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434 lines
16 KiB
HTML
434 lines
16 KiB
HTML
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<meta http-equiv="content-type" content="text/html; charset=utf-8">
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<title>Activity: Subsets</title>
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Activity: Subsets</h1>
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<p class="center">Please read <a
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href="../sets/sets-introduction.html">Introduction
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to Sets</a> first!</p>
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<p>This activity investigates <b>how many subsets</b> a set has.</p>
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<h2>What is a Subset?</h2>
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<p>A <b>subset</b> is a set <b>contained in</b> another set</p>
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<p>It is like you can choose ice cream from the following flavors:</p>
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<p class="center larger">{banana, chocolate, vanilla}</p>
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<p>You could choose any one flavor <b>{banana}</b>, <b>{chocolate}</b>, or <b>{vanilla}</b>,</p>
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<p>Or any two flavors: <b>{banana, chocolate}</b>, <b>{banana, vanilla}</b>, or <b> {chocolate, vanilla}</b>,</p>
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<p>Or all three flavors (no that isn't greedy),</p>
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<p><b>Or</b> you could say "none at all thanks", which is the "empty set": <b>{}</b></p>
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<p> </p>
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<div class="example">
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<h3>Example: The set {alex, billy, casey, dale}</h3>
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<p>Has the subsets:</p>
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<ul>
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<li>{alex}</li>
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<li>{billy}</li>
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<li>etc ...</li>
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</ul>
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<p>It also has the subsets:</p>
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<ul>
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<li>{alex, billy}</li>
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<li>{alex, casey}</li>
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<li>{billy, dale}</li>
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<li>etc ...</li>
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</ul>
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<p>Also:</p>
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<ul>
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<li>{alex, billy, casey}</li>
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<li>{alex, billy, dale}</li>
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<li>etc ...</li>
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</ul>
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<p>And also:</p>
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<ul>
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<li>the whole set: {alex, billy, casey, dale}</li>
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<li>the empty set: {}</li>
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</ul>
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</div>
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<p>Now let's start with the Empty Set and move on up ...</p>
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<h2>The
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Empty Set</h2>
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<p>How many subsets does the empty set have?</p>
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<p>You could choose:</p>
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<ul>
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<li>the whole set: <b>{}</b></li>
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<li>the empty set: <b>{} </b></li>
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</ul>
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<p>But, hang on a minute, in this case those are the same thing!</p>
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<p>So the
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empty set really has <b>just 1 subset</b> (which
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is itself, the empty set).</p>
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<p>It is like asking "There is nothing available, so what do you choose?" Answer "nothing". That is your only choice. Done.</p>
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<h2>A
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Set With One Element</h2>
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<p>The set could be anything, but let's just say it is:</p>
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<p class="center large">{apple}</p>
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<p>How many <b>subsets</b> does the set <span class="large">{apple}</span> have?</p>
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<ul>
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<li>the whole set: {apple}</li>
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<li>the empty set: {}</li>
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</ul>
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<p>And that's all. You can choose the one element, or nothing.</p>
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<p>So any set with <b>one</b> element will have <b>2</b> subsets.</p>
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<h2>A
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Set With Two Elements</h2>
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<p>Let's add another element to our example set:</p>
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<p class="center large">{apple, banana}</p>
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<p>How many subsets does the set <span class="large">{apple, banana}</span> have?</p>
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<p>It could have <b>{apple}</b>, or <b>{banana}</b>, and don't forget:</p>
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<ul>
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<li>the whole set: <b>{apple, banana}</b></li>
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<li>the empty set: <b>{}</b></li>
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</ul>
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<p>So a set with <b>two</b> elements has <b>4</b> subsets.</p>
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<h2>A
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Set With Three Elements</h2>
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<p>How about:</p>
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<p class="center large">{apple, banana, cherry}</p>
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<p>OK, let's be more systematic now, and list the subsets by how many elements they have:</p>
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<p>Subsets with one element: <b>{apple}</b>, <b>{banana}</b>, <b>{cherry}</b></p>
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<p>Subsets with two elements: <b>{apple, banana}</b>, <b>{apple, cherry}</b>, <b>{banana, cherry}</b></p>
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<p>And:</p>
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<ul>
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<li>the whole set: <b>{apple, banana, cherry}</b></li>
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<li>the empty set: <b>{}</b></li>
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</ul>
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<p>In fact we could put it in a table:</p>
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<table border="1" align="center" cellpadding="2"
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cellspacing="2">
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<tbody>
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<tr>
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<td> </td>
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<td>List</td>
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<td align="center">Number of <br>
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subsets</td>
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</tr>
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<tr>
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<td>zero elements</td>
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<td>{}</td>
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<td align="center" class="larger">1</td>
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</tr>
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<tr>
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<td>one element</td>
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<td>{apple}, {banana}, {cherry} </td>
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<td align="center" class="larger">3</td>
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</tr>
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<tr>
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<td>two elements</td>
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<td>{apple, banana}, {apple, cherry}, {banana, cherry}</td>
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<td align="center" class="larger">3</td>
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</tr>
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<tr>
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<td>three elements</td>
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<td>{apple, banana, cherry}</td>
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<td align="center" class="larger">1</td>
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</tr>
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<tr>
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<td colspan="2" rowspan="1" align="right">Total:</td>
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<td align="center" class="larger"><b>8</b></td>
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</tr>
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</tbody>
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</table>
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<p class="center">(Note: did you see a pattern in the numbers there?)</p>
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<h2>Sets
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with Four Elements (Your Turn!)</h2>
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<p>Now try to do the same for this set:</p>
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<p class="center large">{apple, banana, cherry, date}</p>
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<p>Here is a table for you:</p>
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<table border="1" align="center" cellpadding="2"
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cellspacing="2">
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<tbody>
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<tr>
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<td> </td>
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<td width="400">List</td>
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<td align="center">Number of <br>
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subsets</td>
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</tr>
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<tr>
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<td height="50">zero elements</td>
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<td>{}</td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td height="50">one element</td>
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<td> </td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td height="50">two elements</td>
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<td> </td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td height="50">three elements</td>
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<td> </td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td height="50">four elements</td>
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<td> </td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td colspan="2" rowspan="1" align="right">Total:</td>
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<td height="30" align="center"> </td>
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</tr>
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</tbody>
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</table>
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<p class="center">(Note: if you did this right, there will be a pattern to the numbers.)</p>
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<p> </p>
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<h2>Sets
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with Five Elements</h2>
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<p>And now:</p>
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<p class="center large">{apple, banana, cherry, date, egg}</p>
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<p>Here is a table for you:</p>
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<table border="1" align="center" cellpadding="2"
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cellspacing="2">
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<tbody>
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<tr>
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<td> </td>
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<td width="400">List</td>
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<td align="center">Number of <br>
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subsets</td>
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</tr>
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<tr>
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<td height="50">zero elements</td>
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<td>{}</td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td height="50">one element</td>
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<td> </td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td height="50">two elements</td>
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<td> </td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td height="50">three elements</td>
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<td> </td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td height="50">four elements</td>
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<td> </td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td height="50">five elements</td>
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<td> </td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td colspan="2" rowspan="1" align="right">Total:</td>
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<td height="30" align="center"> </td>
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</tr>
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</tbody>
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</table>
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<p class="center">(Was there a pattern to the numbers?)</p>
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<p> </p>
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<h2>Sets
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with Six Elements</h2>
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<p>What about:</p>
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<p class="center large">{apple, banana, cherry, date, egg, fudge}</p>
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<p>OK ... we don't need to complete a table, because...</p>
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<div class="center"><b>... you should be able to
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see a pattern by now! </b></div>
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<h2>Doubling</h2>
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<p>The first thing to notice is that the total number of subsets doubles each time:</p>
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<div class="center80">
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<p class="center larger">A set with <b>n</b> elements has <b>2<sup>n</sup></b> subsets</p>
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</div>
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<p>So you should be able to answer:</p>
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<ul>
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<div class="bigul">
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<li>How many subsets are there for a set of 6 elements? _____</li>
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<li>How many subsets are there for a set of 7 elements? _____</li>
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</div>
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</ul>
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<h2>Another
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Pattern</h2>
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<p>Now let's think about
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subsets and sizes:</p>
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<ul>
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<li>The
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empty
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set has
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just <b>1
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subset</b>: <span class="large hi">1</span></li>
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<li>A set with one element has <b>1 subset</b> with no elements and <b>1
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subset</b> with one element: <span class="large hi">1 1</span></li>
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<li>A set with two
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elements has <b>1 subset</b> with no elements, <b>2 subsets</b> with one element and <b>1 subset</b> with two elements: <span class="large hi">1
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2 1</span></li>
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<li>A set with three
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elements has <b>1 subset</b> with no elements, <b>3 subsets</b> with one
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element, <b>3 subsets</b> with two elements and <b>1 subset</b> with three
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elements: <span class="large hi">1 3 3 1</span></li>
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<li>and so on!</li>
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</ul>
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<p>Do you recognize this
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pattern of numbers?</p>
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<p>They are the numbers from <a
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href="../pascals-triangle.html">Pascal's
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Triangle</a>!</p>
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<div class="center"><img src="../images/pascals-triangle-4.gif" width="334" height="245" alt="pascals triangle" /></div>
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<p>This is <b>very useful</b>, because now you can check if you have the right number of subsets.</p>
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<p>Note: the rows start at 0, and likewise the columns.</p>
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<div class="example">
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<p>Example: For the set <b>{apple, banana, cherry, date, egg}</b> you list subsets of length three:</p>
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<ul>
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<li>{apple, banana, cherry}</li>
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<li>{apple, banana, date}</li>
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<li>{apple, banana, egg}</li>
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<li>{apple, cherry, egg}</li>
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</ul>
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<p>But that is only <b>4</b> subsets, how many should there be?</p>
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<p>Well, you are choosing 3 out of 5, so go to <b>row 5, position 3</b> of Pascal's Triangle (remember to start counting at 0) to find you need <b>10 subsets</b>, so you must think harder!</p>
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<p>In fact these are the results: {apple,banana,cherry} {apple,banana,date} {apple,banana,egg} {apple,cherry,date} {apple,cherry,egg} {apple,date,egg} {banana,cherry,date} {banana,cherry,egg} {banana,date,egg} {cherry,date,egg}</p>
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</div>
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<h2>Calculating The Numbers</h2>
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<p>Is there a way of calculating the numbers such as <b>1, 4, 6, 4 and 1</b> (instead of looking them up in Pascal's Triangle)?</p>
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<p>Yes, we can find the number of ways of selecting each number of
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elements using <a
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href="../combinatorics/combinations-permutations.html">Combinations</a>.</p>
|
|
<p>There are four elements in the set, and:</p>
|
|
<div class="bigul">
|
|
<ul>
|
|
<li>The number of ways of selecting 0 elements from 4 = <sup>4</sup>C<sub>0</sub> = <b>1</b></li>
|
|
<li>The number of ways of
|
|
selecting 1 element from 4 = <sup>4</sup>C<sub>1</sub> = <b>4</b></li>
|
|
<li>The number of ways of selecting 2 elements from 4 = <sup>4</sup>C<sub>2</sub> = <b>6</b></li>
|
|
<li>The number of ways of selecting 3 elements from 4 = <sup>4</sup>C<sub>3</sub> = <b>4</b></li>
|
|
<li>The number of ways of selecting 4 elements from 4 = <sup>4</sup>C<sub>4</sub> = <b>1</b></li>
|
|
<li>
|
|
|
|
|
|
Total number of
|
|
subsets = <b>16</b></li>
|
|
</ul>
|
|
</div>
|
|
<p> </p>
|
|
<p>Can you do the same for a set with five elements?</p>
|
|
<p>Complete the following:</p>
|
|
<div class="bigul">
|
|
<ul>
|
|
<li>The number of ways
|
|
of selecting 0 elements from 5 = <sup>5</sup>C<sub>0</sub> = 1</li>
|
|
<li>The number of ways of
|
|
selecting 1 element from 5 = ___________</li>
|
|
<li>The number of ways of selecting 2 elements from 5 = ___________</li>
|
|
<li>The number of ways of selecting 3 elements from 5 = ___________</li>
|
|
<li>The number of ways of selecting 4 elements from 5 = ___________ </li>
|
|
<li>The
|
|
number of ways of selecting 5 elements from 5 = ___________</li>
|
|
<li>
|
|
|
|
|
|
Total number of subsets = ___________</li>
|
|
</ul>
|
|
</div>
|
|
<h2><br style="font-weight: normal;" />
|
|
Conclusion</h2>
|
|
<p>In this activity you have:<br style="font-weight: normal;" /></p>
|
|
<ul style="font-weight: normal;">
|
|
<li>Discovered a rule for
|
|
determining the total number of subsets for a given set: A set with n
|
|
elements has 2<sup style="font-weight: bold;">n</sup> subsets.</li>
|
|
<li>Found a connection between
|
|
the numbers of subsets of each size with the numbers in Pascal's
|
|
triangle.</li>
|
|
<li>Discovered a quick way to
|
|
calculate these numbers using Combinations.</li>
|
|
</ul>
|
|
<p>More
|
|
importantly you have learned how different branches of mathematics can
|
|
be combined together.</p>
|
|
<p> </p>
|
|
|
|
<div class="related">
|
|
<a href="../sets/sets-introduction.html">Introduction to Sets</a>
|
|
<a href="index.html">Activity Index</a>
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</div>
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