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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p>Read <a href="/assets/courseware/v1/9664aa14064a3e9a58f60f23d06bb363/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session18.pdf" target="[object Object]">Chapter 9.5–9.11 (PDF)</a> of <em>Mathematics for Computer Science</em> for 2.7 Partial Orders and Equivalence.</p>
<p>View the <a href="/assets/courseware/v1/fd07cae350b7c5f7e4a745704b7f7845/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp18.pdf" target="[object Object]">Section 2.7 In-Class Questions (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Lecture Video | Partial Orders</h2>
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<p>Download a copy of the slides for <a href="/assets/courseware/v1/8792bd77462a6010d9c50785cb0e0717/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_PartialOrder.pdf" target="[object Object]">Partial Orders (PDF)<br /></a></p>
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<h2 class="hd hd-2 unit-title">Exercise | Population Partial Order</h2>
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Population Partial Order
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<label class="problem-group-label" for="input_3d21480bfbb2487e84ac0862e2f3abaf_2_1" id="label_3d21480bfbb2487e84ac0862e2f3abaf_2_1">In a population of 10 people, let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> be the "older than" relation and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>T</mi>
</math> be the "taller than" relation.
<br/>
<br/>
1. Which of the following properties guarantee that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> will be a linear order?</label>
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<option value="option_3d21480bfbb2487e84ac0862e2f3abaf_2_1_dummy_default">Select an option</option>
<option value="This is a unique oldest person"> This is a unique oldest person</option>
<option value="There are at most two people with the same age"> There are at most two people with the same age</option>
<option value="No two people are the same age"> No two people are the same age</option>
<option value="There is an age that no one has"> There is an age that no one has</option>
<option value="Some person appears twice in the list"> Some person appears twice in the list</option>
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<legend id="3d21480bfbb2487e84ac0862e2f3abaf_3_1-legend" class="response-fieldset-legend field-group-hd">2. Assume both <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>T</mi>
</math> are linear orders. Which properties are <strong>guaranteed</strong> to be true for the product relation <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
<mo>&#215;<!-- &#215; --></mo>
<mi>T</mi>
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</label>
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</label>
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<input type="checkbox" name="input_3d21480bfbb2487e84ac0862e2f3abaf_3_1[]" id="input_3d21480bfbb2487e84ac0862e2f3abaf_3_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="3d21480bfbb2487e84ac0862e2f3abaf_3_1-choice_6-label" for="input_3d21480bfbb2487e84ac0862e2f3abaf_3_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_3d21480bfbb2487e84ac0862e2f3abaf_3_1"> linear
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<h2 class="hd hd-2 unit-title">Lecture Video | Representing Partial Orders as Subset Relations</h2>
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<h3 class="hd hd-2">Representing Partial Orders As Subset Relations</h3>
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<h2 class="hd hd-2 unit-title">Lecture Video | Equivalence Relations</h2>
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<h2 class="hd hd-2 unit-title">Exercise | Relational Properties</h2>
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Relational Properties
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p> For each of the following relations on people, decide whether the relation is <strong> reflexive</strong>, <strong>irreflexive</strong>, <strong>antisymmetric</strong>, or <strong>transitive</strong>: </p>
<div class="choicegroup capa_inputtype" id="inputtype_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1">
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<legend id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1-legend" class="response-fieldset-legend field-group-hd">1. "are the same age"</legend>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1-choice_0-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1"> reflexive
</label>
</div>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1-choice_1-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1"> irreflexive
</label>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1-choice_2-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1"> antisymmetric
</label>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1-choice_3-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1"> transitive
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<span id="answer_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1"/>
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<span class="status unanswered" id="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_2_1" data-tooltip="Not yet answered.">
<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_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1">
<fieldset aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1">
<legend id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1-legend" class="response-fieldset-legend field-group-hd">2. "is younger than"</legend>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1-choice_0-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1"> reflexive
</label>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1-choice_1-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1"> irreflexive
</label>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1-choice_2-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1"> antisymmetric
</label>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1-choice_3-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1"> transitive
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<span id="answer_ed5b3df76a3e4ee3a986ac06ccb0c2e0_3_1"/>
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<fieldset aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1">
<legend id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1-legend" class="response-fieldset-legend field-group-hd">3. "have the same parents"</legend>
<div class="field">
<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1-choice_0-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1"> reflexive
</label>
</div>
<div class="field">
<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1-choice_1-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1"> irreflexive
</label>
</div>
<div class="field">
<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1-choice_2-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1"> antisymmetric
</label>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1-choice_3-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1"> transitive
</label>
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<span id="answer_ed5b3df76a3e4ee3a986ac06ccb0c2e0_4_1"/>
</fieldset>
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<legend id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1-legend" class="response-fieldset-legend field-group-hd">4. "is a descendent of"</legend>
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</label>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1-choice_1-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1"> irreflexive
</label>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1-choice_2-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1"> antisymmetric
</label>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1-choice_3-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1"> transitive
</label>
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<span id="answer_ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1"/>
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<span class="status unanswered" id="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<fieldset aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1">
<legend id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1-legend" class="response-fieldset-legend field-group-hd">5. "have a parent in common"</legend>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1-choice_0-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1"> reflexive
</label>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1-choice_1-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1"> irreflexive
</label>
</div>
<div class="field">
<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1-choice_2-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1"> antisymmetric
</label>
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<input type="checkbox" name="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1[]" id="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1-choice_3-label" for="input_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_ed5b3df76a3e4ee3a986ac06ccb0c2e0_6_1"> transitive
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<h2 class="hd hd-2 unit-title">Exercise | Properties of Relations</h2>
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Properties of Relations
</h3>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_aabe9e839b2f43239b82c874d665111e_2_1">
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<legend id="aabe9e839b2f43239b82c874d665111e_2_1-legend" class="response-fieldset-legend field-group-hd">Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>S</mi>
</math> be relations on the set <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math>. Which of the following statements about <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>S</mi>
</math> are true?</legend>
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<input type="checkbox" name="input_aabe9e839b2f43239b82c874d665111e_2_1[]" id="input_aabe9e839b2f43239b82c874d665111e_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="aabe9e839b2f43239b82c874d665111e_2_1-choice_0-label" for="input_aabe9e839b2f43239b82c874d665111e_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_aabe9e839b2f43239b82c874d665111e_2_1"> If <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> is reflexive, then the complement of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> is also reflexive.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_aabe9e839b2f43239b82c874d665111e_2_1[]" id="input_aabe9e839b2f43239b82c874d665111e_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="aabe9e839b2f43239b82c874d665111e_2_1-choice_1-label" for="input_aabe9e839b2f43239b82c874d665111e_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_aabe9e839b2f43239b82c874d665111e_2_1"> If <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> is symmetric, then the complement of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> is symmetric.
</label>
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<mi>R</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>S</mi>
</math> are reflexive, then <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
<mo>&#8745;<!-- &#8745; --></mo>
<mi>S</mi>
</math> is also reflexive.
</label>
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<h2 class="hd hd-2 unit-title">Exercise | Equivalence Relations & Partial Orders</h2>
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Equivalence Relations &amp; Partial Orders
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p> For each of the following relations, indicate whether it is an <strong>equivalence relation</strong>, a <strong>partial but not a total order</strong>, a <strong>total order</strong>, or <strong>none of the above.</strong></p>
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<option value="equivalence relation"> equivalence relation</option>
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<option value="total order"> total order</option>
<option value="none of the above"> none of the above</option>
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<option value="equivalence relation"> equivalence relation</option>
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<option value="total order"> total order</option>
<option value="none of the above"> none of the above</option>
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<label class="problem-group-label" for="input_8d673969d9f54b45af88400f999b53b9_4_1" id="label_8d673969d9f54b45af88400f999b53b9_4_1">3. <math xmlns="http://www.w3.org/1998/Math/MathML">
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<option value="equivalence relation"> equivalence relation</option>
<option value="partial but not a total order"> partial but not a total order</option>
<option value="total order"> total order</option>
<option value="none of the above"> none of the above</option>
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