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340 lines
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340 lines
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Power Set</h1>
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<div align="center"><span class="large">A Power Set is a set of <i>all the subsets of a set</i>.</span></div>
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<p align="center"><i><b>OK? Got that? Maybe an example will help...</b></i></p>
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<h2>All The Subsets</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/power-set.svg" alt="power set of {a,b,c}" /></p>
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<p>For the <a href="sets-introduction.html">set</a> {a,b,c}:</p>
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<ul>
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<li>The empty set {} is a subset of {a,b,c}</li>
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<li>And these are subsets: {a}, {b} and {c} </li>
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<li>And these are also subsets: {a,b}, {a,c} and {b,c}</li>
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<li>And {a,b,c} is a subset of {a,b,c} </li>
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</ul>
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<p>And altogether we get the <b>Power Set</b> of {a,b,c}:</p>
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<p align="center"><font size="+1">P(S) =</font> <font size="+1">{</font> {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} <font size="+1">}</font></p>
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<p>Think of it as all the different ways we can select the items (the order of the items doesn't matter), including selecting none, or all.</p>
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<div class="example"><p style="float:right; margin: 0 0 5px 10px;"><img src="../combinatorics/images/ice-cream.jpg" width="100" height="133" alt="ice cream" /></p>
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<h3>Example: The shop has banana, chocolate and lemon ice cream. </h3>
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<p> </p>
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<p>What do you order?</p>
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<ul>
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<li>Nothing at all: {}</li>
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<li>Or maybe just banana: {banana}. Or just {chocolate} or just {lemon}</li>
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<li>Or two together: {banana,chocolate} or {banana,lemon} or {chocolate,lemon}</li>
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<li>Or all three! {banana, chocolate,lemon}</li>
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</ul>
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<p><i>Question: if the shop also has strawberry flavor what are your options? Solution later</i>.</p>
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</div>
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<h2>How Many Subsets</h2>
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<p>Easy! If the original set has <span class="large">n</span> members, then the Power Set will have <span class="large">2<sup>n</sup></span> members</p>
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<div class="example">
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<h3>Example: {a,b,c} has three members (<b>a</b>,<b>b</b> and <b>c</b>).</h3>
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<p>So, the Power Set should have 2<sup>3</sup> = 8, which it does, as we worked out before.</p>
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</div>
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<h3>Notation</h3>
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<p>The number of members of a set is often written as |S|, so when S has n members we can write:</p>
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<div class="large" align="center">|P(S)| = 2<sup>n</sup></div>
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<br />
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<div class="example">
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<h3>Example: for the set S={1,2,3,4,5} how many members will the power set have?</h3>
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<p>Well, S has 5 members, so:</p>
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<p align="center" class="large">|P(S)| = 2<sup>n</sup> = 2<sup>5</sup> = 32</p>
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</div>
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<p>You will see in a minute why the number of members is a power of 2</p>
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<h2>It's Binary!</h2>
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<p>And here is the most amazing thing. To create the Power Set, write down the sequence of <a href="../binary-number-system.html">binary numbers</a> (using <b>n</b> digits), and then let "1" mean "put the matching member into this subset".</p>
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<p>So "101" is replaced by 1 <b>a</b>, 0 <b>b</b> and 1 <b>c</b> to get us {a,c}</p>
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<p>Like this:</p>
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<div class="simple">
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<table width="300" border="0" align="center">
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<tr align="center">
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<th width="10%"> </th>
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<th width="40%">abc</th>
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<th width="40%">Subset</th>
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</tr>
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<tr align="center">
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<td width="10%">0</td>
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<td width="40%">000</td>
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<td width="40%">{ }</td>
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</tr>
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<tr align="center">
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<td width="10%">1</td>
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<td width="40%">001</td>
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<td width="40%">{c}</td>
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</tr>
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<tr align="center">
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<td width="10%">2</td>
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<td width="40%">010</td>
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<td width="40%">{b}</td>
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</tr>
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<tr align="center">
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<td width="10%">3</td>
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<td width="40%">011</td>
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<td width="40%">{b,c}</td>
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</tr>
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<tr align="center">
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<td width="10%">4</td>
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<td width="40%">100</td>
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<td width="40%">{a}</td>
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</tr>
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<tr align="center">
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<td width="10%">5</td>
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<td width="40%">101</td>
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<td width="40%">{a,c}</td>
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</tr>
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<tr align="center">
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<td width="10%">6</td>
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<td width="40%">110</td>
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<td width="40%">{a,b}</td>
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</tr>
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<tr align="center">
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<td width="10%">7</td>
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<td width="40%">111</td>
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<td width="40%">{a,b,c}</td>
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</tr>
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</table>
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</div>
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<p>Well, they are not in a pretty order, but they are all there.</p>
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<h2>Another Example</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../combinatorics/images/ice-cream.jpg" width="100" height="133" alt="ice cream" /></p>
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<p>Let's eat! We have four flavors of ice cream: <b>banana, chocolate, lemon, and strawberry</b>. How many different ways can we have them?</p>
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<p>Let's use letters for the flavors: {b, c, l, s}. Example selections include: </p>
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<ul>
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<li>{} (nothing, you are on a diet)</li>
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<li>{b, c, l, s} (every flavor)</li>
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<li>{b, c} (banana and chocolate are good together)</li>
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<li>etc</li>
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</ul>
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Let's make the table using "binary":
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<div class="simple">
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<table width="300" border="0" align="center">
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<tr align="center">
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<th width="10%"> </th>
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<th width="40%">bcls</th>
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<th width="40%">Subset</th>
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</tr>
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<tr align="center">
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<td width="10%">0</td>
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<td width="40%">0000</td>
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<td width="40%">{}</td>
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</tr>
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<tr align="center">
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<td width="10%">1</td>
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<td width="40%">0001</td>
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<td width="40%">{s}</td>
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</tr>
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<tr align="center">
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<td width="10%">2</td>
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<td width="40%">0010</td>
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<td width="40%">{l}</td>
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</tr>
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<tr align="center">
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<td width="10%">3</td>
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<td width="40%">0011</td>
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<td width="40%">{l,s}</td>
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</tr>
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<tr align="center">
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<td width="10%">...</td>
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<td width="40%"><i>... etc ..</i></td>
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<td width="40%"><i>... etc ...</i></td>
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</tr>
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<tr align="center">
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<td width="10%">12</td>
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<td width="40%">1100</td>
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<td width="40%">{b,c}</td>
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</tr>
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<tr align="center">
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<td width="10%">13</td>
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<td width="40%">1101</td>
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<td width="40%">{b,c,s}</td>
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</tr>
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<tr align="center">
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<td width="10%">14</td>
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<td width="40%">1110</td>
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<td width="40%">{b,c,l}</td>
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</tr>
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<tr align="center">
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<td width="10%">15</td>
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<td width="40%">1111</td>
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<td width="40%">{b,c,l,s}</td>
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</tr>
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</table>
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<p>And the result is (more neatly arranged):</p>
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<p align="center"><font size="+1">P =</font> <font size="+1">{</font> {}, {b}, {c}, {l}, {s}, {b,c}, {b,l}, {b,s}, {c,l}, {c,s}, {l,s}, {b,c,l}, {b,c,s}, <br />
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{b,l,s}, {c,l,s}, {b,c,l,s} <font size="+1">}</font></p>
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</div>
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<br />
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<div class="beach">
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<table border="0" align="center">
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<tr>
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<td><img src="../images/style/yin-yang.gif" width="100" height="100" alt="yin yang" /></td>
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<td><h3>Symmetry</h3>
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<p>In the table above, did you notice that the first subset is empty and the last has every member? </p>
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<p>But did you also notice that the second subset has "s", and the second last subset has everything <i>except</i> "s"?</p></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td><img src="../numbers/images/binary-symmetry.svg" alt="binary symmetry" /></td>
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<td><p>In fact when we mirror that table about the middle we see there is a kind of symmetry. </p>
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<p>This is because the binary numbers (that we used to help us get all those combinations) have a beautiful and elegant pattern.</p></td>
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</tr>
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</table>
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</div>
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<h2>A Prime Example</h2>
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<p>The Power Set can be useful in unexpected areas. </p>
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<p>I wanted to find all factors (not just the <a href="../prime-factorization.html">prime factors</a>, but <i>all</i> factors) of a number. </p>
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<p>I could test all possible numbers: I could check 2, 3, 4, 5, 6, 7, etc... </p>
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<p>That took a <b>long time</b> for large numbers.</p>
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<p>But could I try to combine the prime factors?</p>
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<p>Let me see, the prime factors of 510 are 2×3×5×17 (using <a href="../numbers/prime-factorization-tool.html">prime factor tool</a>).</p>
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<p>So, <b>all the factors</b> of 510 are: </p>
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<ul>
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<li>2, 3, 5 and 17,</li>
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<li>2×3, 2×5 and 2×17 as well, and </li>
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<li>2×3×5 and 2×3×17 and ... </li>
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<li>.. aha! Just like ice cream I needed a Power Set!</li>
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</ul>
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<p>And this is what I got:</p>
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<div class="simple">
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<table border="0" align="center">
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<tr align="center">
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<th> </th>
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<th>2,3,5,17</th>
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<th>Subset</th>
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<th>Factors of 510</th>
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</tr>
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<tr align="center">
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<td>0</td>
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<td>0000</td>
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<td>{ }</td>
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<td>1</td>
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</tr>
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<tr align="center">
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<td>1</td>
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<td>0001</td>
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<td>{17}</td>
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<td>17</td>
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</tr>
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<tr align="center">
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<td>2</td>
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<td>0010</td>
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<td>{5}</td>
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<td>5</td>
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</tr>
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<tr align="center">
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<td>3</td>
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<td>0011</td>
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<td>{5,17}</td>
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<td>5 × 17 = 85</td>
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</tr>
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<tr align="center">
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<td>4</td>
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<td>0100</td>
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<td>{3}</td>
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<td>3</td>
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</tr>
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<tr align="center">
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<td>5</td>
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<td>0101</td>
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<td>{3,17}</td>
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<td>3 × 17 = 51</td>
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</tr>
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<tr align="center">
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<td> </td>
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<td><i>... etc ...</i></td>
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<td><i>... etc ...</i></td>
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<td><i>... etc ...</i></td>
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</tr>
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<tr align="center">
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<td>15</td>
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<td>1111</td>
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<td>{2,3,5,17}</td>
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<td>2 × 3 × 5 × 17 = 510</td>
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</tr>
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</table>
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</div>
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<p><br />
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And the result? The factors of 510 are 1, 2, 3, 5, 6, 10, 15, 17, 30, 34, 51, 85, 102, 170, 255 and 510 (and −1, −2, −3, etc as well). See the <i> <a href="../numbers/factors-all-tool.html">All Factors Tool</a></i>. </p>
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<h2>Automated</h2>
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<p>I couldn't resist making Power Sets available to you in an automated way.</p>
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<p>So, when you need a power set, try <a href="power-set-maker.html">Power Set Maker</a>.</p>
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<script type="text/javascript">getQ(2028, 2029, 2030, 8274, 8275, 8276, 8277, 8278, 2031, 8279);</script> </div>
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<div class="related"> <a href="sets-introduction.html">Introduction to Sets</a> <a href="../binary-digits.html">Binary Digits</a>
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<a href="set-calculator.html">Set Calculator</a>
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<a href="../geometry/symmetry.html">Symmetry</a> <a href="power-set-maker.html">Power Set Maker</a> <a href="index.html">Sets Index</a> </div>
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