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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p>Read <a href="/assets/courseware/v1/c7daf1d22c338a19806c9fd8d6df48f8/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session7.pdf" target="[object Object]">Chapter 4.3–4.5 (PDF)</a> of <em>Mathematics for Computer Science</em> for 1.7 Binary Relations.</p>
<p>View the <a href="/assets/courseware/v1/62df39bbb4a601c196f93c01c09acc58/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp7.pdf" target="[object Object]">Section 1.7 In-Class Questions (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Lecture Video | Relations</h2>
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<p>Download a copy of the slides for <a href="/assets/courseware/v1/6de4a42872e33ec582bbe8c9a28f0d00/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS16_Relations_Slides.pdf" target="[object Object]">Relations (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Exercise| Range of a Relation</h2>
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Range of a Relation
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<legend id="e304c3b6555f45c5921a9d9543c97bdd_2_1-legend" class="response-fieldset-legend field-group-hd">The range of a relation is:</legend>
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<h2 class="hd hd-2 unit-title">Lecture Video | Relational Mappings</h2>
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<h2 class="hd hd-2 unit-title">Exercise | Total Injection</h2>
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<h2 class="hd hd-2 unit-title">Lecture Video | Finite Cardinality</h2>
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<h2 class="hd hd-2 unit-title">Exercise | A inj B</h2>
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<h3 class="hd hd-3 problem-header" id="f6f0eb8709794e648f443dff5917ef21-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@f6f0eb8709794e648f443dff5917ef21-problem-progress" tabindex="-1">
A inj B
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@f6f0eb8709794e648f443dff5917ef21-problem-progress"></div>
<div class="problem">
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_f6f0eb8709794e648f443dff5917ef21_2_1">
<fieldset aria-describedby="status_f6f0eb8709794e648f443dff5917ef21_2_1">
<legend id="f6f0eb8709794e648f443dff5917ef21_2_1-legend" class="response-fieldset-legend field-group-hd">For finite sets <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>,<math xmlns="http://www.w3.org/1998/Math/MathML">
<mspace width="thickmathspace"/>
<mi>A</mi>
<mtext>&#160;inj&#160;</mtext>
<mi>B</mi>
</math> is equivalent to:</legend>
<div class="field">
<input type="checkbox" name="input_f6f0eb8709794e648f443dff5917ef21_2_1[]" id="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="f6f0eb8709794e648f443dff5917ef21_2_1-choice_0-label" for="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_f6f0eb8709794e648f443dff5917ef21_2_1"> There exists an injective relation from <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> to <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_f6f0eb8709794e648f443dff5917ef21_2_1[]" id="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="f6f0eb8709794e648f443dff5917ef21_2_1-choice_1-label" for="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_f6f0eb8709794e648f443dff5917ef21_2_1"> There exists a total injective relation from <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> to <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_f6f0eb8709794e648f443dff5917ef21_2_1[]" id="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="f6f0eb8709794e648f443dff5917ef21_2_1-choice_2-label" for="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_f6f0eb8709794e648f443dff5917ef21_2_1"> There exists an injective function from <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> to <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_f6f0eb8709794e648f443dff5917ef21_2_1[]" id="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="f6f0eb8709794e648f443dff5917ef21_2_1-choice_3-label" for="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_f6f0eb8709794e648f443dff5917ef21_2_1"> There exists a surjective relation from <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math> to <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_f6f0eb8709794e648f443dff5917ef21_2_1[]" id="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="f6f0eb8709794e648f443dff5917ef21_2_1-choice_4-label" for="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_f6f0eb8709794e648f443dff5917ef21_2_1"> There exists a total surjective relation from <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math> to <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_f6f0eb8709794e648f443dff5917ef21_2_1[]" id="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="f6f0eb8709794e648f443dff5917ef21_2_1-choice_5-label" for="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_f6f0eb8709794e648f443dff5917ef21_2_1"> There exists a surjective function from <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math> to <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_f6f0eb8709794e648f443dff5917ef21_2_1[]" id="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="f6f0eb8709794e648f443dff5917ef21_2_1-choice_6-label" for="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_f6f0eb8709794e648f443dff5917ef21_2_1"> There exists a bijection from <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> to some subset of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_f6f0eb8709794e648f443dff5917ef21_2_1[]" id="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="f6f0eb8709794e648f443dff5917ef21_2_1-choice_7-label" for="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_f6f0eb8709794e648f443dff5917ef21_2_1"> There exists a bijection from <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math> to some subset of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_f6f0eb8709794e648f443dff5917ef21_2_1[]" id="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_8" class="field-input input-checkbox" value="choice_8"/><label id="f6f0eb8709794e648f443dff5917ef21_2_1-choice_8-label" for="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_f6f0eb8709794e648f443dff5917ef21_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo stretchy="false">&#8739;<!-- &#8739; --></mo>
<mi>A</mi>
<mo>&#8739;<!-- &#8739; --></mo>
<mspace width="thinmathspace"/>
<mo>&#8804;<!-- &#8804; --></mo>
<mspace width="thinmathspace"/>
<mo>&#8739;<!-- &#8739; --></mo>
<mi>B</mi>
<mo stretchy="false">&#8739;<!-- &#8739; --></mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_f6f0eb8709794e648f443dff5917ef21_2_1[]" id="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_9" class="field-input input-checkbox" value="choice_9"/><label id="f6f0eb8709794e648f443dff5917ef21_2_1-choice_9-label" for="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_9" class="response-label field-label label-inline" aria-describedby="status_f6f0eb8709794e648f443dff5917ef21_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo stretchy="false">&#8739;<!-- &#8739; --></mo>
<mi>A</mi>
<mo>&#8739;<!-- &#8739; --></mo>
<mspace width="thinmathspace"/>
<mo>&gt;</mo>
<mspace width="thinmathspace"/>
<mo>&#8739;<!-- &#8739; --></mo>
<mi>B</mi>
<mo stretchy="false">&#8739;<!-- &#8739; --></mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_f6f0eb8709794e648f443dff5917ef21_2_1[]" id="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_10" class="field-input input-checkbox" value="choice_10"/><label id="f6f0eb8709794e648f443dff5917ef21_2_1-choice_10-label" for="input_f6f0eb8709794e648f443dff5917ef21_2_1_choice_10" class="response-label field-label label-inline" aria-describedby="status_f6f0eb8709794e648f443dff5917ef21_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo stretchy="false">&#8739;<!-- &#8739; --></mo>
<mi>A</mi>
<mo>&#8739;<!-- &#8739; --></mo>
<mspace width="thinmathspace"/>
<mo>=</mo>
<mspace width="thinmathspace"/>
<mo>&#8739;<!-- &#8739; --></mo>
<mi>B</mi>
<mo stretchy="false">&#8739;<!-- &#8739; --></mo>
</math>
</label>
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<span id="answer_f6f0eb8709794e648f443dff5917ef21_2_1"/>
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<span id="solution_f6f0eb8709794e648f443dff5917ef21_solution_1"/>
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<h2 class="hd hd-2 unit-title">Exercise | Total Relations</h2>
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<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@7087670967254d2fb92c144088df2816">
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data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@7087670967254d2fb92c144088df2816" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@7087670967254d2fb92c144088df2816/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="7087670967254d2fb92c144088df2816-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@7087670967254d2fb92c144088df2816-problem-progress" tabindex="-1">
Total Relations
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@7087670967254d2fb92c144088df2816-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_7087670967254d2fb92c144088df2816_2_1">
<fieldset aria-describedby="status_7087670967254d2fb92c144088df2816_2_1">
<legend id="7087670967254d2fb92c144088df2816_2_1-legend" class="response-fieldset-legend field-group-hd">A relation <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> from a set <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>A</mi>
</math> to a set <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
</math> is <b>total</b> iff (check all that apply):</legend>
<div class="field">
<input type="checkbox" name="input_7087670967254d2fb92c144088df2816_2_1[]" id="input_7087670967254d2fb92c144088df2816_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="7087670967254d2fb92c144088df2816_2_1-choice_0-label" for="input_7087670967254d2fb92c144088df2816_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_7087670967254d2fb92c144088df2816_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>B</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_7087670967254d2fb92c144088df2816_2_1[]" id="input_7087670967254d2fb92c144088df2816_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="7087670967254d2fb92c144088df2816_2_1-choice_1-label" for="input_7087670967254d2fb92c144088df2816_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_7087670967254d2fb92c144088df2816_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>A</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_7087670967254d2fb92c144088df2816_2_1[]" id="input_7087670967254d2fb92c144088df2816_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="7087670967254d2fb92c144088df2816_2_1-choice_2-label" for="input_7087670967254d2fb92c144088df2816_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_7087670967254d2fb92c144088df2816_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_7087670967254d2fb92c144088df2816_2_1[]" id="input_7087670967254d2fb92c144088df2816_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="7087670967254d2fb92c144088df2816_2_1-choice_3-label" for="input_7087670967254d2fb92c144088df2816_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_7087670967254d2fb92c144088df2816_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>(&#8805;<!-- &#8805; --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math> - out
</label>
</div>
<div class="field">
<input type="checkbox" name="input_7087670967254d2fb92c144088df2816_2_1[]" id="input_7087670967254d2fb92c144088df2816_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="7087670967254d2fb92c144088df2816_2_1-choice_4-label" for="input_7087670967254d2fb92c144088df2816_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_7087670967254d2fb92c144088df2816_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
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</math>
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<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>A</mi>
</math>
</label>
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<h2 class="hd hd-2 unit-title">Exercise | Surjective Relations</h2>
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<h3 class="hd hd-3 problem-header" id="cfaf103cdaeb46f2900f3e38d614734e-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@cfaf103cdaeb46f2900f3e38d614734e-problem-progress" tabindex="-1">
Surjective 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_cfaf103cdaeb46f2900f3e38d614734e_2_1">
<fieldset aria-describedby="status_cfaf103cdaeb46f2900f3e38d614734e_2_1">
<legend id="cfaf103cdaeb46f2900f3e38d614734e_2_1-legend" class="response-fieldset-legend field-group-hd">A relation <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi> R</mi>
</math> is <strong>surjective</strong> iff (check all that apply):</legend>
<div class="field">
<input type="checkbox" name="input_cfaf103cdaeb46f2900f3e38d614734e_2_1[]" id="input_cfaf103cdaeb46f2900f3e38d614734e_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="cfaf103cdaeb46f2900f3e38d614734e_2_1-choice_0-label" for="input_cfaf103cdaeb46f2900f3e38d614734e_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_cfaf103cdaeb46f2900f3e38d614734e_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#8722;<!-- &#8722; --></mo>
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<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>B</mi>
</math>
</label>
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<div class="field">
<input type="checkbox" name="input_cfaf103cdaeb46f2900f3e38d614734e_2_1[]" id="input_cfaf103cdaeb46f2900f3e38d614734e_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="cfaf103cdaeb46f2900f3e38d614734e_2_1-choice_1-label" for="input_cfaf103cdaeb46f2900f3e38d614734e_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_cfaf103cdaeb46f2900f3e38d614734e_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
</mrow>
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<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>A</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_cfaf103cdaeb46f2900f3e38d614734e_2_1[]" id="input_cfaf103cdaeb46f2900f3e38d614734e_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="cfaf103cdaeb46f2900f3e38d614734e_2_1-choice_2-label" for="input_cfaf103cdaeb46f2900f3e38d614734e_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_cfaf103cdaeb46f2900f3e38d614734e_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>B</mi>
</math>
</label>
</div>
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<input type="checkbox" name="input_cfaf103cdaeb46f2900f3e38d614734e_2_1[]" id="input_cfaf103cdaeb46f2900f3e38d614734e_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="cfaf103cdaeb46f2900f3e38d614734e_2_1-choice_3-label" for="input_cfaf103cdaeb46f2900f3e38d614734e_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_cfaf103cdaeb46f2900f3e38d614734e_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>A</mi>
</math>
</label>
</div>
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<h2 class="hd hd-2 unit-title">Exercise | Inverse Relations</h2>
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<h3 class="hd hd-3 problem-header" id="b9c2a9a2c1b64b43b66009b220c68fce-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@b9c2a9a2c1b64b43b66009b220c68fce-problem-progress" tabindex="-1">
Inverse Relations
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@b9c2a9a2c1b64b43b66009b220c68fce-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_b9c2a9a2c1b64b43b66009b220c68fce_2_1">
<fieldset aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_2_1">
<legend id="b9c2a9a2c1b64b43b66009b220c68fce_2_1-legend" class="response-fieldset-legend field-group-hd">The inverse, <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
</mrow>
</msup>
</math>, of a binary relation, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
<mo>:</mo>
<mi>A</mi>
<mo stretchy="false">&#8594;<!-- &#8594; --></mo>
<mi>B</mi>
</math>, is the relation from<math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>B</mi>
<mo stretchy="false">&#8594;<!-- &#8594; --></mo>
<mi>A</mi>
</math> defined by
<br/>
<br/>
<div align="center">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>b</mi>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
</mrow>
</msup>
<mi>a</mi>
<mspace width="thickmathspace"/>
<mtext>&#160;IFF&#160;</mtext>
<mspace width="thickmathspace"/>
<mi>a</mi>
<mi>R</mi>
<mi>b</mi>
</math>
</div>
<br/>
<br/>
In other words, you get the diagram for <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
</mrow>
</msup>
</math> from <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R </mi>
</math> by "reversing the arrows" in the diagram describing <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math>. Many of the relational properties of <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
</mrow>
</msup>
</math> correspond to different properties of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math>. For example, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> is total iff <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
</mrow>
</msup>
</math> is a surjection. How about the following relational properties?
<br/>
<br/>
1. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> is a function iff <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
</mrow>
</msup>
</math> is</legend>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_2_1-choice_0-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_2_1"> a function
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_2_1-choice_1-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_2_1"> a surjection
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_2_1-choice_2-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_2_1"> an injection
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_2_1-choice_3-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_2_1"> a bijection
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_2_1-choice_4-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_2_1"> total
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_2_1-choice_5-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_2_1"> none of the above
</label>
</div>
<span id="answer_b9c2a9a2c1b64b43b66009b220c68fce_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_b9c2a9a2c1b64b43b66009b220c68fce_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_b9c2a9a2c1b64b43b66009b220c68fce_3_1">
<fieldset aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_3_1">
<legend id="b9c2a9a2c1b64b43b66009b220c68fce_3_1-legend" class="response-fieldset-legend field-group-hd"><math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>2. R</mi>
</math> is a surjection iff <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
</mrow>
</msup>
</math> is</legend>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_3_1-choice_0-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_3_1"> a function
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_3_1-choice_1-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_3_1"> a surjection
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_3_1-choice_2-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_3_1"> an injection
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_3_1-choice_3-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_3_1"> a bijection
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_3_1-choice_4-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_3_1"> total
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_3_1-choice_5-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_3_1"> none of the above
</label>
</div>
<span id="answer_b9c2a9a2c1b64b43b66009b220c68fce_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_b9c2a9a2c1b64b43b66009b220c68fce_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_b9c2a9a2c1b64b43b66009b220c68fce_4_1">
<fieldset aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_4_1">
<legend id="b9c2a9a2c1b64b43b66009b220c68fce_4_1-legend" class="response-fieldset-legend field-group-hd"><math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>3. R</mi>
</math> is an injection iff <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
</mrow>
</msup>
</math> is</legend>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_4_1-choice_0-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_4_1"> a function
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_4_1-choice_1-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_4_1"> a surjection
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_4_1-choice_2-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_4_1"> an injection
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_4_1-choice_3-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_4_1"> a bijection
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_4_1-choice_4-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_4_1"> total
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_4_1-choice_5-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_4_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_4_1"> none of the above
</label>
</div>
<span id="answer_b9c2a9a2c1b64b43b66009b220c68fce_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_b9c2a9a2c1b64b43b66009b220c68fce_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_b9c2a9a2c1b64b43b66009b220c68fce_5_1">
<fieldset aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_5_1">
<legend id="b9c2a9a2c1b64b43b66009b220c68fce_5_1-legend" class="response-fieldset-legend field-group-hd"><math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>4. R</mi>
</math> is a bijection iff <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
</mrow>
</msup>
</math> is</legend>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_5_1-choice_0-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_5_1"> a function
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_5_1-choice_1-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_5_1"> a surjection
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_5_1-choice_2-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_5_1"> an injection
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_5_1-choice_3-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_5_1"> a bijection
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_5_1-choice_4-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_5_1"> total
</label>
</div>
<div class="field">
<input type="radio" name="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1" id="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="b9c2a9a2c1b64b43b66009b220c68fce_5_1-choice_5-label" for="input_b9c2a9a2c1b64b43b66009b220c68fce_5_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_b9c2a9a2c1b64b43b66009b220c68fce_5_1"> none of the above
</label>
</div>
<span id="answer_b9c2a9a2c1b64b43b66009b220c68fce_5_1"/>
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<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@366fa0318abf4e999d6f81bc1e7f5649" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-has-score="False" data-block-type="vertical" data-runtime-version="1" data-request-token="af15a2e6dbd511ef904e12ab06d4b43d" data-graded="False" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">Exercise | In-, Sur-, and Bijections</h2>
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<h3 class="hd hd-3 problem-header" id="f29b881715734ff4ad000e9ef363a57b-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@f29b881715734ff4ad000e9ef363a57b-problem-progress" tabindex="-1">
In-, Sur-, and Bijections
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@f29b881715734ff4ad000e9ef363a57b-problem-progress"></div>
<div class="problem">
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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_f29b881715734ff4ad000e9ef363a57b_2_1" id="label_f29b881715734ff4ad000e9ef363a57b_2_1">For each of the following real-valued functions on <strong>the real numbers </strong><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</math>, indicate whether it is a <em>bijection</em>, a <em>surjection but not a bijection</em>, an <em>injection but not a bijection</em>, or <em>neither an injection nor a surjection</em>.
<br/>
<br/>
1. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</math></label>
<select name="input_f29b881715734ff4ad000e9ef363a57b_2_1" id="input_f29b881715734ff4ad000e9ef363a57b_2_1" aria-describedby="status_f29b881715734ff4ad000e9ef363a57b_2_1">
<option value="option_f29b881715734ff4ad000e9ef363a57b_2_1_dummy_default">Select an option</option>
<option value="a bijection"> a bijection</option>
<option value="a surjection but not a bijection"> a surjection but not a bijection</option>
<option value="an injection but not a bijection"> an injection but not a bijection</option>
<option value="neither an injection nor a surjection"> neither an injection nor a surjection</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_f29b881715734ff4ad000e9ef363a57b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_f29b881715734ff4ad000e9ef363a57b_2_1"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="inputtype option-input ">
<label class="problem-group-label" for="input_f29b881715734ff4ad000e9ef363a57b_3_1" id="label_f29b881715734ff4ad000e9ef363a57b_3_1">2. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>2</mn>
<mi>x</mi>
</math></label>
<select name="input_f29b881715734ff4ad000e9ef363a57b_3_1" id="input_f29b881715734ff4ad000e9ef363a57b_3_1" aria-describedby="status_f29b881715734ff4ad000e9ef363a57b_3_1">
<option value="option_f29b881715734ff4ad000e9ef363a57b_3_1_dummy_default">Select an option</option>
<option value="a bijection"> a bijection</option>
<option value="a surjection but not a bijection"> a surjection but not a bijection</option>
<option value="an injection but not a bijection"> an injection but not a bijection</option>
<option value="neither an injection nor a surjection"> neither an injection nor a surjection</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_f29b881715734ff4ad000e9ef363a57b_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_f29b881715734ff4ad000e9ef363a57b_3_1"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="inputtype option-input ">
<label class="problem-group-label" for="input_f29b881715734ff4ad000e9ef363a57b_4_1" id="label_f29b881715734ff4ad000e9ef363a57b_4_1">3. <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</math></label>
<select name="input_f29b881715734ff4ad000e9ef363a57b_4_1" id="input_f29b881715734ff4ad000e9ef363a57b_4_1" aria-describedby="status_f29b881715734ff4ad000e9ef363a57b_4_1">
<option value="option_f29b881715734ff4ad000e9ef363a57b_4_1_dummy_default">Select an option</option>
<option value="a bijection"> a bijection</option>
<option value="a surjection but not a bijection"> a surjection but not a bijection</option>
<option value="an injection but not a bijection"> an injection but not a bijection</option>
<option value="neither an injection nor a surjection"> neither an injection nor a surjection</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_f29b881715734ff4ad000e9ef363a57b_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_f29b881715734ff4ad000e9ef363a57b_4_1"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="inputtype option-input ">
<label class="problem-group-label" for="input_f29b881715734ff4ad000e9ef363a57b_5_1" id="label_f29b881715734ff4ad000e9ef363a57b_5_1">4. <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</math></label>
<select name="input_f29b881715734ff4ad000e9ef363a57b_5_1" id="input_f29b881715734ff4ad000e9ef363a57b_5_1" aria-describedby="status_f29b881715734ff4ad000e9ef363a57b_5_1">
<option value="option_f29b881715734ff4ad000e9ef363a57b_5_1_dummy_default">Select an option</option>
<option value="a bijection"> a bijection</option>
<option value="a surjection but not a bijection"> a surjection but not a bijection</option>
<option value="an injection but not a bijection"> an injection but not a bijection</option>
<option value="neither an injection nor a surjection"> neither an injection nor a surjection</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_f29b881715734ff4ad000e9ef363a57b_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_f29b881715734ff4ad000e9ef363a57b_5_1"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div class="inputtype option-input ">
<label class="problem-group-label" for="input_f29b881715734ff4ad000e9ef363a57b_6_1" id="label_f29b881715734ff4ad000e9ef363a57b_6_1">5. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>sin</mi>
<mo>&#8289;<!-- --></mo>
<mi>x</mi>
</math></label>
<select name="input_f29b881715734ff4ad000e9ef363a57b_6_1" id="input_f29b881715734ff4ad000e9ef363a57b_6_1" aria-describedby="status_f29b881715734ff4ad000e9ef363a57b_6_1">
<option value="option_f29b881715734ff4ad000e9ef363a57b_6_1_dummy_default">Select an option</option>
<option value="a bijection"> a bijection</option>
<option value="a surjection but not a bijection"> a surjection but not a bijection</option>
<option value="an injection but not a bijection"> an injection but not a bijection</option>
<option value="neither an injection nor a surjection"> neither an injection nor a surjection</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_f29b881715734ff4ad000e9ef363a57b_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_f29b881715734ff4ad000e9ef363a57b_6_1"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 6" role="group"><div class="inputtype option-input ">
<label class="problem-group-label" for="input_f29b881715734ff4ad000e9ef363a57b_7_1" id="label_f29b881715734ff4ad000e9ef363a57b_7_1">6. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>x</mi>
<mi>sin</mi>
<mo>&#8289;<!-- --></mo>
<mi>x</mi>
</math></label>
<select name="input_f29b881715734ff4ad000e9ef363a57b_7_1" id="input_f29b881715734ff4ad000e9ef363a57b_7_1" aria-describedby="status_f29b881715734ff4ad000e9ef363a57b_7_1">
<option value="option_f29b881715734ff4ad000e9ef363a57b_7_1_dummy_default">Select an option</option>
<option value="a bijection"> a bijection</option>
<option value="a surjection but not a bijection"> a surjection but not a bijection</option>
<option value="an injection but not a bijection"> an injection but not a bijection</option>
<option value="neither an injection nor a surjection"> neither an injection nor a surjection</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_f29b881715734ff4ad000e9ef363a57b_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_f29b881715734ff4ad000e9ef363a57b_7_1"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 7" role="group"><div class="inputtype option-input ">
<label class="problem-group-label" for="input_f29b881715734ff4ad000e9ef363a57b_8_1" id="label_f29b881715734ff4ad000e9ef363a57b_8_1">7. <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>e</mi>
<mi>x</mi>
</msup>
</math></label>
<select name="input_f29b881715734ff4ad000e9ef363a57b_8_1" id="input_f29b881715734ff4ad000e9ef363a57b_8_1" aria-describedby="status_f29b881715734ff4ad000e9ef363a57b_8_1">
<option value="option_f29b881715734ff4ad000e9ef363a57b_8_1_dummy_default">Select an option</option>
<option value="a bijection"> a bijection</option>
<option value="a surjection but not a bijection"> a surjection but not a bijection</option>
<option value="an injection but not a bijection"> an injection but not a bijection</option>
<option value="neither an injection nor a surjection"> neither an injection nor a surjection</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_f29b881715734ff4ad000e9ef363a57b_8_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_f29b881715734ff4ad000e9ef363a57b_8_1"/>
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<span id="solution_f29b881715734ff4ad000e9ef363a57b_solution_1"/>
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<h2 class="hd hd-2 unit-title">Exercise | Mapping Lemma: Sizes of Domains and Codomains</h2>
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<p>For any binary relation <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>R</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→<!-- → --></mo> <mi>B</mi> </math> and subset <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>S</mi> <mo>⊆<!-- ⊆ --></mo> <mi>A</mi> </math>, let <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </math> be the image of <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>S</mi> </math> under <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>R</mi> </math>. An example of such an image is the doubling function with domain and codomain equal to the real numbers:</p>
<div align="center"><math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>s</mi> <mi>R</mi> <mi>t</mi> <mtext> IFF </mtext> <mi>t</mi> <mo>=</mo> <mn>2</mn> <mi>s</mi> </math></div>
<p></p>
<p>such that <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>R</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>11</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>22</mn> <mo fence="false" stretchy="false">}</mo> </math>. Another example, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>R</mi> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </math>, is the set of all even integers. For any finite set, we let <math xmlns="http://www.w3.org/1998/Math/MathML"> <mo stretchy="false">|</mo> <mi>S</mi> <mo stretchy="false">|</mo> </math> denote the size (number of elements) of <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>S</mi> </math>.</p>
<p></p>
<p>Now assume <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>R</mi> </math> is some <strong>total function</strong> and <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>A</mi> </math> is finite. Fill in the blanks to produce the <strong>strongest</strong> correct version of the following statements:</p>
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<h3 class="hd hd-3 problem-header" id="7bf2864da8be4032b233e3fe9cf04691-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@7bf2864da8be4032b233e3fe9cf04691-problem-progress" tabindex="-1">
Questions
</h3>
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<div class="problem">
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_7bf2864da8be4032b233e3fe9cf04691_2_1">
<fieldset aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_2_1">
<legend id="7bf2864da8be4032b233e3fe9cf04691_2_1-legend" class="response-fieldset-legend field-group-hd">1. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>R</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</math> _____ <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</math></legend>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_2_1" id="input_7bf2864da8be4032b233e3fe9cf04691_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="7bf2864da8be4032b233e3fe9cf04691_2_1-choice_0-label" for="input_7bf2864da8be4032b233e3fe9cf04691_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&lt;</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_2_1" id="input_7bf2864da8be4032b233e3fe9cf04691_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="7bf2864da8be4032b233e3fe9cf04691_2_1-choice_1-label" for="input_7bf2864da8be4032b233e3fe9cf04691_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8804;<!-- &#8804; --></mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_2_1" id="input_7bf2864da8be4032b233e3fe9cf04691_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="7bf2864da8be4032b233e3fe9cf04691_2_1-choice_2-label" for="input_7bf2864da8be4032b233e3fe9cf04691_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_2_1"> <choice correct="false"><math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>=</mo>
</math></choice>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_2_1" id="input_7bf2864da8be4032b233e3fe9cf04691_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="7bf2864da8be4032b233e3fe9cf04691_2_1-choice_3-label" for="input_7bf2864da8be4032b233e3fe9cf04691_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8805;<!-- &#8805; --></mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_2_1" id="input_7bf2864da8be4032b233e3fe9cf04691_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="7bf2864da8be4032b233e3fe9cf04691_2_1-choice_4-label" for="input_7bf2864da8be4032b233e3fe9cf04691_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&gt;</mo>
</math>
</label>
</div>
<span id="answer_7bf2864da8be4032b233e3fe9cf04691_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_7bf2864da8be4032b233e3fe9cf04691_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="solution-span">
<span id="solution_7bf2864da8be4032b233e3fe9cf04691_solution_1"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_7bf2864da8be4032b233e3fe9cf04691_3_1">
<fieldset aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_3_1">
<legend id="7bf2864da8be4032b233e3fe9cf04691_3_1-legend" class="response-fieldset-legend field-group-hd">2. If <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> is a surjection, then <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</math> ____<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</math></legend>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_3_1" id="input_7bf2864da8be4032b233e3fe9cf04691_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="7bf2864da8be4032b233e3fe9cf04691_3_1-choice_0-label" for="input_7bf2864da8be4032b233e3fe9cf04691_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&lt;</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_3_1" id="input_7bf2864da8be4032b233e3fe9cf04691_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="7bf2864da8be4032b233e3fe9cf04691_3_1-choice_1-label" for="input_7bf2864da8be4032b233e3fe9cf04691_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8804;<!-- &#8804; --></mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_3_1" id="input_7bf2864da8be4032b233e3fe9cf04691_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="7bf2864da8be4032b233e3fe9cf04691_3_1-choice_2-label" for="input_7bf2864da8be4032b233e3fe9cf04691_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>=</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_3_1" id="input_7bf2864da8be4032b233e3fe9cf04691_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="7bf2864da8be4032b233e3fe9cf04691_3_1-choice_3-label" for="input_7bf2864da8be4032b233e3fe9cf04691_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8805;<!-- &#8805; --></mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_3_1" id="input_7bf2864da8be4032b233e3fe9cf04691_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="7bf2864da8be4032b233e3fe9cf04691_3_1-choice_4-label" for="input_7bf2864da8be4032b233e3fe9cf04691_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&gt;</mo>
</math>
</label>
</div>
<span id="answer_7bf2864da8be4032b233e3fe9cf04691_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_7bf2864da8be4032b233e3fe9cf04691_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_7bf2864da8be4032b233e3fe9cf04691_4_1">
<fieldset aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_4_1">
<legend id="7bf2864da8be4032b233e3fe9cf04691_4_1-legend" class="response-fieldset-legend field-group-hd">3. If <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> is a surjection, then <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>R</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</math> ____<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</math></legend>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_4_1" id="input_7bf2864da8be4032b233e3fe9cf04691_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="7bf2864da8be4032b233e3fe9cf04691_4_1-choice_0-label" for="input_7bf2864da8be4032b233e3fe9cf04691_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&lt;</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_4_1" id="input_7bf2864da8be4032b233e3fe9cf04691_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="7bf2864da8be4032b233e3fe9cf04691_4_1-choice_1-label" for="input_7bf2864da8be4032b233e3fe9cf04691_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8804;<!-- &#8804; --></mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_4_1" id="input_7bf2864da8be4032b233e3fe9cf04691_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="7bf2864da8be4032b233e3fe9cf04691_4_1-choice_2-label" for="input_7bf2864da8be4032b233e3fe9cf04691_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>=</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_4_1" id="input_7bf2864da8be4032b233e3fe9cf04691_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="7bf2864da8be4032b233e3fe9cf04691_4_1-choice_3-label" for="input_7bf2864da8be4032b233e3fe9cf04691_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8805;<!-- &#8805; --></mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_4_1" id="input_7bf2864da8be4032b233e3fe9cf04691_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="7bf2864da8be4032b233e3fe9cf04691_4_1-choice_4-label" for="input_7bf2864da8be4032b233e3fe9cf04691_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&gt;</mo>
</math>
</label>
</div>
<span id="answer_7bf2864da8be4032b233e3fe9cf04691_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_7bf2864da8be4032b233e3fe9cf04691_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_7bf2864da8be4032b233e3fe9cf04691_5_1">
<fieldset aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_5_1">
<legend id="7bf2864da8be4032b233e3fe9cf04691_5_1-legend" class="response-fieldset-legend field-group-hd">4.If <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi> R </mi>
</math> is an injection, then <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>R</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</math> ____<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</math>.</legend>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_5_1" id="input_7bf2864da8be4032b233e3fe9cf04691_5_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="7bf2864da8be4032b233e3fe9cf04691_5_1-choice_0-label" for="input_7bf2864da8be4032b233e3fe9cf04691_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_5_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&lt;</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_5_1" id="input_7bf2864da8be4032b233e3fe9cf04691_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="7bf2864da8be4032b233e3fe9cf04691_5_1-choice_1-label" for="input_7bf2864da8be4032b233e3fe9cf04691_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_5_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8804;<!-- &#8804; --></mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_5_1" id="input_7bf2864da8be4032b233e3fe9cf04691_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="7bf2864da8be4032b233e3fe9cf04691_5_1-choice_2-label" for="input_7bf2864da8be4032b233e3fe9cf04691_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_5_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>=</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_5_1" id="input_7bf2864da8be4032b233e3fe9cf04691_5_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="7bf2864da8be4032b233e3fe9cf04691_5_1-choice_3-label" for="input_7bf2864da8be4032b233e3fe9cf04691_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_5_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8805;<!-- &#8805; --></mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_5_1" id="input_7bf2864da8be4032b233e3fe9cf04691_5_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="7bf2864da8be4032b233e3fe9cf04691_5_1-choice_4-label" for="input_7bf2864da8be4032b233e3fe9cf04691_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_5_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&gt;</mo>
</math>
</label>
</div>
<span id="answer_7bf2864da8be4032b233e3fe9cf04691_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_7bf2864da8be4032b233e3fe9cf04691_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_7bf2864da8be4032b233e3fe9cf04691_6_1">
<fieldset aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_6_1">
<legend id="7bf2864da8be4032b233e3fe9cf04691_6_1-legend" class="response-fieldset-legend field-group-hd">5. If <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>R</mi>
</math> is a bijection, then <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</math> _____<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</math></legend>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_6_1" id="input_7bf2864da8be4032b233e3fe9cf04691_6_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="7bf2864da8be4032b233e3fe9cf04691_6_1-choice_0-label" for="input_7bf2864da8be4032b233e3fe9cf04691_6_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_6_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&lt;</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_6_1" id="input_7bf2864da8be4032b233e3fe9cf04691_6_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="7bf2864da8be4032b233e3fe9cf04691_6_1-choice_1-label" for="input_7bf2864da8be4032b233e3fe9cf04691_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_6_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8804;<!-- &#8804; --></mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_6_1" id="input_7bf2864da8be4032b233e3fe9cf04691_6_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="7bf2864da8be4032b233e3fe9cf04691_6_1-choice_2-label" for="input_7bf2864da8be4032b233e3fe9cf04691_6_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_6_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>=</mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_6_1" id="input_7bf2864da8be4032b233e3fe9cf04691_6_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="7bf2864da8be4032b233e3fe9cf04691_6_1-choice_3-label" for="input_7bf2864da8be4032b233e3fe9cf04691_6_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_6_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8805;<!-- &#8805; --></mo>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_7bf2864da8be4032b233e3fe9cf04691_6_1" id="input_7bf2864da8be4032b233e3fe9cf04691_6_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="7bf2864da8be4032b233e3fe9cf04691_6_1-choice_4-label" for="input_7bf2864da8be4032b233e3fe9cf04691_6_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_7bf2864da8be4032b233e3fe9cf04691_6_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&gt;</mo>
</math>
</label>
</div>
<span id="answer_7bf2864da8be4032b233e3fe9cf04691_6_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_7bf2864da8be4032b233e3fe9cf04691_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Questions" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_7bf2864da8be4032b233e3fe9cf04691" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_7bf2864da8be4032b233e3fe9cf04691">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="7bf2864da8be4032b233e3fe9cf04691-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="7bf2864da8be4032b233e3fe9cf04691-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="7bf2864da8be4032b233e3fe9cf04691-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="7bf2864da8be4032b233e3fe9cf04691-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>