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<li><a href="#boolean-values" id="toc-boolean-values" class="nav-link active" data-scroll-target="#boolean-values"> <span class="header-section-number">4.1</span> Boolean values</a></li>
<li><a href="#algebra-of-inequalities" id="toc-algebra-of-inequalities" class="nav-link" data-scroll-target="#algebra-of-inequalities"> <span class="header-section-number">4.2</span> Algebra of inequalities</a>
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<h1 class="title d-none d-lg-block"><span class="chapter-number">4</span>&nbsp; <span class="chapter-title">Inequalities, Logical expressions</span></h1>
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<p>In this section we use the following package:</p>
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">CalculusWithJulia </span><span class="co"># loads the `SpecialFunctions` package</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<section id="boolean-values" class="level2" data-number="4.1">
<h2 data-number="4.1" class="anchored" data-anchor-id="boolean-values"><span class="header-section-number">4.1</span> Boolean values</h2>
<p>In mathematics it is common to test if an expression is true or false. For example, is the point <span class="math inline">\((1,2)\)</span> inside the disc <span class="math inline">\(x^2 + y^2 \leq 1\)</span>? We would check this by substituting <span class="math inline">\(1\)</span> for <span class="math inline">\(x\)</span> and <span class="math inline">\(2\)</span> for <span class="math inline">\(y\)</span>, evaluating both sides of the inequality and then assessing if the relationship is true or false. In this case, we end up with a comparison of <span class="math inline">\(5 \leq 1\)</span>, which we of course know is false.</p>
<p><code>Julia</code> provides numeric comparisons that allow this notation to be exactly mirrored:</p>
<div class="cell" data-hold="true" data-execution_count="4">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>x, y <span class="op">=</span> <span class="fl">1</span>, <span class="fl">2</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>x<span class="op">^</span><span class="fl">2</span> <span class="op">+</span> y<span class="op">^</span><span class="fl">2</span> <span class="op">&lt;=</span> <span class="fl">1</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="5">
<pre><code>false</code></pre>
</div>
</div>
<p>The response is <code>false</code>, as expected. <code>Julia</code> provides <a href="http://en.wikipedia.org/wiki/Boolean_data_type">Boolean</a> values <code>true</code> and <code>false</code> for such questions. The same process is followed as was described mathematically.</p>
<p>The set of numeric comparisons is nearly the same as the mathematical counterparts: <code>&lt;</code>, <code>&lt;=</code>, <code>==</code>, <code>&gt;=</code>, <code>&gt;</code>. The syntax for less than or equal can also be represented with the Unicode <code>≤</code> (generated by <code>\le[tab]</code>). Similarly, for greater than or equal, there is <code>\ge[tab]</code>.</p>
<div class="callout-warning callout callout-style-default callout-captioned">
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<p>The use of <code>==</code> is necessary, as <code>=</code> is used for assignment and mutation.”)</p>
</div>
</div>
<p>The <code>!</code> operator takes a boolean value and negates it. It uses prefix notation:</p>
<div class="cell" data-execution_count="5">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>!<span class="cn">true</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="6">
<pre><code>false</code></pre>
</div>
</div>
<p>For convenience, <code>a != b</code> can be used in place of <code>!(a == b)</code>.</p>
</section>
<section id="algebra-of-inequalities" class="level2" data-number="4.2">
<h2 data-number="4.2" class="anchored" data-anchor-id="algebra-of-inequalities"><span class="header-section-number">4.2</span> Algebra of inequalities</h2>
<p>To illustrate, let’s see that the algebra of expressions works as expected.</p>
<p>For example, if <span class="math inline">\(a &lt; b\)</span> then for any <span class="math inline">\(c\)</span> it is also true that <span class="math inline">\(a + c &lt; b + c\)</span>.</p>
<p>We can’t “prove” this through examples, but we can investigate it by the choice of various values of <span class="math inline">\(a\)</span>, <span class="math inline">\(b\)</span>, and <span class="math inline">\(c\)</span>. For example:</p>
<div class="cell" data-hold="true" data-execution_count="6">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a>a,b,c <span class="op">=</span> <span class="fl">1</span>,<span class="fl">2</span>,<span class="fl">3</span></span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>a <span class="op">&lt;</span> b, a <span class="op">+</span> c <span class="op">&lt;</span> b <span class="op">+</span> c</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="7">
<pre><code>(true, true)</code></pre>
</div>
</div>
<p>Or in reverse:</p>
<div class="cell" data-hold="true" data-execution_count="7">
<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a>a,b,c <span class="op">=</span> <span class="fl">3</span>,<span class="fl">2</span>,<span class="fl">1</span></span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a>a <span class="op">&lt;</span> b, a <span class="op">+</span> c <span class="op">&lt;</span> b <span class="op">+</span> c</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="8">
<pre><code>(false, false)</code></pre>
</div>
</div>
<p>Trying other choices will show that the two answers are either both <code>false</code> or both <code>true</code>.</p>
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<p>Well, almost… When <code>Inf</code> or <code>NaN</code> are involved, this may not hold, for example <code>1 + Inf &lt; 2 + Inf</code> is actually <code>false</code>. As would be <code>1 + (typemax(1)-1) &lt; 2 + (typemax(1)-1)</code>.</p>
</div>
</div>
<p>So adding or subtracting most any finite value from an inequality will preserve the inequality, just as it does for equations.</p>
<p>What about addition and multiplication?</p>
<p>Consider the case <span class="math inline">\(a &lt; b\)</span> and <span class="math inline">\(c &gt; 0\)</span>. Then <span class="math inline">\(ca &lt; cb\)</span>. Here we investigate using <span class="math inline">\(3\)</span> random values (which will be positive):</p>
<div class="cell" data-hold="true" data-execution_count="8">
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>a,b,c <span class="op">=</span> <span class="fu">rand</span>(<span class="fl">3</span>) <span class="co"># 3 random numbers in [0,1)</span></span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a>a <span class="op">&lt;</span> b, c<span class="op">*</span>a <span class="op">&lt;</span> c<span class="op">*</span>b</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="9">
<pre><code>(true, true)</code></pre>
</div>
</div>
<p>Whenever these two commands are run, the two logical values should be identical, even though the specific values of <code>a</code>, <code>b</code>, and <code>c</code> will vary.</p>
<p>The restriction that <span class="math inline">\(c &gt; 0\)</span> is needed. For example, if <span class="math inline">\(c = -1\)</span>, then we have <span class="math inline">\(a &lt; b\)</span> if and only if <span class="math inline">\(-a &gt; -b\)</span>. That is the inequality is “flipped.”</p>
<div class="cell" data-hold="true" data-execution_count="9">
<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a>a,b <span class="op">=</span> <span class="fu">rand</span>(<span class="fl">2</span>)</span>
<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a>a <span class="op">&lt;</span> b, <span class="op">-</span>a <span class="op">&gt;</span> <span class="op">-</span>b</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="10">
<pre><code>(false, false)</code></pre>
</div>
</div>
<p>Again, whenever this is run, the two logical values should be the same. The values <span class="math inline">\(a\)</span> and <span class="math inline">\(-a\)</span> are the same distance from <span class="math inline">\(0\)</span>, but on opposite sides. Hence if <span class="math inline">\(0 &lt; a &lt; b\)</span>, then <span class="math inline">\(b\)</span> is farther from <span class="math inline">\(0\)</span> than <span class="math inline">\(a\)</span>, so <span class="math inline">\(-b\)</span> will be farther from <span class="math inline">\(0\)</span> than <span class="math inline">\(-a\)</span>, which in this case says <span class="math inline">\(-b &lt; -a\)</span>, as expected.</p>
<p>Finally, we have the case of division. The relation of <span class="math inline">\(x\)</span> and <span class="math inline">\(1/x\)</span> (for <span class="math inline">\(x &gt; 0\)</span>) is that the farther <span class="math inline">\(x\)</span> is from <span class="math inline">\(0\)</span>, the closer <span class="math inline">\(1/x\)</span> is to <span class="math inline">\(0\)</span>. So large values of <span class="math inline">\(x\)</span> make small values of <span class="math inline">\(1/x\)</span>. This leads to this fact for <span class="math inline">\(a,b &gt; 0\)</span>: <span class="math inline">\(a &lt; b\)</span> if and only if <span class="math inline">\(1/a &gt; 1/b\)</span>.</p>
<p>We can check with random values again:</p>
<div class="cell" data-hold="true" data-execution_count="10">
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a>a,b <span class="op">=</span> <span class="fu">rand</span>(<span class="fl">2</span>)</span>
<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a>a <span class="op">&lt;</span> b, <span class="fl">1</span><span class="op">/</span>a <span class="op">&gt;</span> <span class="fl">1</span><span class="op">/</span>b</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="11">
<pre><code>(true, true)</code></pre>
</div>
</div>
<p>In summary we investigated numerically that the following hold:</p>
<ul>
<li><code>a &lt; b</code> if and only if <code>a + c &lt; b + c</code> for all finite <code>a</code>, <code>b</code>, and <code>c</code>.</li>
<li><code>a &lt; b</code> if and only if <code>c*a &lt; c*b</code> for all finite <code>a</code> and <code>b</code>, and finite, positive <code>c</code>.</li>
<li><code>a &lt; b</code> if and only if <code>-a &gt; -b</code> for all finite <code>a</code> and <code>b</code>.</li>
<li><code>a &lt; b</code> if and only if <code>1/a &gt; 1/b</code> for all finite, positive <code>a</code> and <code>b</code>.</li>
</ul>
<section id="examples" class="level3" data-number="4.2.1">
<h3 data-number="4.2.1" class="anchored" data-anchor-id="examples"><span class="header-section-number">4.2.1</span> Examples</h3>
<p>We now show some inequalities highlighted on this <a href="http://en.wikipedia.org/wiki/Inequality_%28mathematics%29">Wikipedia</a> page.</p>
<p>Numerically investigate the fact <span class="math inline">\(e^x \geq 1 + x\)</span> by showing it is true for three different values of <span class="math inline">\(x\)</span>. We pick <span class="math inline">\(x=-1\)</span>, <span class="math inline">\(0\)</span>, and <span class="math inline">\(1\)</span>:</p>
<div class="cell" data-hold="true" data-execution_count="11">
<div class="sourceCode cell-code" id="cb16"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb16-1"><a href="#cb16-1" aria-hidden="true" tabindex="-1"></a>x <span class="op">=</span> <span class="op">-</span><span class="fl">1</span>; <span class="fu">exp</span>(x) <span class="op">&gt;=</span> <span class="fl">1</span> <span class="op">+</span> x</span>
<span id="cb16-2"><a href="#cb16-2" aria-hidden="true" tabindex="-1"></a>x <span class="op">=</span> <span class="fl">0</span>; <span class="fu">exp</span>(x) <span class="op">&gt;=</span> <span class="fl">1</span> <span class="op">+</span> x</span>
<span id="cb16-3"><a href="#cb16-3" aria-hidden="true" tabindex="-1"></a>x <span class="op">=</span> <span class="fl">1</span>; <span class="fu">exp</span>(x) <span class="op">&gt;=</span> <span class="fl">1</span> <span class="op">+</span> x</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="12">
<pre><code>true</code></pre>
</div>
</div>
<p>Now, let’s investigate that for any distinct real numbers, <span class="math inline">\(a\)</span> and <span class="math inline">\(b\)</span>, that</p>
<p><span class="math display">\[
\frac{e^b - e^a}{b - a} &gt; e^{(a+b)/2}
\]</span></p>
<p>For this, we use <code>rand(2)</code> to generate two random numbers in <span class="math inline">\([0,1)\)</span>:</p>
<div class="cell" data-hold="true" data-execution_count="12">
<div class="sourceCode cell-code" id="cb18"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb18-1"><a href="#cb18-1" aria-hidden="true" tabindex="-1"></a>a, b <span class="op">=</span> <span class="fu">rand</span>(<span class="fl">2</span>)</span>
<span id="cb18-2"><a href="#cb18-2" aria-hidden="true" tabindex="-1"></a>(<span class="fu">exp</span>(b) <span class="op">-</span> <span class="fu">exp</span>(a)) <span class="op">/</span> (b<span class="op">-</span>a) <span class="op">&gt;</span> <span class="fu">exp</span>((a<span class="op">+</span>b)<span class="op">/</span><span class="fl">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="13">
<pre><code>true</code></pre>
</div>
</div>
<p>This should evaluate to <code>true</code> for any random choice of <code>a</code> and <code>b</code> returned by <code>rand(2)</code>.</p>
<p>Finally, let’s investigate the fact that the harmonic mean, <span class="math inline">\(2/(1/a + 1/b)\)</span> is less than or equal to the geometric mean, <span class="math inline">\(\sqrt{ab}\)</span>, which is less than or equal to the quadratic mean, <span class="math inline">\(\sqrt{a^2 + b^2}/\sqrt{2}\)</span>, using two randomly chosen values:</p>
<div class="cell" data-hold="true" data-execution_count="13">
<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a>a, b <span class="op">=</span> <span class="fu">rand</span>(<span class="fl">2</span>)</span>
<span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a>h <span class="op">=</span> <span class="fl">2</span> <span class="op">/</span> (<span class="fl">1</span><span class="op">/</span>a <span class="op">+</span> <span class="fl">1</span><span class="op">/</span>b)</span>
<span id="cb20-3"><a href="#cb20-3" aria-hidden="true" tabindex="-1"></a>g <span class="op">=</span> (a <span class="op">*</span> b) <span class="op">^</span> (<span class="fl">1</span> <span class="op">/</span> <span class="fl">2</span>)</span>
<span id="cb20-4"><a href="#cb20-4" aria-hidden="true" tabindex="-1"></a>q <span class="op">=</span> <span class="fu">sqrt</span>((a<span class="op">^</span><span class="fl">2</span> <span class="op">+</span> b<span class="op">^</span><span class="fl">2</span>) <span class="op">/</span> <span class="fl">2</span>)</span>
<span id="cb20-5"><a href="#cb20-5" aria-hidden="true" tabindex="-1"></a>h <span class="op">&lt;=</span> g, g <span class="op">&lt;=</span> q</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="14">
<pre><code>(true, true)</code></pre>
</div>
</div>
</section>
</section>
<section id="chaining-combining-expressions-absolute-values" class="level2" data-number="4.3">
<h2 data-number="4.3" class="anchored" data-anchor-id="chaining-combining-expressions-absolute-values"><span class="header-section-number">4.3</span> Chaining, combining expressions: absolute values</h2>
<p>The absolute value notation can be defined through cases:</p>
<p><span class="math display">\[
\lvert x\rvert = \begin{cases}
x &amp; x \geq 0\\
-x &amp; \text{otherwise}.
\end{cases}
\]</span></p>
<p>The interpretation of <span class="math inline">\(\lvert x\rvert\)</span>, as the distance on the number line of <span class="math inline">\(x\)</span> from <span class="math inline">\(0\)</span>, means that many relationships are naturally expressed in terms of absolute values. For example, a simple shift: <span class="math inline">\(\lvert x -c\rvert\)</span> is related to the distance <span class="math inline">\(x\)</span> is from the number <span class="math inline">\(c\)</span>. As common as they are, the concept can still be confusing when inequalities are involved.</p>
<p>For example, the expression <span class="math inline">\(\lvert x - 5\rvert &lt; 7\)</span> has solutions which are all values of <span class="math inline">\(x\)</span> within <span class="math inline">\(7\)</span> units of <span class="math inline">\(5\)</span>. This would be the values <span class="math inline">\(-2&lt; x &lt; 12\)</span>. If this isn’t immediately intuited, then formally <span class="math inline">\(\lvert x - 5\rvert &lt;7\)</span> is a compact representation of a chain of inequalities: <span class="math inline">\(-7 &lt; x-5 &lt; 7\)</span>. (Which is really two combined inequalities: <span class="math inline">\(-7 &lt; x-5\)</span> <em>and</em> <span class="math inline">\(x-5 &lt; 7\)</span>.) We can “add” <span class="math inline">\(5\)</span> to each side to get <span class="math inline">\(-2 &lt; x &lt; 12\)</span>, using the fact that adding by a finite number does not change the inequality sign.</p>
<p>Julia’s precedence for logical expressions, allows such statements to mirror the mathematical notation:</p>
<div class="cell" data-execution_count="14">
<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a>x <span class="op">=</span> <span class="fl">18</span></span>
<span id="cb22-2"><a href="#cb22-2" aria-hidden="true" tabindex="-1"></a><span class="fu">abs</span>(x <span class="op">-</span> <span class="fl">5</span>) <span class="op">&lt;</span> <span class="fl">7</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="15">
<pre><code>false</code></pre>
</div>
</div>
<p>This is to be expected, but we could also have written:</p>
<div class="cell" data-execution_count="15">
<div class="sourceCode cell-code" id="cb24"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a><span class="op">-</span><span class="fl">7</span> <span class="op">&lt;</span> x <span class="op">-</span> <span class="fl">5</span> <span class="op">&lt;</span> <span class="fl">7</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="16">
<pre><code>false</code></pre>
</div>
</div>
<p>Read aloud this would be “minus <span class="math inline">\(7\)</span> is less than <span class="math inline">\(x\)</span> minus <span class="math inline">\(5\)</span> <strong>and</strong> <span class="math inline">\(x\)</span> minus <span class="math inline">\(5\)</span> is less than <span class="math inline">\(7\)</span>”.</p>
<p>The “and” equations can be combined as above with a natural notation. However, an equation like <span class="math inline">\(\lvert x - 5\rvert &gt; 7\)</span> would emphasize an <strong>or</strong> and be “<span class="math inline">\(x\)</span> minus <span class="math inline">\(5\)</span> less than minus <span class="math inline">\(7\)</span> <strong>or</strong> <span class="math inline">\(x\)</span> minus <span class="math inline">\(5\)</span> greater than <span class="math inline">\(7\)</span>”. Expressing this requires some new notation.</p>
<p>The <em>boolean shortcut operators</em> <code>&amp;&amp;</code> and <code>||</code> implement “and” and “or.” (There are also <em>bitwise</em> boolean operators <code>&amp;</code> and <code>|</code>, but we only describe the former.)</p>
<p>Thus we could write <span class="math inline">\(-7 &lt; x-5 &lt; 7\)</span> as</p>
<div class="cell" data-execution_count="16">
<div class="sourceCode cell-code" id="cb26"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a>(<span class="op">-</span><span class="fl">7</span> <span class="op">&lt;</span> x <span class="op">-</span> <span class="fl">5</span>) <span class="op">&amp;&amp;</span> (x <span class="op">-</span> <span class="fl">5</span> <span class="op">&lt;</span> <span class="fl">7</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="17">
<pre><code>false</code></pre>
</div>
</div>
<p>and could write <span class="math inline">\(\lvert x-5\rvert &gt; 7\)</span> as</p>
<div class="cell" data-execution_count="17">
<div class="sourceCode cell-code" id="cb28"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a>(x <span class="op">-</span> <span class="fl">5</span> <span class="op">&lt;</span> <span class="op">-</span><span class="fl">7</span>) <span class="op">||</span> (x <span class="op">-</span> <span class="fl">5</span> <span class="op">&gt;</span> <span class="fl">7</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="18">
<pre><code>true</code></pre>
</div>
</div>
<p>(The first expression is false for <span class="math inline">\(x=18\)</span> and the second expression true, so the “or”ed result is <code>true</code> and the “and” result if <code>false</code>.)</p>
<section id="example" class="level5">
<h5 class="anchored" data-anchor-id="example">Example</h5>
<p>One of <a href="http://en.wikipedia.org/wiki/De_Morgan%27s_laws">DeMorgan’s Laws</a> states that “not (A and B)” is the same as “(not A) or (not B)”. This is a kind of distributive law for “not”, but note how the “and” changes to “or”. We can verify this law systematically. For example, the following shows it true for <span class="math inline">\(1\)</span> of the <span class="math inline">\(4\)</span> possible cases for the pair <code>A</code>, <code>B</code> to take:</p>
<div class="cell" data-execution_count="18">
<div class="sourceCode cell-code" id="cb30"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a>A,B <span class="op">=</span> <span class="cn">true</span>, <span class="cn">false</span> <span class="co">## also true, true; false, true; and false, false</span></span>
<span id="cb30-2"><a href="#cb30-2" aria-hidden="true" tabindex="-1"></a>!(A <span class="op">&amp;&amp;</span> B) <span class="op">==</span> !A <span class="op">||</span> !B</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="19">
<pre><code>true</code></pre>
</div>
</div>
</section>
</section>
<section id="precedence" class="level2" data-number="4.4">
<h2 data-number="4.4" class="anchored" data-anchor-id="precedence"><span class="header-section-number">4.4</span> Precedence</h2>
<p>The question of when parentheses are needed and when they are not is answered by the <a href="http://julia.readthedocs.org/en/latest/manual/mathematical-operations/#operator-precedence">precedence</a> rules implemented. Earlier, we wrote</p>
<div class="cell" data-execution_count="19">
<div class="sourceCode cell-code" id="cb32"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a>(x <span class="op">-</span> <span class="fl">5</span> <span class="op">&lt;</span> <span class="op">-</span><span class="fl">7</span>) <span class="op">||</span> (x <span class="op">-</span> <span class="fl">5</span> <span class="op">&gt;</span> <span class="fl">7</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="20">
<pre><code>true</code></pre>
</div>
</div>
<p>To represent <span class="math inline">\(\lvert x-5\rvert &gt; 7\)</span>. Were the parentheses necessary? Let’s just check.</p>
<div class="cell" data-execution_count="20">
<div class="sourceCode cell-code" id="cb34"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb34-1"><a href="#cb34-1" aria-hidden="true" tabindex="-1"></a>x <span class="op">-</span> <span class="fl">5</span> <span class="op">&lt;</span> <span class="op">-</span><span class="fl">7</span> <span class="op">||</span> x <span class="op">-</span> <span class="fl">5</span> <span class="op">&gt;</span> <span class="fl">7</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="21">
<pre><code>true</code></pre>
</div>
</div>
<p>So no, they were not in this case.</p>
<p>An operator (such as <code>&lt;</code>, <code>&gt;</code>, <code>||</code> above) has an associated associativity and precedence. The associativity is whether an expression like <code>a - b - c</code> is <code>(a-b) - c</code> or <code>a - (b-c)</code>. The former being left associative, the latter right. Of issue here is <em>precedence</em>, as in with two or more different operations, which happens first, second, <span class="math inline">\(\dots\)</span>.</p>
<p>The table in the manual on <a href="https://docs.julialang.org/en/v1/manual/mathematical-operations/#Operator-Precedence-and-Associativity">operator precedence and associativity</a> shows that for these operations “control flow” (the <code>&amp;&amp;</code> above) is lower than “comparisons” (the <code>&lt;</code>, <code>&gt;</code>), which are lower than “Addition” (the <code>-</code> above). So the expression without parentheses would be equivalent to:</p>
<div class="cell" data-execution_count="21">
<div class="sourceCode cell-code" id="cb36"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb36-1"><a href="#cb36-1" aria-hidden="true" tabindex="-1"></a>((x<span class="op">-</span><span class="fl">5</span>) <span class="op">&lt;</span> <span class="op">-</span><span class="fl">7</span>) <span class="op">&amp;&amp;</span> ((x<span class="op">-</span><span class="fl">5</span>) <span class="op">&gt;</span> <span class="fl">7</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="22">
<pre><code>false</code></pre>
</div>
</div>
<p>(This is different than the precedence of the bitwise boolean operators, which have <code>&amp;</code> with “Multiplication” and <code>|</code> with “Addition”, so <code>x-5 &lt; 7 | x - 5 &gt; 7</code> would need parentheses.)</p>
<p>A thorough understanding of the precedence rules can help eliminate unnecessary parentheses, but in most cases it is easier just to put them in.</p>
</section>
<section id="arithmetic-with" class="level2" data-number="4.5">
<h2 data-number="4.5" class="anchored" data-anchor-id="arithmetic-with"><span class="header-section-number">4.5</span> Arithmetic with</h2>
<p>For convenience, basic arithmetic can be performed with Boolean values, <code>false</code> becomes <span class="math inline">\(0\)</span> and true <span class="math inline">\(1\)</span>. For example, both these expressions make sense:</p>
<div class="cell" data-execution_count="22">
<div class="sourceCode cell-code" id="cb38"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb38-1"><a href="#cb38-1" aria-hidden="true" tabindex="-1"></a><span class="cn">true</span> <span class="op">+</span> <span class="cn">true</span> <span class="op">+</span> <span class="cn">false</span>, <span class="cn">false</span> <span class="op">*</span> <span class="fl">1000</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="23">
<pre><code>(2, 0)</code></pre>
</div>
</div>
<p>The first example shows a common means used to count the number of <code>true</code> values in a collection of Boolean values - just add them.</p>
<p>This can be cleverly exploited. For example, the following expression returns <code>x</code> when it is positive and <span class="math inline">\(0\)</span> otherwise:</p>
<div class="cell" data-execution_count="23">
<div class="sourceCode cell-code" id="cb40"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb40-1"><a href="#cb40-1" aria-hidden="true" tabindex="-1"></a>(x <span class="op">&gt;</span> <span class="fl">0</span>) <span class="op">*</span> x</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="24">
<pre><code>18</code></pre>
</div>
</div>
<p>There is a built in function, <code>max</code> that can be used for this: <code>max(0, x)</code>.</p>
<p>This expression returns <code>x</code> if it is between <span class="math inline">\(-10\)</span> and <span class="math inline">\(10\)</span> and otherwise <span class="math inline">\(-10\)</span> or <span class="math inline">\(10\)</span> depending on whether <span class="math inline">\(x\)</span> is negative or positive.</p>
<div class="cell" data-execution_count="24">
<div class="sourceCode cell-code" id="cb42"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a>(x <span class="op">&lt;</span> <span class="op">-</span><span class="fl">10</span>)<span class="fu">*</span>(<span class="op">-</span><span class="fl">10</span>) <span class="op">+</span> (x <span class="op">&gt;=</span> <span class="op">-</span><span class="fl">10</span>)<span class="fu">*</span>(x <span class="op">&lt;</span> <span class="fl">10</span>) <span class="op">*</span> x <span class="op">+</span> (x<span class="op">&gt;=</span><span class="fl">10</span>)<span class="op">*</span><span class="fl">10</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="25">
<pre><code>10</code></pre>
</div>
</div>
<p>The <code>clamp(x, a, b)</code> performs this task more generally, and is used as in <code>clamp(x, -10, 10)</code>.</p>
</section>
<section id="questions" class="level2" data-number="4.6">
<h2 data-number="4.6" class="anchored" data-anchor-id="questions"><span class="header-section-number">4.6</span> Questions</h2>
<section id="question" class="level6">
<h6 class="anchored" data-anchor-id="question">Question</h6>
<p>Is <code>e^pi</code> or <code>pi^e</code> greater?</p>
<div class="cell" data-hold="true" data-execution_count="25">
<div class="cell-output cell-output-display" data-execution_count="26">
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="12435863512260717783" data-controltype="">
<div class="form-group ">
<div class="controls">
<div class="form" id="controls_12435863512260717783">
<div style="padding-top: 5px">
<div class="form-check">
<label class="form-check-label" for="radio_12435863512260717783_1">
<input class="form-check-input" type="radio" name="radio_12435863512260717783" id="radio_12435863512260717783_1" value="1">
<span class="label-body px-1">
<code>e^pi</code> is less than <code>pi^e</code>
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_12435863512260717783_2">
<input class="form-check-input" type="radio" name="radio_12435863512260717783" id="radio_12435863512260717783_2" value="2">
<span class="label-body px-1">
<code>e^pi</code> is greater than <code>pi^e</code>
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_12435863512260717783_3">
<input class="form-check-input" type="radio" name="radio_12435863512260717783" id="radio_12435863512260717783_3" value="3">
<span class="label-body px-1">
<code>e^pi</code> is equal to <code>pi^e</code>
</span>
</label>
</div>
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</section>
<section id="question-1" class="level6">
<h6 class="anchored" data-anchor-id="question-1">Question</h6>
<p>Is <span class="math inline">\(\sin(1000)\)</span> positive?</p>
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Yes
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No
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</section>
<section id="question-2" class="level6">
<h6 class="anchored" data-anchor-id="question-2">Question</h6>
<p>Suppose you know <span class="math inline">\(0 &lt; a &lt; b\)</span>. What can you say about the relationship between <span class="math inline">\(-1/a\)</span> and <span class="math inline">\(-1/b\)</span>?</p>
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\(-1/a &gt; -1/b\)
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\(-1/a \geq -1/b\)
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\(-1/a &lt; -1/b\)
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</section>
<section id="question-3" class="level6">
<h6 class="anchored" data-anchor-id="question-3">Question</h6>
<p>Suppose you know <span class="math inline">\(a &lt; 0 &lt; b\)</span>, is it true that <span class="math inline">\(1/a &gt; 1/b\)</span>?</p>
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Yes, it is always true.
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It is never true, as \(1/a\) is negative and \(1/b\) is positive
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It can sometimes be true, though not always.
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</section>
<section id="question-4" class="level6">
<h6 class="anchored" data-anchor-id="question-4">Question</h6>
<p>The <code>airyai</code> <a href="http://en.wikipedia.org/wiki/Airy_function">function</a> is a special function named after a British Astronomer who realized the function’s value in his studies of the rainbow. The <code>SpecialFunctions</code> package must be loaded to include this function, which is done with the accompanying package <code>CalculusWithJulia</code>.</p>
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<div class="sourceCode cell-code" id="cb44"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb44-1"><a href="#cb44-1" aria-hidden="true" tabindex="-1"></a><span class="fu">airyai</span>(<span class="fl">0</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>0.3550280538878172</code></pre>
</div>
</div>
<p>It is known that this function is always positive for <span class="math inline">\(x &gt; 0\)</span>, though not so for negative values of <span class="math inline">\(x\)</span>. Which of these indicates the first negative value : <code>airyai(-1) &lt;0</code>, <code>airyai(-2) &lt; 0</code>, …, or <code>airyai(-5) &lt; 0</code>?</p>
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<code>airyai(-1) &lt; 0</code>
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<code>airyai(-2) &lt; 0</code>
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<code>airyai(-3) &lt; 0</code>
</span>
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<code>airyai(-4) &lt; 0</code>
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<code>airyai(-5) &lt; 0</code>
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</section>
<section id="question-5" class="level6">
<h6 class="anchored" data-anchor-id="question-5">Question</h6>
<p>By trying three different values of <span class="math inline">\(x &gt; 0\)</span> which of these could possibly be always true:</p>
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<code>x^x &lt;= (1/e)^(1/e)</code>
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<code>x^x &gt;= (1/e)^(1/e)</code>
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<code>x^x == (1/e)^(1/e)</code>
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<section id="question-6" class="level6">
<h6 class="anchored" data-anchor-id="question-6">Question</h6>
<p>Student logic says <span class="math inline">\((x+y)^p = x^p + y^p\)</span>. Of course, this isn’t correct for all <span class="math inline">\(p\)</span> and <span class="math inline">\(x\)</span>. By trying a few points, which is true when <span class="math inline">\(x,y &gt; 0\)</span> and <span class="math inline">\(0 &lt; p &lt; 1\)</span>:</p>
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<code>(x+y)^p &gt; x^p + y^p</code>
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<code>(x+y)^p &lt; x^p + y^p</code>
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<code>(x+y)^p == x^p + y^p</code>
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<section id="question-7" class="level6">
<h6 class="anchored" data-anchor-id="question-7">Question</h6>
<p>According to Wikipedia, one of the following inequalities is always true for <span class="math inline">\(a, b &gt; 0\)</span> (as proved by I. Ilani in JSTOR, AMM, Vol.97, No.1, 1990). Which one?</p>
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<code>a^b + b^a &lt;= 1</code>
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<code>a^a + b^b &lt;= a^b + b^a</code>
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<code>a^a + b^b &gt;= a^b + b^a</code>
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<section id="question-8" class="level6">
<h6 class="anchored" data-anchor-id="question-8">Question</h6>
<p>Is <span class="math inline">\(3\)</span> in the set <span class="math inline">\(\lvert x - 2\rvert &lt; 1/2\)</span>?</p>
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Yes
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<input class="form-check-input" type="radio" name="radio_12528338007097546375" id="radio_12528338007097546375_2" value="2">
<span class="label-body px-1">
No
</span>
</label>
</div>
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</div>
</section>
<section id="question-9" class="level6">
<h6 class="anchored" data-anchor-id="question-9">Question</h6>
<p>Which of the following is equivalent to <span class="math inline">\(\lvert x - a\rvert &gt; b\)</span>:</p>
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<div class="form-group ">
<div class="controls">
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<div class="form-check">
<label class="form-check-label" for="radio_18360175618415997205_1">
<input class="form-check-input" type="radio" name="radio_18360175618415997205" id="radio_18360175618415997205_1" value="1">
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\(-b &lt; x - a &lt; b\)
</span>
</label>
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<input class="form-check-input" type="radio" name="radio_18360175618415997205" id="radio_18360175618415997205_2" value="2">
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\(-b &lt; x-a \text{ and } x - a &lt; b\)
</span>
</label>
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<label class="form-check-label" for="radio_18360175618415997205_3">
<input class="form-check-input" type="radio" name="radio_18360175618415997205" id="radio_18360175618415997205_3" value="3">
<span class="label-body px-1">
\(x - a &lt; -b \text{ or } x - a &gt; b\)
</span>
</label>
</div>
</div>
</div>
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</section>
<section id="question-10" class="level6">
<h6 class="anchored" data-anchor-id="question-10">Question</h6>
<p>If <span class="math inline">\(\lvert x - \pi\rvert &lt; 1/10\)</span> is <span class="math inline">\(\lvert \sin(x) - \sin(\pi)\rvert &lt; 1/10\)</span>?</p>
<p>Guess an answer based on a few runs of</p>
<div class="sourceCode cell-code" id="cb46"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb46-1"><a href="#cb46-1" aria-hidden="true" tabindex="-1"></a>x <span class="op">=</span> <span class="cn">pi</span> <span class="op">+</span> <span class="fl">1</span><span class="op">/</span><span class="fl">10</span> <span class="op">*</span> (<span class="fl">2</span><span class="fu">rand</span>()<span class="op">-</span><span class="fl">1</span>)</span>
<span id="cb46-2"><a href="#cb46-2" aria-hidden="true" tabindex="-1"></a><span class="fu">abs</span>(x <span class="op">-</span> <span class="cn">pi</span>) <span class="op">&lt;</span> <span class="fl">1</span><span class="op">/</span><span class="fl">10</span>, <span class="fu">abs</span>(<span class="fu">sin</span>(x) <span class="op">-</span> <span class="fu">sin</span>(<span class="cn">pi</span>)) <span class="op">&lt;</span> <span class="fl">1</span><span class="op">/</span><span class="fl">10</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<input class="form-check-input" type="radio" name="radio_4048552582332687781" id="radio_4048552582332687781_1" value="1">
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true
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<input class="form-check-input" type="radio" name="radio_4048552582332687781" id="radio_4048552582332687781_2" value="2">
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false
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<section id="question-11" class="level6">
<h6 class="anchored" data-anchor-id="question-11">Question</h6>
<p>Does <code>12</code> satisfy <span class="math inline">\(\lvert x - 3\rvert + \lvert x-9\rvert &gt; 12\)</span>?</p>
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<span class="label-body px-1">
Yes
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</label>
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<div class="form-check">
<label class="form-check-label" for="radio_5241220318146713321_2">
<input class="form-check-input" type="radio" name="radio_5241220318146713321" id="radio_5241220318146713321_2" value="2">
<span class="label-body px-1">
No
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</section>
<section id="question-12" class="level6">
<h6 class="anchored" data-anchor-id="question-12">Question</h6>
<p>Which of these will show DeMorgan’s law holds when both values are <code>false</code>:</p>
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<code>!(false &amp;&amp; false) == (false || false)</code>
</span>
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<code>!(false &amp;&amp; false) == (!false || !false)</code>
</span>
</label>
</div>
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<label class="form-check-label" for="radio_1229196818275069853_3">
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<code>!(false &amp;&amp; false) == (!false &amp;&amp; !false)</code>
</span>
</label>
</div>
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</div>
</section>
<section id="question-13" class="level6">
<h6 class="anchored" data-anchor-id="question-13">Question</h6>
<p>For floating point numbers there are two special values <code>Inf</code> and <code>NaN</code>. For which of these is the answer always <code>false</code>:</p>
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<code>NaN &lt; 3.0</code> and <code>3.0 &lt;= NaN</code>
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_4352975797473589501_2">
<input class="form-check-input" type="radio" name="radio_4352975797473589501" id="radio_4352975797473589501_2" value="2">
<span class="label-body px-1">
<code>Inf &lt; 3.0</code> and <code>3.0 &lt;= Inf</code>
</span>
</label>
</div>
</div>
</div>
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</section>
<section id="question-14" class="level6">
<h6 class="anchored" data-anchor-id="question-14">Question</h6>
<p>The IEEE 754 standard is about floating point numbers, for which there are the special values <code>Inf</code>, <code>-Inf</code>, <code>NaN</code>, and, surprisingly, <code>-0.0</code> (as a floating point number and not <code>-0</code>, an integer). Here are 4 facts that seem reasonable:</p>
<ul>
<li>Positive zero is equal but not greater than negative zero.</li>
<li><code>Inf</code> is equal to itself and greater than everything else except <code>NaN</code>.</li>
<li><code>-Inf</code> is equal to itself and less then everything else except <code>NaN</code>.</li>
<li><code>NaN</code> is not equal to, not less than, and not greater than anything, including itself.</li>
</ul>
<p>Do all four seem to be the case within <code>Julia</code>? Find your answer by trial and error.</p>
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<input class="form-check-input" type="radio" name="radio_12689194611054504382" id="radio_12689194611054504382_1" value="1">
<span class="label-body px-1">
Yes
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_12689194611054504382_2">
<input class="form-check-input" type="radio" name="radio_12689194611054504382" id="radio_12689194611054504382_2" value="2">
<span class="label-body px-1">
No
</span>
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</div>
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</section>
<section id="question-15" class="level6">
<h6 class="anchored" data-anchor-id="question-15">Question</h6>
<p>The <code>NaN</code> value is meant to signal an error in computation. <code>Julia</code> has value to indicate some data is missing or unavailable. This is <code>missing</code>. For <code>missing</code> values we have these computations:</p>
<div class="cell" data-execution_count="42">
<div class="sourceCode cell-code" id="cb47"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb47-1"><a href="#cb47-1" aria-hidden="true" tabindex="-1"></a><span class="cn">true</span> <span class="op">&amp;&amp;</span> <span class="cn">missing</span>, <span class="cn">true</span> <span class="op">||</span> <span class="cn">missing</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>(missing, true)</code></pre>
</div>
</div>
<p>We see the value of <code>true || missing</code> is <code>true</code>. Why?</p>
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Since the second value is "<code>missing</code>", only the first is used. So <code>false || missing</code> would also be <code>false</code>
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<input class="form-check-input" type="radio" name="radio_12647630933029730517" id="radio_12647630933029730517_2" value="2">
<span class="label-body px-1">
In the manual we can read that "In the expression <code>a || b</code>, the subexpression <code>b</code> is only evaluated if <code>a</code> evaluates to false." In this case <code>a</code> is <code>true</code> and so <code>a</code> is returned.
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<p>The value for <code>true &amp;&amp; missing</code> is <code>missing</code>, not a boolean value. What happens?</p>
<div class="cell" data-hold="true" data-execution_count="44">
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In the manual we can read that "In the expression <code>a &amp;&amp; b</code>, the subexpression <code>b</code> is only evaluated if <code>a</code> evaluates to true." In this case, <code>a</code> is <code>false</code> so <code>b</code> is evaluated and returned. As <code>b</code> is just <code>missing</code> that is the return value.
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Since the second value is "<code>missing</code>" all such answers would be missing.
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