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130 lines
6.3 KiB
HTML
130 lines
6.3 KiB
HTML
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<head>
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<title>Closure</title>
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<meta name="description" content="Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents." />
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<h1 class="center">Closure</h1>
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<table border="0" align="center">
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<tr>
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<td><img src="images/closure.svg" alt="sets: closed and not closed" /></td>
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<td width="20" align="center"> </td>
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<td align="center">Closure is when an <b>operation </b> (such as "adding") <br />
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on members of a <b>set</b> (such as "real numbers") <br />
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always makes<b> a member of the same set</b>. </td>
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</tr>
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</table>
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<p>So the result stays in the same set.</p>
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<div class="example">
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<h3>Example: when we add two <a href="../numbers/real-numbers.html">real numbers</a> we get another real number</h3>
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<p class="center">3.1 + 0.5 = 3.6</p>
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<p>This is always true, so: <b>real numbers are closed under addition</b></p>
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</div>
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<div class="example">
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<h3>Example: subtracting two <a href="../whole-numbers.html">whole numbers</a> might <b>not</b> make a whole number</h3>
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<p class="center">4 − 9 = −5</p>
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<p class="center">−5 is <b>not</b> a whole number (whole numbers can't be negative)</p>
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<p>So: <b>whole numbers are <span class="large">not</span> closed under subtraction</b></p>
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</div>
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<p>This is a <b>general idea</b>, and can apply to any sort of operation on any kind of set!</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/shirt.gif" width="120" height="90" alt="shirt" /></p>
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<h3>Example: the set of shirts</h3>
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<p>For the operation "wash", the shirt is still a shirt after washing.</p>
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<ul>
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<li>So shirts are <b>closed</b> under the operation "wash"</li>
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</ul>
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<p>For the operation "rip", a small rip may be OK, but a shirt ripped in half ceases to be a shirt!</p>
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<ul>
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<li>So shirts are <b>not closed</b> under the operation "rip"</li>
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</ul>
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</div>
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<h2>Sets</h2>
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<p>A <a href="sets-introduction.html">set</a> is a collection of things (usually numbers). Examples:</p>
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<ul>
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<li>Set of even numbers: {..., -4, -2, 0, 2, 4, ...}</li>
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<li> Set of odd numbers: {..., -3, -1, 1, 3, ...}</li>
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<li> Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}</li>
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<li> Positive multiples of 3 that are less than 10: {3, 6, 9}</li>
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</ul>
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<p>Let's look more closely at one set:</p>
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<div class="example">
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<h3>Example: Odd numbers {..., -3, -1, 1, 3, ...}</h3>
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<p>Is the set of odd numbers closed under the simple operations <span class="larger">+ − × ÷</span> ?</p>
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<ul>
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<li>Adding? 3 + 7 = 10 but 10 is even, not odd, so <b>no</b></li>
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<li>Subtracting? 11 <span class="center">− </span>3 = 8 but 8 is even, not odd, so <b>no</b></li>
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<li>Multiplying? 5 × 7 = 35 yes ... in fact <a href="../numbers/even-odd.html">multiplying odd numbers</a> <b>always</b> produces odd numbers, so <b>odd numbers are closed under multiplication</b></li>
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<li>Dividing? 33/3 = 11 which looks good! But try 33/5 = 6.6 which is not odd, so <b>no</b> </li>
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</ul>
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</div>
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<p>As we just saw, just one case where it does <b>NOT</b> work is enough to say it is <b>NOT</b> closed.</p>
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<p>But to say it <b>IS</b> closed, we must know it is <b>ALWAYS</b> closed (just one example could fool us).</p>
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<p> </p>
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<div class="related"><a href="sets-introduction.html">Introduction to Sets</a>
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<a href="index.html">Sets Index</a>
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</div>
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Copyright © 2017 MathsIsFun.com
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