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<h2 class="hd hd-2 unit-title">Homework</h2>
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<p><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">The exercises below will count towards your grade. <strong>You have only one chance to answer these questions. </strong>Take your time, and think carefully before answering.</span></p>
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Problem 1a
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<p>Three properties a relation might have are <i> reflexivity</i>, <i>symmetry</i>, and <i>transitivity.</i> </p>
<p> If \(A\) is a set and \(R\) is a relation that holds amongst members of that set: </p>
<ul>
<li>
<p> \(R\) is reflexive on \(A\) if and only if for any \(a\in A\), \(aRa\). </p>
</li>
<li>
<p> \(R\) is symmetric on \(A\) if and only if for any \(a,b\in A\), if \(aRb\) then \(bRa\). </p>
</li>
<li>
<p> \(R\) is transitive on \(A\) if and only if for any \(a,b,c\in A\), if \(aRb\) and \(bRc\) then \(aRc\). </p>
</li>
</ul>
<p> Here is an example. Let \(A\) be the set of people, and let \(R\) be the relation &#8220;having the same birthday as&#8221;. \(R\) is reflexive because everyone has the same birthday as herself. \(R\) is symmetric because whenever \(a\) has the same birthday as \(b\), \(b\) will have the same birthday as \(a\). And \(R\) is transitive because for any people \(a\), \(b\), and \(c\), if \(a\) has the same birthday as \(b\), and \(b\) has the same birthday as \(c\), then \(a\) will have the same birthday as \(c\). </p>
<p> Now consider the relation &#8220;is less than or equal to&#8221;. </p>
<p>Is this relation reflexive, symmetric, and transitive on the set of natural numbers?</p>
<p>Reflexive? </p>
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Problem 1b
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<p> There are ten guests at a dinner party, sitting around a large table. Consider the relation \(R\) such that guest \(a\) bears \(R\) to guest \(b\) just in case \(a\)'s seat is immediately adjacent to \(b\)'s seat. (Note that a seat is not adjacent to itself.)</p>
<p>Is \(R\) reflexive, symmetric, and transitive on the set of guests?</p>
<p>Reflexive?</p>
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<h3 class="hd hd-3 problem-header" id="8e7294f17cad424798168b1273ff7776-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@8e7294f17cad424798168b1273ff7776-problem-progress" tabindex="-1">
Problem 2
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<p>Recall that a function from set \(A\) to set \(B\) is an assignment of each member of \(A\) to some member of \(B\). For a function \(f\) from \(A\) to \(B\) to be an <em>injection</em> is for \(f\) never to assign the same element of \(B\) to two different elements of \(A\). For a function \(f\) from \(A\) to \(B\) to be a <em>surjection</em> is for there to be no element of \(B\) to which \(f\) fails to assign some element of \(A\). For a function to be a <em>bijection</em> is for it to be both an injection and a surjection.
</p>
<p>Describe each of the following functions \(f\) from \(A\) to \(B\):</p>
<br> </br>
<p>\(A = \{\dots,-2,-1,0,1,2,\dots\}\)</p>
<p> \(B = \{\dots,-2,-1,0,1,2,\dots\}\)</p>
<p>\(f(x)=x+2\) </p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8e7294f17cad424798168b1273ff7776_2_1">
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<div class="field">
<input type="radio" name="input_8e7294f17cad424798168b1273ff7776_2_1" id="input_8e7294f17cad424798168b1273ff7776_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="8e7294f17cad424798168b1273ff7776_2_1-choice_0-label" for="input_8e7294f17cad424798168b1273ff7776_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_8e7294f17cad424798168b1273ff7776_2_1"> \(f\) is an injection (but not a surjection)
</label>
</div>
<div class="field">
<input type="radio" name="input_8e7294f17cad424798168b1273ff7776_2_1" id="input_8e7294f17cad424798168b1273ff7776_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="8e7294f17cad424798168b1273ff7776_2_1-choice_1-label" for="input_8e7294f17cad424798168b1273ff7776_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8e7294f17cad424798168b1273ff7776_2_1"> \(f\) is a surjection (but not an injection)
</label>
</div>
<div class="field">
<input type="radio" name="input_8e7294f17cad424798168b1273ff7776_2_1" id="input_8e7294f17cad424798168b1273ff7776_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="8e7294f17cad424798168b1273ff7776_2_1-choice_2-label" for="input_8e7294f17cad424798168b1273ff7776_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_8e7294f17cad424798168b1273ff7776_2_1"> \(f\) is a bijection
</label>
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<div class="field">
<input type="radio" name="input_8e7294f17cad424798168b1273ff7776_2_1" id="input_8e7294f17cad424798168b1273ff7776_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="8e7294f17cad424798168b1273ff7776_2_1-choice_3-label" for="input_8e7294f17cad424798168b1273ff7776_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_8e7294f17cad424798168b1273ff7776_2_1"> \(f\) is neither a bijection nor a surjection
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<p>\(A =\{1,2,3,\dots,10\}\)</p>
<p>\(B = \{1,2,3,\dots,100\}\)</p>
<p>\(f(x)=x^2\)</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8e7294f17cad424798168b1273ff7776_3_1">
<fieldset aria-describedby="status_8e7294f17cad424798168b1273ff7776_3_1">
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<input type="radio" name="input_8e7294f17cad424798168b1273ff7776_3_1" id="input_8e7294f17cad424798168b1273ff7776_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="8e7294f17cad424798168b1273ff7776_3_1-choice_0-label" for="input_8e7294f17cad424798168b1273ff7776_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_8e7294f17cad424798168b1273ff7776_3_1"> \(f\) is an injection (but not a surjection)
</label>
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<input type="radio" name="input_8e7294f17cad424798168b1273ff7776_3_1" id="input_8e7294f17cad424798168b1273ff7776_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="8e7294f17cad424798168b1273ff7776_3_1-choice_1-label" for="input_8e7294f17cad424798168b1273ff7776_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8e7294f17cad424798168b1273ff7776_3_1"> \(f\) is a surjection (but not an injection)
</label>
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<input type="radio" name="input_8e7294f17cad424798168b1273ff7776_3_1" id="input_8e7294f17cad424798168b1273ff7776_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="8e7294f17cad424798168b1273ff7776_3_1-choice_2-label" for="input_8e7294f17cad424798168b1273ff7776_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_8e7294f17cad424798168b1273ff7776_3_1"> \(f\) is a bijection
</label>
</div>
<div class="field">
<input type="radio" name="input_8e7294f17cad424798168b1273ff7776_3_1" id="input_8e7294f17cad424798168b1273ff7776_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="8e7294f17cad424798168b1273ff7776_3_1-choice_3-label" for="input_8e7294f17cad424798168b1273ff7776_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_8e7294f17cad424798168b1273ff7776_3_1"> \(f\) is neither a bijection nor a surjection
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<p>\(A =\{-5,-4,-3, -2,-1,0,1,2, 3, 4,5\}\)</p>
<p>\(B=\{0, 1, 4, 9, 16, 25\}\) </p>
<p>\(f(x)=x^2\)</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8e7294f17cad424798168b1273ff7776_4_1">
<fieldset aria-describedby="status_8e7294f17cad424798168b1273ff7776_4_1">
<div class="field">
<input type="radio" name="input_8e7294f17cad424798168b1273ff7776_4_1" id="input_8e7294f17cad424798168b1273ff7776_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="8e7294f17cad424798168b1273ff7776_4_1-choice_0-label" for="input_8e7294f17cad424798168b1273ff7776_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_8e7294f17cad424798168b1273ff7776_4_1"> \(f\) is an injection (but not a surjection)
</label>
</div>
<div class="field">
<input type="radio" name="input_8e7294f17cad424798168b1273ff7776_4_1" id="input_8e7294f17cad424798168b1273ff7776_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="8e7294f17cad424798168b1273ff7776_4_1-choice_1-label" for="input_8e7294f17cad424798168b1273ff7776_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8e7294f17cad424798168b1273ff7776_4_1"> \(f\) is a surjection (but not an injection)
</label>
</div>
<div class="field">
<input type="radio" name="input_8e7294f17cad424798168b1273ff7776_4_1" id="input_8e7294f17cad424798168b1273ff7776_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="8e7294f17cad424798168b1273ff7776_4_1-choice_2-label" for="input_8e7294f17cad424798168b1273ff7776_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_8e7294f17cad424798168b1273ff7776_4_1"> \(f\) is a bijection
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<div class="field">
<input type="radio" name="input_8e7294f17cad424798168b1273ff7776_4_1" id="input_8e7294f17cad424798168b1273ff7776_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="8e7294f17cad424798168b1273ff7776_4_1-choice_3-label" for="input_8e7294f17cad424798168b1273ff7776_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_8e7294f17cad424798168b1273ff7776_4_1"> \(f\) is neither a bijection nor a surjection
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Problem 3
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<p> A set is <i>dense</i> (relative to a given ordering) if and only if there is a member of the set between any two members of the set (according to that ordering).</p>
<p>Does the fact that a set is dense (relative to some ordering) entail that the set is bigger than the set of the natural numbers? </p>
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Problem 4
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<p>Between which of these infinite sets is it possible to construct a bijection: </p>
<p>The set of natural numbers and the set of integers?</p>
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<p>The set of prime numbers and the set of real numbers?</p>
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<p>The set of rational numbers and the set of real numbers between 0 and 1?</p>
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<p>The set of real numbers and the power set of the real numbers?</p>
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Problem 5
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<p>Suppose that Hilbert's Hotel is completely full. We showed that when as many new guests as there are natural numbers show up, the new guests can be accommodated.</p>
<p>Now suppose that as many new guests as there are <i>real</i> numbers show up. Can they be accommodated? (In other words: can we fit <i>uncountably</i> new guests in Hilbert's Hotel?)</p>
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Problem 6
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>There is a bijection between the set of natural numbers and the set of prime numbers. To see this, note that one can assign the smallest prime number (i.e. the zeroth-smallest prime number) to \(0\), the next smallest prime number (i.e. the first-smallest prime number) to \(1\), and, in general, assigning the \(n\)th smallest prime to \(n\). </p>
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<h3 class="hd hd-3 problem-header" id="37ad3f2a0a3e43d5801b974820cf887b-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@37ad3f2a0a3e43d5801b974820cf887b-problem-progress" tabindex="-1">
Problem 7
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<legend id="37ad3f2a0a3e43d5801b974820cf887b_2_1-legend" class="response-fieldset-legend field-group-hd">Is there a bijection between the set of natural numbers, \(\mathbb{N}\), and the set of all pairs of natural numbers, \(\{ \langle n, m \rangle : n, m \in \mathbb{N}\}\)?</legend>
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<input type="radio" name="input_37ad3f2a0a3e43d5801b974820cf887b_2_1" id="input_37ad3f2a0a3e43d5801b974820cf887b_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="37ad3f2a0a3e43d5801b974820cf887b_2_1-choice_0-label" for="input_37ad3f2a0a3e43d5801b974820cf887b_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_37ad3f2a0a3e43d5801b974820cf887b_2_1"> Yes
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<legend id="37ad3f2a0a3e43d5801b974820cf887b_3_1-legend" class="response-fieldset-legend field-group-hd">Is there a bijection between the set of natural numbers, \(\mathbb{N}\), and the set \( \{x : x\) is a function that assigns each natural number to a member of \( \{ 1, 2, 3, 4, 5\} \} \)?</legend>
<p class="question-description" id="description_37ad3f2a0a3e43d5801b974820cf887b_2_1">Hint: Can you derive a contradiction from the assumption that there is such a bijection? How about from the assumption that there isn't?</p>
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<h3 class="hd hd-3 problem-header" id="b5dc5c0cd7054512bf6e565977e24a15-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@b5dc5c0cd7054512bf6e565977e24a15-problem-progress" tabindex="-1">
Problem 8
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Consider the infinite tree in the image below. (When fully spelled out, the tree contains one row for each natural number. The zero-th row contains one node, the first row contains two nodes, the second row contains four nodes, and, in general, the nth row contains \(2^n\) nodes.)</p>
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<img src="/assets/courseware/v1/69c17eb1959fa0fbf9adbe55b4bdcfe7/asset-v1:MITx+24.118x+2T2020+type@asset+block/tree.png" alt="tree" width="500"/>
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<legend id="b5dc5c0cd7054512bf6e565977e24a15_2_1-legend" class="response-fieldset-legend field-group-hd">Is there a bijection between the set of nodes of the tree and the set of natural numbers?</legend>
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<legend id="b5dc5c0cd7054512bf6e565977e24a15_3_1-legend" class="response-fieldset-legend field-group-hd">Is there a bijection between the set of paths of this tree and set of natural numbers?</legend>
<p class="question-description" id="description_b5dc5c0cd7054512bf6e565977e24a15_2_1">A path is an infinite sequence of nodes which starts at the top of the tree and contains a node at every row, with each node connected to its successor by an edge. Paths can be represented as infinite sequences of zeroes and ones.</p>
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