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236 lines
7.2 KiB
HTML
236 lines
7.2 KiB
HTML
<!DOCTYPE html>
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<!-- Mirrored from www.mathsisfun.com/algebra/open-sentences.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:01:46 GMT -->
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Open Sentences</h1>
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<p class="center"><i>An example of an open sentence: x + 3 = 6</i></p>
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<h2>First ... what is a "Sentence" ?</h2>
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<p>Just like an English sentence, in mathematics a sentence says something:</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/sentence-closed.svg" alt="closed sentence">
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<h3>English:</h3>
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<div class="bigul">
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<ul>
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<li><i>The sun is shining.</i></li>
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<li><i>Hawaii is in the Pacific Ocean.</i></li></ul>
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</div>
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<h3>Mathematics:</h3>
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<div class="bigul">
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<ul>
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<li><i>3 + 3 = 6</i></li>
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<li><i>10 is an even number</i></li></ul></div>
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</p>
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<h2>Now ... what is a "Closed Sentence" or an "Open Sentence" ?</h2>
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<div class="simple"><br>
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<table align="center" width="90%" border="0">
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<tbody>
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<tr>
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<td class="large">Closed</td>
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<td>A closed sentence is <b>always true</b> (or <b>always false</b>). </td>
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</tr>
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<tr>
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<td class="large">Open</td>
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<td>A sentence is open when it is <b>not known</b> if it is true or false.</td>
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</tr>
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</tbody></table><br>
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</div>
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<h3>Examples:</h3>
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<table align="center" cellpadding="2" border="0">
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<tbody>
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<tr>
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<td style="text-align:center;"><i><b>8</b> is an even number</i></td>
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<td style="width:20px;"> </td>
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<td>is <b>closed</b> (it is <b>always true</b>)</td>
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</tr>
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<tr>
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<td style="text-align:center;"><i><b>9</b> is an even number</i></td>
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<td> </td>
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<td>is <b>closed</b> (it is <b>always false</b>)</td>
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</tr>
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<tr>
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<td style="text-align:center;"><i><b>n</b> is an even number</i></td>
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<td> </td>
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<td>is <b>open</b> (could be true or false, depending on the value of <b>n</b>)</td>
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</tr>
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</tbody></table>
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<p>In that last example:</p>
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<ul>
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<li>if <b>n</b> was 4 the sentence would be true,</li>
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<li>if <b>n</b> was 5 the sentence would be false,</li>
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<li>etc ...</li>
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</ul>
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<p>But we didn't say what value <b>n</b> has!</p>
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<p>So "<i><b>n</b> is an even number</i>" may be <b>true or false</b>. So it is <b>open</b>.</p>
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<h2>Open Sentence</h2>
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<p>So, we get this definition:</p>
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<div class="def">
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<p>An open sentence can be <b>either true or false</b> depending on what values are used.</p>
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</div>
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<h2>Variables</h2>
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<p>The value we don't know is called a <span class="large">variable</span> (also called an <span class="large">unknown</span>)</p>
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<p>In this example of an open sentence, <b>x</b> is a variable:</p>
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<p class="center"><span class="large">x + 3 = 8</span> <i><br>
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</i></p>
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<p>In this example, <b>w</b> and <b>q</b> are both variables:</p>
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<p class="center large">w + q = 2</p>
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<h2>Solving</h2>
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<p><b>Solving</b> means finding a value for the variable that makes the sentence <b>true</b>.</p>
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<div class="example">
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<h3>Example: Solve x + 3 = 8</h3>
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<p>Let us subtract 3 from both sides:</p>
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<p class="center">x + 3 − <b>3</b> = 8 − <b>3</b> </p>
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<p class="center"><b>x = 5</b></p>
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<p>Check: <b>5</b> + 3 = 8 is <b>true</b></p>
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<p>So we have solved <b>x + 3 = 8</b> by making <b>x = 5</b></p>
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</div>
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<p> </p>
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<h2>Some More Examples</h2>
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<p>Here are some more examples of closed and open sentences for you:</p>
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<h3>Closed Sentences:</h3>
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<table align="center" cellpadding="2" border="0">
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<tbody>
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<tr>
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<td>A square has four corners</td>
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<td style="width:20px;"> </td>
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<td>always <b>true</b></td>
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</tr>
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<tr>
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<td>6 is less than 5</td>
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<td> </td>
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<td>always <b>false</b></td>
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</tr>
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<tr>
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<td>−3 is a negative number</td>
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<td> </td>
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<td>always <b>true</b></td>
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</tr>
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</tbody></table>
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<h3> </h3>
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<h3>Open Sentences:</h3>
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<table align="center" cellpadding="2" border="0">
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<tbody>
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<tr>
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<td>A triangle has <b>n</b> sides</td>
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<td style="width:20px;"> </td>
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<td>Can be true or false (depending on the value of <b>n</b>)</td>
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</tr>
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<tr>
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<td><b>z</b> is a positive number</td>
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<td> </td>
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<td>Can be true or false (depending on the value of <b>z</b>)</td>
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</tr>
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<tr>
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<td>3y = 4x + 2</td>
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<td> </td>
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<td>Can be true or false (depending on the values of <b>x</b> and<b> y</b>)</td>
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</tr>
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<tr>
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<td>a + b = c + d</td>
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<td> </td>
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<td>Can be true or false (depending on the values of <b>a,b,c,d</b>)</td>
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</tr>
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</tbody></table>
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<p> </p>
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<div class="questions">1721, 1722, 1723, 1724, 6377, 6378, 6379, 6380, 6381, 6382, 6383, 6384, 6385, 6386, 6387</div>
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<div class="related">
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<a href="definitions.html">Algebra Definitions</a>
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<div id="copyrt">Copyright © 2022 Rod Pierce</div>
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