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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p>Read<a href="/assets/courseware/v1/639e73819789708b2180776f5143bdf5/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session2.pdf" target="[object Object]"> Chapter 1.7–1.9 (PDF)</a> of <em>Mathematics for Computer Science</em> for 1.2 Proof Methods.</p>
<p>View the <a href="/assets/courseware/v1/8e67c0038c10a24ad4a870a1948127ca/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp2.pdf" target="[object Object]">Section 1.2 In-Class Questions (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Lecture Video | Proof by Contradiction</h2>
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<p>Download a copy of the slides for <a href="/assets/courseware/v1/552a4b64fbced8dc5a912a14ed49475a/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS16_ProofContrad.pdf" target="[object Object]">Proof by Contradiction (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Exercise | Proof by Contradiction Practice</h2>
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Proof by Contradiction
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<legend id="dd2f166033af4f1280aa612f3c1f4b57_2_1-legend" class="response-fieldset-legend field-group-hd">Proof by contradiction often involves clever application of proven knowledge to arrive at a contradiction. In the example proof of <math>
<msqrt>
<mn>2</mn>
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<h2 class="hd hd-2 unit-title">Exercise | When to Prove by Cases</h2>
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When to Prove by Cases
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<input type="checkbox" name="input_c4f729374ee04dee893410cb804b8814_2_1[]" id="input_c4f729374ee04dee893410cb804b8814_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="c4f729374ee04dee893410cb804b8814_2_1-choice_1-label" for="input_c4f729374ee04dee893410cb804b8814_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c4f729374ee04dee893410cb804b8814_2_1"> The predicate in each case is easier to prove.
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<h2 class="hd hd-2 unit-title">Exercise | Friends and Strangers</h2>
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Friends and Strangers [optional]
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<legend id="f70691b3feca4b84a4f5d4f9721028e5_2_1-legend" class="response-fieldset-legend field-group-hd">The PROOF about three friends or strangers on the slides showed that the <math>
<mi>R</mi>
<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
</math> is related to 6 by which of the following relations?</legend>
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<mo>&lt;</mo>
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<mo>&#8804;<!-- &#8804; --></mo>
</math>
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<mo>&gt;</mo>
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<h3 class="hd hd-3 problem-header" id="4d2545737014482eb40ddf497427323d-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@4d2545737014482eb40ddf497427323d-problem-progress" tabindex="-1">
A Bogus Proof by Cases
</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_4d2545737014482eb40ddf497427323d_2_1">
<fieldset aria-describedby="status_4d2545737014482eb40ddf497427323d_2_1">
<legend id="4d2545737014482eb40ddf497427323d_2_1-legend" class="response-fieldset-legend field-group-hd">Which step(s) contain the logical error?
<br/>
<br/>
<strong>Bogus Claim</strong>: For any integer <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
</math>, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mspace width="thickmathspace"/>
<mspace width="thickmathspace"/>
<mspace width="thickmathspace"/>
<mn>2</mn>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
<mo>&gt;</mo>
<mi>a</mi>
</math></legend>
<div class="field">
<input type="checkbox" name="input_4d2545737014482eb40ddf497427323d_2_1[]" id="input_4d2545737014482eb40ddf497427323d_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="4d2545737014482eb40ddf497427323d_2_1-choice_0-label" for="input_4d2545737014482eb40ddf497427323d_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_4d2545737014482eb40ddf497427323d_2_1"> This proof is by case analysis.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4d2545737014482eb40ddf497427323d_2_1[]" id="input_4d2545737014482eb40ddf497427323d_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="4d2545737014482eb40ddf497427323d_2_1-choice_1-label" for="input_4d2545737014482eb40ddf497427323d_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_4d2545737014482eb40ddf497427323d_2_1"> There are two cases: <b> Case 1:</b> <i>a is positive.</i> <b> Case 2: </b><i> a is negative </i>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4d2545737014482eb40ddf497427323d_2_1[]" id="input_4d2545737014482eb40ddf497427323d_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="4d2545737014482eb40ddf497427323d_2_1-choice_2-label" for="input_4d2545737014482eb40ddf497427323d_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_4d2545737014482eb40ddf497427323d_2_1"> One of these cases must always hold, because an integer is either positive or
negative.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4d2545737014482eb40ddf497427323d_2_1[]" id="input_4d2545737014482eb40ddf497427323d_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="4d2545737014482eb40ddf497427323d_2_1-choice_3-label" for="input_4d2545737014482eb40ddf497427323d_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_4d2545737014482eb40ddf497427323d_2_1"> <b> Case 1:</b> Suppose <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi> </math> is positive.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4d2545737014482eb40ddf497427323d_2_1[]" id="input_4d2545737014482eb40ddf497427323d_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="4d2545737014482eb40ddf497427323d_2_1-choice_4-label" for="input_4d2545737014482eb40ddf497427323d_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_4d2545737014482eb40ddf497427323d_2_1"> Since <em>a</em> is an integer, we must have that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
<mo>&#8805;<!-- &#8804; --></mo>
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4d2545737014482eb40ddf497427323d_2_1[]" id="input_4d2545737014482eb40ddf497427323d_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="4d2545737014482eb40ddf497427323d_2_1-choice_5-label" for="input_4d2545737014482eb40ddf497427323d_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_4d2545737014482eb40ddf497427323d_2_1"> Hence, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>2</mn>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
<mo>=</mo>
<mn>2</mn>
<mi>a</mi>
<mo>&#8901;<!-- &#8901; --></mo>
<mi>a</mi>
<mo>&#8805;<!-- &#8805; --></mo>
<mn>2</mn>
<mi>a</mi>
<mo>&#8901;<!-- &#8901; --></mo>
<mn>1</mn>
<mo>&gt;</mo>
<mi>a</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4d2545737014482eb40ddf497427323d_2_1[]" id="input_4d2545737014482eb40ddf497427323d_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="4d2545737014482eb40ddf497427323d_2_1-choice_6-label" for="input_4d2545737014482eb40ddf497427323d_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_4d2545737014482eb40ddf497427323d_2_1"> This implies the claim holds in Case 1.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4d2545737014482eb40ddf497427323d_2_1[]" id="input_4d2545737014482eb40ddf497427323d_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="4d2545737014482eb40ddf497427323d_2_1-choice_7-label" for="input_4d2545737014482eb40ddf497427323d_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_4d2545737014482eb40ddf497427323d_2_1"> <b> Case 2: </b> Suppose <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
</math> is negative.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4d2545737014482eb40ddf497427323d_2_1[]" id="input_4d2545737014482eb40ddf497427323d_2_1_choice_8" class="field-input input-checkbox" value="choice_8"/><label id="4d2545737014482eb40ddf497427323d_2_1-choice_8-label" for="input_4d2545737014482eb40ddf497427323d_2_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_4d2545737014482eb40ddf497427323d_2_1"> Since <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
</math> is an integer, we must have that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
<mo>&#8804;<!-- &#8804; --></mo>
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4d2545737014482eb40ddf497427323d_2_1[]" id="input_4d2545737014482eb40ddf497427323d_2_1_choice_9" class="field-input input-checkbox" value="choice_9"/><label id="4d2545737014482eb40ddf497427323d_2_1-choice_9-label" for="input_4d2545737014482eb40ddf497427323d_2_1_choice_9" class="response-label field-label label-inline" aria-describedby="status_4d2545737014482eb40ddf497427323d_2_1"> Hence, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>2</mn>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
<mo>&#8805;<!-- &#8805; --></mo>
<mn>2</mn>
<mo>&#8901;<!-- &#8901; --></mo>
<mo stretchy="false">(</mo>
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
<mo>&#8901;<!-- &#8901; --></mo>
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mo>&gt;</mo>
<mo>&#8722;<!-- &#8722; --></mo>
<mn>1</mn>
<mo>&#8805;<!-- &#8805; --></mo>
<mi>a</mi>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4d2545737014482eb40ddf497427323d_2_1[]" id="input_4d2545737014482eb40ddf497427323d_2_1_choice_10" class="field-input input-checkbox" value="choice_10"/><label id="4d2545737014482eb40ddf497427323d_2_1-choice_10-label" for="input_4d2545737014482eb40ddf497427323d_2_1_choice_10" class="response-label field-label label-inline" aria-describedby="status_4d2545737014482eb40ddf497427323d_2_1"> This implies the claim holds in Case 2.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4d2545737014482eb40ddf497427323d_2_1[]" id="input_4d2545737014482eb40ddf497427323d_2_1_choice_11" class="field-input input-checkbox" value="choice_11"/><label id="4d2545737014482eb40ddf497427323d_2_1-choice_11-label" for="input_4d2545737014482eb40ddf497427323d_2_1_choice_11" class="response-label field-label label-inline" aria-describedby="status_4d2545737014482eb40ddf497427323d_2_1"> The claim therefore holds in both cases.
</label>
</div>
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<h2 class="hd hd-2 unit-title">Exercise | A Bogus Proof by Contradiction</h2>
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<h3 class="hd hd-3 problem-header" id="faf1a41eda224f34a7cea5e5366a49c0-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@faf1a41eda224f34a7cea5e5366a49c0-problem-progress" tabindex="-1">
A Bogus Proof by Contradiction
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@faf1a41eda224f34a7cea5e5366a49c0-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_faf1a41eda224f34a7cea5e5366a49c0_2_1">
<fieldset aria-describedby="status_faf1a41eda224f34a7cea5e5366a49c0_2_1 description_faf1a41eda224f34a7cea5e5366a49c0_1_1">
<legend id="faf1a41eda224f34a7cea5e5366a49c0_2_1-legend" class="response-fieldset-legend field-group-hd">Which step(s) contain the logical error?</legend>
<p class="question-description" id="description_faf1a41eda224f34a7cea5e5366a49c0_1_1"><p> <b>Bogus Claim:</b> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msqrt>
<mn>4 </mn>
</msqrt>
</math> is irrational</p></p>
<div class="field">
<input type="checkbox" name="input_faf1a41eda224f34a7cea5e5366a49c0_2_1[]" id="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="faf1a41eda224f34a7cea5e5366a49c0_2_1-choice_0-label" for="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_faf1a41eda224f34a7cea5e5366a49c0_2_1 description_faf1a41eda224f34a7cea5e5366a49c0_1_1"> This proof is by contradiction.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_faf1a41eda224f34a7cea5e5366a49c0_2_1[]" id="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="faf1a41eda224f34a7cea5e5366a49c0_2_1-choice_1-label" for="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_faf1a41eda224f34a7cea5e5366a49c0_2_1 description_faf1a41eda224f34a7cea5e5366a49c0_1_1"> Suppose the claim is false: <math xmlns="http://www.w3.org/1998/Math/MathML">
<msqrt>
<mn>4</mn>
</msqrt>
</math> is rational.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_faf1a41eda224f34a7cea5e5366a49c0_2_1[]" id="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="faf1a41eda224f34a7cea5e5366a49c0_2_1-choice_2-label" for="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_faf1a41eda224f34a7cea5e5366a49c0_2_1 description_faf1a41eda224f34a7cea5e5366a49c0_1_1"> We can write
<math xmlns="http://www.w3.org/1998/Math/MathML">
<msqrt>
<mn>4</mn>
</msqrt>
<mo>=</mo>
<mfrac>
<mi>n</mi>
<mi>d</mi>
</mfrac>
</math> where <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>d</mi>
</math> are integers with no common factors.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_faf1a41eda224f34a7cea5e5366a49c0_2_1[]" id="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="faf1a41eda224f34a7cea5e5366a49c0_2_1-choice_3-label" for="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_faf1a41eda224f34a7cea5e5366a49c0_2_1 description_faf1a41eda224f34a7cea5e5366a49c0_1_1"> Squaring both sides and moving <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>d</mi>
<mn>2</mn>
</msup>
</math> to the left-hand side, we get <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>4</mn>
<msup>
<mi>d</mi>
<mn>2</mn>
</msup>
<mo>=</mo>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_faf1a41eda224f34a7cea5e5366a49c0_2_1[]" id="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="faf1a41eda224f34a7cea5e5366a49c0_2_1-choice_4-label" for="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_faf1a41eda224f34a7cea5e5366a49c0_2_1 description_faf1a41eda224f34a7cea5e5366a49c0_1_1"> This implies that <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</math> is a multiple of 4, and thus 4 is a factor of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_faf1a41eda224f34a7cea5e5366a49c0_2_1[]" id="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="faf1a41eda224f34a7cea5e5366a49c0_2_1-choice_5-label" for="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_faf1a41eda224f34a7cea5e5366a49c0_2_1 description_faf1a41eda224f34a7cea5e5366a49c0_1_1"> Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</math> is a multiple of 16.
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<input type="checkbox" name="input_faf1a41eda224f34a7cea5e5366a49c0_2_1[]" id="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="faf1a41eda224f34a7cea5e5366a49c0_2_1-choice_6-label" for="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_faf1a41eda224f34a7cea5e5366a49c0_2_1 description_faf1a41eda224f34a7cea5e5366a49c0_1_1"> But since <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>4</mn>
<msup>
<mi>d</mi>
<mn>2</mn>
</msup>
<mo>=</mo>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</math>,
<math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>d</mi>
<mn>2</mn>
</msup>
</math>must also be a multiple of 4, and thus 4 is a factor of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>d</mi>
</math>.
</label>
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<input type="checkbox" name="input_faf1a41eda224f34a7cea5e5366a49c0_2_1[]" id="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="faf1a41eda224f34a7cea5e5366a49c0_2_1-choice_7-label" for="input_faf1a41eda224f34a7cea5e5366a49c0_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_faf1a41eda224f34a7cea5e5366a49c0_2_1 description_faf1a41eda224f34a7cea5e5366a49c0_1_1"> The numerator and demominator have a common factor of 4, which is a contradiction. Hence, <math xmlns="http://www.w3.org/1998/Math/MathML">
<msqrt>
<mn>4</mn>
</msqrt>
</math> must be irrational <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>&#9724;<!-- &#9724; --></mi>
</math>.
</label>
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