class="answer"/>
<div id="input_ca3732ea2be3483a8317cdf0dd3e578b_9_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<div class="solution-span">
<span id="solution_ca3732ea2be3483a8317cdf0dd3e578b_solution_4"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 9" role="group"><p><br/><br/>\( [\frac{1}{4},\frac{5}{6}] - \{\frac{1}{3},\frac{1}{2}\} \)</p>
<div class="choicegroup capa_inputtype" id="inputtype_ca3732ea2be3483a8317cdf0dd3e578b_10_1">
<fieldset aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_10_1">
<div class="field">
<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_10_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_10_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_10_1-choice_0-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_10_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_10_1"> Borel Set
</label>
</div>
<div class="field">
<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_10_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_10_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_10_1-choice_1-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_10_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_10_1"> Not a Borel Set
</label>
</div>
<span id="answer_ca3732ea2be3483a8317cdf0dd3e578b_10_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ca3732ea2be3483a8317cdf0dd3e578b_10_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 10" role="group"><div id="formulaequationinput_ca3732ea2be3483a8317cdf0dd3e578b_11_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_ca3732ea2be3483a8317cdf0dd3e578b_11_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_11_1" data-input-id="ca3732ea2be3483a8317cdf0dd3e578b_11_1" value="" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_11_1" size="20"/>
<span class="trailing_text" id="trailing_text_ca3732ea2be3483a8317cdf0dd3e578b_11_1"/>
<span class="status unanswered" id="status_ca3732ea2be3483a8317cdf0dd3e578b_11_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ca3732ea2be3483a8317cdf0dd3e578b_11_1" class="answer"/>
<div id="input_ca3732ea2be3483a8317cdf0dd3e578b_11_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<div class="solution-span">
<span id="solution_ca3732ea2be3483a8317cdf0dd3e578b_solution_5"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 11" role="group"><p><br/><br/>\( \{\frac{1}{2^1},\frac{1}{2^2},\frac{1}{2^3}, ...\} \)</p>
<div class="choicegroup capa_inputtype" id="inputtype_ca3732ea2be3483a8317cdf0dd3e578b_12_1">
<fieldset aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_12_1">
<div class="field">
<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_12_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_12_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_12_1-choice_0-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_12_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_12_1"> Borel Set
</label>
</div>
<div class="field">
<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_12_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_12_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_12_1-choice_1-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_12_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_12_1"> Not a Borel Set
</label>
</div>
<span id="answer_ca3732ea2be3483a8317cdf0dd3e578b_12_1"/>
</fieldset>
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<div class="unanswered">
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\(\)
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<span id="solution_ca3732ea2be3483a8317cdf0dd3e578b_solution_6"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 13" role="group"><p><br/><br/>\( [\frac{4}{5},\frac{5}{6}] \cup \{\frac{1}{2^1},\frac{1}{2^2},\frac{1}{2^3}, ...\} \)</p>
<div class="choicegroup capa_inputtype" id="inputtype_ca3732ea2be3483a8317cdf0dd3e578b_14_1">
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<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_14_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_14_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_14_1-choice_0-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_14_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_14_1"> Borel Set
</label>
</div>
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<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_14_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_14_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_14_1-choice_1-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_14_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_14_1"> Not a Borel Set
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<span id="solution_ca3732ea2be3483a8317cdf0dd3e578b_solution_7"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 15" role="group"><p><br/><br/>\( [0,\frac{1}{2}] - \{\frac{1}{2^1},\frac{1}{2^2},\frac{1}{2^3}, ...\} \)</p>
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</label>
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<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_16_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_16_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_16_1-choice_1-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_16_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_16_1"> Not a Borel Set
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<span id="solution_ca3732ea2be3483a8317cdf0dd3e578b_solution_8"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 17" role="group"><p><br/><br/>a Vitali set</p>
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</label>
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<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_18_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_18_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_18_1-choice_1-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_18_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_18_1"> Not a Borel Set
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<span id="solution_ca3732ea2be3483a8317cdf0dd3e578b_solution_9"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 19" role="group"><p><br/><br/>the complement of a Vitali set</p>
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</label>
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<input type="radio" name="input_ca3732ea2be3483a8317cdf0dd3e578b_20_1" id="input_ca3732ea2be3483a8317cdf0dd3e578b_20_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ca3732ea2be3483a8317cdf0dd3e578b_20_1-choice_1-label" for="input_ca3732ea2be3483a8317cdf0dd3e578b_20_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ca3732ea2be3483a8317cdf0dd3e578b_20_1"> Not a Borel Set
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<h3 class="hd hd-3 problem-header" id="fa858269092e40dfb4964877f160b097-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@fa858269092e40dfb4964877f160b097-problem-progress" tabindex="-1">
Problem 2
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<p>Consider the following procedure for picking points at random from \([0, 1]\). We use <em>two</em> coins, coin \(O\) and coin \(E\). Coin \(O\) has a \(\frac{1}{5}\) chance of landing heads. Coin \(E\) has a \(\frac{1}{8}\) chance of landing heads. You then do as follows: </p>
<p style="padding-left:60px;">You toss a coin once for each natural number. For the first toss, you use coin \(O\). For the second toss, you use coin \(E\). For the third toss, you use coin \(O\). In general, you use coin \(O\) on the \(n\)-th toss if \(n\) is odd, and coin \(E\) on the \(n\)-th toss if \(n\) is even. Each time the coin lands Heads you write down a \(0\), and each time it lands Tails you write down a \(1\). Then you pick whichever number in \([0,1]\) has as its binary expansion '\(0\)' followed by a decimal point followed by the infinite sequence of zeroes and ones you got from your coin tosses.</p>
<p>Answer the following questions, inputting your answer as a whole number, or as a fraction of whole numbers.</p>
<p>Using this procedure, what is the probability of selecting a number in \([0, \frac{1}{2}]\)?</p>
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</div><p> Using this procedure, what is probability of selecting a number in the interval \((0, 1]\)? Note that \((0, 1] = [0,1]-\{0\}\).</p>
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</div><p> Using this procedure, what is probability of selecting a number greater than or equal to \(\frac{15}{16}\)?</p>
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Problem 3
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<p>Is the following claim true or false?</p>
<p style="padding-left:60px;">There is a definite probability that the <b>Standard Coin-Toss Procedure</b> from the "<a href="/courses/course-v1:MITx+24.118x+2T2020/jump_to_id/7dd6ba11bb8940e7a58d9178c01a02b0" target="[object Object]">Uniform and Non-Uniform Measures</a>" section will output a number in \([\frac{1}{2^{n+1}},\frac{1}{2^n})\), for \(1 \leq n\). </p>
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</div><p>Now suppose the set \(A\) is the union of finitely many intervals of the form \([\frac{1}{2^{n+1}},\frac{1}{2^n})\), for \(1 \leq n\).</p>
<p>Is the following claim true or false?</p>
<p style="padding-left:60px;">There is a definite probability that the <b>Standard Coin-Toss Procedure</b> from the "<a href="/courses/course-v1:MITx+24.118x+2T2020/jump_to_id/7dd6ba11bb8940e7a58d9178c01a02b0" target="[object Object]">Uniform and Non-Uniform Measures</a>" section will output a number in \(A\). (You may assume that the Standard Coin Toss Procedure delivers a probability function.) </p>
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<h3 class="hd hd-3 problem-header" id="b38ed6560b4640d6a44d6bf24c7e8dc4-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@b38ed6560b4640d6a44d6bf24c7e8dc4-problem-progress" tabindex="-1">
Problem 4
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<p>Let the \(A_1, A_2, A_3\ldots\) be subsets of \([0, 1]\) such that:</p>
<p style="padding-left:30px;"> \[A_1 = \{\frac{1}{2}\}\] </p>
<p style="padding-left:30px;"> \[A_2 = \{\frac{1}{4}, \frac{1}{8}\}\] </p>
<p style="padding-left:30px;"> \[A_3 = \{\frac{1}{16}, \frac{1}{32}, \frac{1}{64}, \frac{1}{128}\}\] </p>
<p style="padding-left:30px;"> \[... \textrm{etc.}\] </p>
<p>Suppose we are picking numbers from \([0,1]\) using the <b>Standard Coin-Toss Procedure</b> from the "<a href="/jump_to_7dd6ba11bb8940e7a58d9178c01a02b0" target="[object Object]">Uniform and Non-Uniform Measures</a>". Consider the probability of picking a number in the the union of the \(A_i\)s.:</p>
<p style="padding-left:30px;"> \[\bigcup_{i\in \mathbb{N}}(A_i)\] </p>
<p>Assuming <i>Countable Additivity</i>, what is the probability of picking a point in that union? (Enter your answer as a fraction)</p>
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Problem 5
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<p>Recall that a <i>choice set</i> for set \(A\) is a set containing exactly one element from each member of \(A\). And recall the <i>Axiom of Choice</i>:</p>
<p style="padding-left:60px;"><b>Axiom of Choice</b>: Any set of non-empty, non-overlapping sets has a choice set.</p>
<p>On a first reading, the Axiom of Choice is likely to sound trivial. This exercise is aimed at helping you understand why it is not. (It is a variant of an explanation given long ago by British philosopher Bertrand Russell.)</p>
<p>Here are two standard set-theoretic axioms:</p>
<p style="padding-left:60px;"><b>Union</b>: If a set \(A\) exists, then so does its union, \(\bigcup A\). (Recall that \(\bigcup A\) is the set \(\{x : x\) is a member of some element of \( A\} \) (i.e. the set of members of members of \(A\).)</p>
<p style="padding-left:60px;"><b>Separation</b>: Let \(\phi(x)\) be any formula of the form ''\(x\) is such and such'' (for instance, ''\(x\) is a natural number''). Then if set \(A\) exists, the following set also exists: \( \{x : x \in A \) and \( \phi(x)\}\) (i.e. the set of objects that are members of \(A\) and satisfy condition \(\phi(x)\)).</p>
<p>Here is an example of how Separation might be used. Let \(\phi(x)\) be the formula ''\(x\) is an octopus''. Separation entails that if the set \(A = \{x : \text{\(x\) is an animal}\}\) exists, so does
\( \{x : x \in A \text{ and $x$ is an octopus}\} \), which is the set of octopuses.</p>
<p>Now, let \(S\) be an infinite set, each member of which is a set of two shoes: a right shoe and a left shoe. Assume that no two elements of \(S\) have any shoes in common. Now suppose you'd like to have a choice set for \(S\). The Axiom of Choice guarantees that a choice set exists, but it doesn't give you much information about what it looks like. Let's see if we can do better than that. It follows from Union that \(\bigcup S\) exists.</p>
<p>Question: Is there an application of Separation to \(\bigcup S\) that delivers a choice set for \(S\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_209580bde9de423a93229560ae910e5d_2_1">
<fieldset aria-describedby="status_209580bde9de423a93229560ae910e5d_2_1">
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<input type="radio" name="input_209580bde9de423a93229560ae910e5d_2_1" id="input_209580bde9de423a93229560ae910e5d_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="209580bde9de423a93229560ae910e5d_2_1-choice_0-label" for="input_209580bde9de423a93229560ae910e5d_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_209580bde9de423a93229560ae910e5d_2_1"> Yes
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<div class="field">
<input type="radio" name="input_209580bde9de423a93229560ae910e5d_2_1" id="input_209580bde9de423a93229560ae910e5d_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="209580bde9de423a93229560ae910e5d_2_1-choice_1-label" for="input_209580bde9de423a93229560ae910e5d_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_209580bde9de423a93229560ae910e5d_2_1"> No
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<span id="answer_209580bde9de423a93229560ae910e5d_2_1"/>
</fieldset>
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</div><p>Next, let \(S\) be an infinite set, each member of which is a set of two socks. (Assume that all socks are alike, and, in particular, that there is no such thing as a ''right'' sock or a ''left'' sock.) Assume that no two elements of \(S\) have any socks in common. Now suppose you'd like to have a choice set for \(S\). The Axiom of Choice guarantees that a choice set exists, but it doesn't give you much information about what it looks like. Let's see if we can do better than that. It follows from Union that \(\bigcup S\) exists.</p>
<p>Question: Is there an application of Separation to \(\bigcup S\) that delivers a choice set for \(S\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_209580bde9de423a93229560ae910e5d_3_1">
<fieldset aria-describedby="status_209580bde9de423a93229560ae910e5d_3_1">
<div class="field">
<input type="radio" name="input_209580bde9de423a93229560ae910e5d_3_1" id="input_209580bde9de423a93229560ae910e5d_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="209580bde9de423a93229560ae910e5d_3_1-choice_0-label" for="input_209580bde9de423a93229560ae910e5d_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_209580bde9de423a93229560ae910e5d_3_1"> Yes
</label>
</div>
<div class="field">
<input type="radio" name="input_209580bde9de423a93229560ae910e5d_3_1" id="input_209580bde9de423a93229560ae910e5d_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="209580bde9de423a93229560ae910e5d_3_1-choice_1-label" for="input_209580bde9de423a93229560ae910e5d_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_209580bde9de423a93229560ae910e5d_3_1"> No
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<span id="answer_209580bde9de423a93229560ae910e5d_3_1"/>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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