Hykilpikonna
115 lines
7.7 KiB
HTML
115 lines
7.7 KiB
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<header id="title-block-header">
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<h1 class="title">C.2 Inequalities</h1>
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<section>
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<p>In this course we will deal heavily with the manipulation of <em>inequalities</em>. While many of these operations are very similar to manipulating equalities, there are enough differences to warrant a comprehensive list.</p>
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<div id="theorem:inequalities_basics" class="theorem">
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<p>(<em>Arithmetic manipulations</em>) For all real numbers <span class="math inline">\(a\)</span>, <span class="math inline">\(b\)</span>, and <span class="math inline">\(c\)</span>, the following are true:</p>
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<ol type="a">
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<li>If <span class="math inline">\(a \leq b\)</span> and <span class="math inline">\(b \leq c\)</span>, then <span class="math inline">\(a \leq c\)</span>.</li>
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<li>If <span class="math inline">\(a \leq b\)</span>, then <span class="math inline">\(a + c \leq b + c\)</span>.</li>
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<li>If <span class="math inline">\(a \leq b\)</span> and <span class="math inline">\(c > 0\)</span>, then <span class="math inline">\(ac \leq bc\)</span>.</li>
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<li>If <span class="math inline">\(a \leq b\)</span> and <span class="math inline">\(c < 0\)</span>, then <span class="math inline">\(ac \geq bc\)</span>.</li>
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<li>If <span class="math inline">\(0 < a \leq b\)</span>, then <span class="math inline">\(\frac{1}{a} \geq \frac{1}{b}\)</span>.</li>
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<li>If <span class="math inline">\(a \leq b < 0\)</span>, then <span class="math inline">\(\frac{1}{a} \geq \frac{1}{b}\)</span>.</li>
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<p>Moreover, if we replace any of the “if” inequalities with a strict inequality (i.e., change <span class="math inline">\(\leq\)</span> to <span class="math inline">\(<\)</span>), then the corresponding “then” inequality is also strict.<label for="sn-0" class="margin-toggle sidenote-number"></label><input type="checkbox" id="sn-0" class="margin-toggle"/><span class="sidenote">For example, the following is true: “If <span class="math inline">\(a < b\)</span>, then <span class="math inline">\(a + c < b + c\)</span>.”</span></p>
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<p>The previous theorem tells us that basic operations like adding a number or multiplying by a positive number preserves inequalities. However, other operations like multiplying by a negative number or taking reciprocals <em>reverses</em> the direction of the inequality, which is something we didn’t have to worry about when dealing with equalities. But it turns out that, at least for non-negative numbers, most of our familiar functions preserve inequalities.</p>
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<div class="definition" data-terms="increasing">
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<p>Let <span class="math inline">\(f : \R^{\geq 0} \to \R^{\geq 0}\)</span>. We say that <span class="math inline">\(f\)</span> is when for all <span class="math inline">\(x, y \in \R^{\geq 0}\)</span>, if <span class="math inline">\(x < y\)</span> then <span class="math inline">\(f(x) < f(y)\)</span>.</p>
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<p>Most common functions are strictly increasing:</p>
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<ul>
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<li>Raising to a positive power, e.g., <span class="math inline">\(f(x) = x^2\)</span> or <span class="math inline">\(f(x) = x^{3.14}\)</span>.<label for="sn-1" class="margin-toggle sidenote-number"></label><input type="checkbox" id="sn-1" class="margin-toggle"/><span class="sidenote"> Remember that we’re restricting ourselves to the <span class="math inline">\(\R^{\geq 0}\)</span> for the domain of these functions! <span class="math inline">\(f(x) = x^2\)</span> is not increasing on the domain <span class="math inline">\(\R\)</span>, for example.</span></li>
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<li>Logarithms with a base greater than one, e.g., <span class="math inline">\(f(x) = \log_3(x + 1)\)</span>.</li>
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<li>Exponential functions with a base greater than one, e.g., <span class="math inline">\(f(x) = 2^x\)</span>.</li>
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</ul>
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<p>Moreover, adding two strictly increasing functions, or multiplying a strictly increasing function by a positive constant or another always-positive strictly increasing function, results in another strictly increasing function. So for example, we know that <span class="math inline">\(f(x) = 300x^2 + x \log_3 x + 2^{x+100}\)</span> is also strictly increasing.</p>
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</div>
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<p>It should be clear from this definition that the following property holds, which enables us to manipulate inequalities using a host of common functions.</p>
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<div id="theorem:inequalities_functions" class="theorem">
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<p>For all non-negative real numbers <span class="math inline">\(a\)</span> and <span class="math inline">\(b\)</span>, and all strictly increasing functions <span class="math inline">\(f: \R^{\geq 0} \TO \R^{\geq 0}\)</span>, if <span class="math inline">\(a \leq b\)</span>, then <span class="math inline">\(f(a) \leq f(b)\)</span>.</p>
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<p>Moreover, if <span class="math inline">\(a < b\)</span>, then <span class="math inline">\(f(a) < f(b)\)</span>.</p>
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<p>It is this theorem that allows us to perform several common operations on inequalities as a “step” in a computation. For example, if we know <span class="math inline">\(0 < a \leq b\)</span>, then we can conclude that <span class="math inline">\(a^2 \leq b^2\)</span>, or <span class="math inline">\(\log_2(a) \leq \log_2(b)\)</span>, because both of the functions <span class="math inline">\(x^2\)</span> and <span class="math inline">\(\log_2(x)\)</span> are strictly increasing functions.</p>
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</section>
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<a href="https://www.teach.cs.toronto.edu/~csc110y/fall/notes/">CSC110 Course Notes Home</a>
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