Discrete Mathematics | CS 139: Theory of Computation

<h1 id="discrete-mathematics">Discrete Mathematics</h1>

<hr />

<h2 id="administrivia">Administrivia</h2>
<ol>
  <li>A few of you haven’t submitted your reading journals for today.
    <ul>
      <li>These are due at <strong>8:00 AM</strong> the morning of the corresponding class period</li>
      <li>For today only, I’ll give credit if you submit one before 10pm tonight</li>
    </ul>
  </li>
  <li class="fragment">Assignment 1
    <ul>
      <li>Due <strong>Tuesday, February 4th</strong> (before class)</li>
    </ul>
  </li>
</ol>

<h2 id="questionsabout-anything">Questions…about anything?</h2>

<h2 id="course-themes">Course Themes</h2>
<ol>
  <li>Complexity Theory
    <ul>
      <li>The study of the <strong>difficulty</strong> of computational problems</li>
    </ul>
  </li>
  <li>Computability Theory
    <ul>
      <li>The study of the <strong>limits</strong> of computation</li>
    </ul>
  </li>
  <li>Automata Theory
    <ul>
      <li>The study of abstract <strong>machines</strong></li>
    </ul>
  </li>
</ol>

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<p>We will cover these topics in a 3, 2, 1 order</p>

<h1 id="review">REVIEW</h1>

<h2 id="exercise-1">Exercise 1</h2>
<p>Let <code class="language-plaintext highlighter-rouge">$\sim:\mathbb{R}\times\mathbb{R}\rightarrow\{\text{true}, \text{false}\}$</code> be the binary relation:
<code class="language-plaintext highlighter-rouge">$$a\sim b \iff a-b \in \mathbb{Z}.$$</code></p>

<hr />

<ol>
  <li>Intuitively, what is the relation <code class="language-plaintext highlighter-rouge">$\sim$</code> doing?
    <ul>
      <li class="fragment">Is <code class="language-plaintext highlighter-rouge">$5\sim 6$</code>?</li>
      <li class="fragment">Is <code class="language-plaintext highlighter-rouge">$-7.3\sim 3.3$</code>?</li>
      <li class="fragment">Is <code class="language-plaintext highlighter-rouge">$\pi\sim e$</code>?</li>
    </ul>
  </li>
  <li class="fragment">Is <code class="language-plaintext highlighter-rouge">$\sim$</code>  an <strong>equivalence relation</strong>? Why or why not?</li>
</ol>

<h2 id="exercise-2">Exercise 2</h2>
<p><strong>Claim:</strong> For all sets <code class="language-plaintext highlighter-rouge">$A$</code> and <code class="language-plaintext highlighter-rouge">$B$</code>, the following holds:
<code class="language-plaintext highlighter-rouge">$$A\setminus B \subseteq A\cap B$$</code></p>

<hr />

<p>Is this claim true? Why or why not?</p>

<h2 id="exercise-3">Exercise 3</h2>
<p><strong>Claim:</strong> For all sets <code class="language-plaintext highlighter-rouge">$A$</code> and <code class="language-plaintext highlighter-rouge">$B$</code>, the following holds:
<code class="language-plaintext highlighter-rouge">$$x\not\in B \implies x\not\in A\setminus (A\setminus B)$$</code></p>

<hr />

<p>Is this claim true? Why or why not?</p>

<h2 id="exercise-4">Exercise 4</h2>
<p><strong>Claim:</strong> Every undirected graph <code class="language-plaintext highlighter-rouge">$G = (V,E)$</code> with at least two vertices has two vertices with the same degree.</p>

<hr />

<p>Is this claim true? Why or why not?</p>

<h1 id="assignment-1">Assignment 1</h1>
<p>Due <strong>Tuesday, February 4th</strong> (before class)</p>

<ol>
  <li>Read the <strong>syllabus</strong></li>
  <li>Complete a <strong>questionnaire</strong></li>
  <li>Review of <strong>prereqs</strong></li>
</ol>