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352 lines
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Introduction to Sets</h1>
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<div class="center80">
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<p>Forget everything you know about numbers.</p>
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<p>In fact, forget you even know what a number is.</p>
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<p>This is where mathematics starts.</p>
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<p>Instead of math with numbers, we will now think about math with "things".</p>
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</div>
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<h2>Definition</h2>
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<p>What is a set? Well, simply put, it's <b>a collection</b>.</p>
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<p>First we specify a common property among "things" (we define this word later) and then we gather up all the "things" that have this common property.</p>
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<p style="float:left; margin: 0 20px 5px 0;"><img src="images/set-clothes.svg" alt="set of clothes"></p>
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<p>For example, the items you wear: hat, shirt, jacket, pants, and so on.</p>
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<p>I'm sure you could come up with at least a hundred.</p>
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<p>This is known as a <b>set</b>.</p>
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<table width="100%" border="0">
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<tbody>
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<tr>
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<td>
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<p align="right">Or another example is <b>types of fingers</b>.</p>
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<p align="right">This set includes index, middle, ring, and pinky.</p></td>
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<td> </td>
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<td><img src="images/set-fingers.jpg" alt="set of fingers" height="96" width="120"></td>
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</tr>
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</tbody></table>
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<p>So it is just things grouped together with a certain property in common.</p>
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<h2>Notation</h2>
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<p>There is a fairly simple notation for sets. We simply list each element (or "member") separated by a comma, and then put some curly brackets around the whole thing:</p>
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<p class="center"><img src="images/set.svg" alt="Set Notation"></p>
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<div class="def">
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<p class="center">The curly brackets <span class="large">{ }</span> are sometimes called "set brackets" or "braces".</p>
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</div>
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<p>This is the notation for the two previous examples:</p>
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<p class="indent"><b>{socks, shoes, watches, shirts, ...}</b><br>
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<b>{index, middle, ring, pinky}</b></p>
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<p>Notice how the first example has the "..." (three dots together).</p>
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<div class="def">
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<p class="center">The three dots <span class="huge">...</span> are called an ellipsis, and mean "continue on".</p>
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</div>
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<p>So that means the first example continues on ... for infinity.</p>
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<p><i>(OK, there isn't <b>really</b> an infinite amount of things you could wear, but I'm not entirely sure about that! After an hour of thinking of different things, I'm still not sure. So let's just say it is infinite for this example.) </i></p>
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<p>So:</p>
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<ul>
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<li>The first set <b>{socks, shoes, watches, shirts, ...}</b> we call an <b>infinite set</b>,</li>
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<li>the second set <b>{index, middle, ring, pinky}</b> we call a <b>finite set</b>.</li>
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</ul>
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<p>But sometimes the "..." can be used in the middle to save writing long lists:</p>
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<div class="example">
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<p>Example: the set of letters:</p>
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<p class="center"><b>{a, b, c, ..., x, y, z}</b></p>
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<p class="center">In this case it is a <b>finite set</b> (there are only 26 letters, right?)</p>
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</div>
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<h2>Numerical Sets</h2>
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<p>So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers?</p>
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<div class="indent">Set of even numbers: {..., −4, −2, 0, 2, 4, ...}<br>
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Set of odd numbers: {..., −3, −1, 1, 3, ...}<br>
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Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}<br>
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Positive multiples of 3 that are less than 10: {3, 6, 9}</div>
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<p>And so on. We can come up with all different types of sets.</p>
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<p>We can also define a set by its properties, such as <b>{x|x>0}</b> which means "the set of all x's, such that x is greater than 0", see <a href="set-builder-notation.html">Set-Builder Notation</a> to learn more.</p>
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<p>And we can have sets of numbers that have no common property, they are just <b>defined</b> that way. For example:</p>
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<div class="indent">{2, 3, 6, 828, 3839, 8827}<br>
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{4, 5, 6, 10, 21}<br>
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{2, 949, 48282, 42882959, 119484203}</div>
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<p>Are all sets that I just randomly banged on my keyboard to produce.</p>
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<h2>Why are Sets Important?</h2>
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<p>Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when we apply sets in different situations do they become the powerful building block of mathematics that they are.</p>
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<p class="indent"><i>Math can get amazingly complicated quite fast. Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis, Linear Algebra, Number Theory, and the list goes on. But there is one thing that all of these share in common: <b>Sets</b></i>.</p>
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<h2>Universal Set</h2>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td><img src="images/star.jpg" alt="star" height="74" width="99"></td>
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<td> </td>
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<td>
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<p>At the start we used the word "things" in quotes.</p>
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<p>We call this the <b>universal set</b>. It's a set that contains everything. Well, not <i>exactly</i> everything. <b>Everything that is relevant to our question.</b></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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<td> </td>
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</tr>
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<tr>
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<td><img src="images/various-integers.svg" alt="various integers"></td>
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<td> </td>
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<td>
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<p>In Number Theory the universal set is <b>all the integers</b>, as Number Theory is simply the study of integers.</p></td>
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</tr>
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<tr>
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<td><img src="images/various-reals.svg" alt="various reals"></td>
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<td> </td>
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<td>
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<p>But in <a href="../calculus/index.html">Calculus</a> (also known as real analysis), the universal set is almost always the <a href="../numbers/real-numbers.html">real numbers</a>.</p></td>
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</tr>
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<tr>
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<td><img src="images/various-complex.svg" alt="various complex numbers"></td>
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<td> </td>
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<td>And in complex analysis, you guessed it, the universal set is the <a href="../numbers/complex-numbers.html">complex numbers</a>.</td>
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</tr>
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</tbody></table>
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<h2>Some More Notation</h2>
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<div class="simple">
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<table align="center" width="93%" border="0">
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<tbody>
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<tr>
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<td><img src="images/aa.gif" alt="A= {a,...}" height="80" width="60"></td>
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<td> When talking about sets, it is fairly standard to use Capital Letters to represent the set, and lowercase letters to represent an element in that set.<br>
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<br>
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So for example, A is a set, and a is an element in A. Same with B and b, and C and c.</td>
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</tr>
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</tbody></table>
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</div>
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<p>Now you don't have to listen to the standard, you can use something like <i><b>m</b></i> to represent a set without breaking any mathematical laws (watch out, you can get <b><font face="Times New Roman, Times, serif" size="+1">π</font></b> years in math jail for dividing by 0), but this notation is pretty nice and easy to follow, so why not?</p>
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<p>Also, when we say an element <b>a</b> is in a set <b>A</b>, we use the symbol <img src="images/element-symbol.png" alt="element symbol" height="12" width="12"> to show it.<br>
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And if something is not in a set use <img src="images/not-element-symbol.png" alt="not element symbol" height="14" width="12">.</p>
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<div class="example">
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<p>Example: Set <b>A</b> is {1,2,3}. We can see that <b>1 <img src="images/element-symbol.png" alt="element symbol" height="12" width="12"> A</b>, but <b>5 <img src="images/not-element-symbol.png" alt="not element symbol" height="14" width="12"> A</b></p>
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</div>
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<h2>Equality</h2>
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<p>Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, so we may have to examine them closely!</p>
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<div class="example">
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<p>Example: Are A and B equal where:</p>
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<ul>
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<li>A is the set whose members are the first four positive whole numbers</li>
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<li>B = {4, 2, 1, 3}</li>
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</ul>
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<p>Let's check. They both contain 1. They both contain 2. And 3, And 4. And we have checked every element of both sets, so: <b>Yes, they are equal!</b></p>
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</div>
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<p>And the equals sign (=) is used to show equality, so we write:</p>
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<p class="center large">A = B</p>
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<div class="example">
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<h3>Example: Are these sets equal?</h3>
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<ul>
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<li>A is {1, 2, 3}</li>
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<li>B is {3, 1, 2}</li>
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</ul>
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<p>Yes, they are equal!</p>
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<p>They both contain exactly the members 1, 2 and 3.</p>
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<p>It doesn't matter <i>where</i> each member appears, so long as it is there.</p>
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</div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/subset.svg" alt="subset"></p>
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<h2>Subsets</h2>
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<p>When we define a set, if we take pieces of that set, we can form what is called a <b>subset</b>.</p>
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<div class="example">
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<h3>Example: the set {1, 2, 3, 4, 5}</h3>
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<p>A <b>subset</b> of this is {1, 2, 3}. Another subset is {3, 4} or even another is {1}, etc.</p>
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<p>But {1, 6} is <b>not</b> a subset, since it has an element (6) which is not in the parent set.</p>
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</div>
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<p>In general:</p>
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<div class="def">
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<p class="center">A is a <b>subset </b>of B if and only if every element of A is in B.</p>
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</div>
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<p>So let's use this definition in some examples.</p>
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<div class="example">
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<h3>Example: Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?</h3>
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<p>1 is in A, and 1 is in B as well. So far so good.</p>
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<p>3 is in A and 3 is also in B.</p>
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<p>4 is in A, and 4 is in B.</p>
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<p>That's all the elements of A, and every single one is in B, so we're done.</p>
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<p class="center larger">Yes, A is a subset of B</p>
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<p>Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter, we only look at the elements in A.</p>
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</div>
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<p>Let's try a harder example.</p>
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<div class="example">
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<h3>Example: Let <b>A be all multiples of 4</b> and <b>B be all multiples of 2</b>.<br>
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Is A a subset of B? And is B a subset of A?</h3>
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<p>Well, we can't check every element in these sets, because they have an infinite number of elements. So we need to get an idea of what the elements look like in each, and then compare them.</p>
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<p>The sets are:</p>
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<ul>
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<li>A = {..., −8, −4, 0, 4, 8, ...}</li>
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<li>B = {..., −8, −6, −4, −2, 0, 2, 4, 6, 8, ...}</li>
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</ul>
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<p>By pairing off members of the two sets, we can see that every member of A is also a member of B, but not every member of B is a member of A:</p>
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<p> </p>
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<p class="center"><img src="images/sets-example2.gif" alt="pairing off A and B" height="91" width="357"></p>
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<p>So:</p>
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<p class="center larger">A is a subset of B, but B is not a subset of A</p>
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</div>
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<h2>Proper Subsets</h2>
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<p>If we look at the defintion of subsets and let our mind wander a bit, we come to a weird conclusion.</p>
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<p>Let <b>A</b> be a set. Is every element of <b>A</b> in <b>A</b>? </p>
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<p class="center">Well, umm, <i>yes of course</i>, right?</p>
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<p>So that means that <b><i>A is a subset of A</i></b>. It is a subset of itself!</p>
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<p>This doesn't seem very <i>proper</i>, does it? If we want our subsets to be <i>proper</i> we introduce (what else but) <b>proper subsets</b>:</p>
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<div class="def">
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<p class="center">A is a <b>proper</b> subset of B if and only if every element of A is also in B, and there exists <b>at least one element</b> in B that is <b>not</b> in A.</p>
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</div>
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<p>This little piece at the end is there to make sure that A is not a proper subset of itself: we say that B must have at least one extra element.</p>
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<div class="example">
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<h3>Example:</h3>
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<p>{1, 2, 3} is a <b>subset</b> of {1, 2, 3}, but is <b>not </b>a<b> proper subset</b> of {1, 2, 3}.</p>
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</div>
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<div class="example">
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<h3>Example:</h3>
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<p>{1, 2, 3} <b>is </b>a<b> proper subset</b> of {1, 2, 3, 4} because the element 4 is not in the first set.</p>
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</div>
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<p>Notice that when A is a proper subset of B then it is also a subset of B.</p>
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<h2>Even More Notation</h2>
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<p>When we say that A is a subset of B, we write A <img src="images/subset-symbol.png" alt="subset symbol" height="16" width="16"> B.</p>
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<p>Or we can say that A is not a subset of B by A <img src="images/not-subset-symbol.png" alt="not subset symbol" height="16" width="16"> B ("A is not a subset of B")</p>
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<p>When we talk about proper subsets, we take out the line underneath and so it becomes A <img src="images/proper-subset-symbol.png" alt="proper subset symbol" height="12" width="15"> B or if we want to say the opposite, A <img src="images/not-proper-subset-symbol.png" alt="not proper subset symbol" height="13" width="16"> B.</p>
|
||
<h2>Empty (or Null) Set</h2>
|
||
<p>This is probably the weirdest thing about sets.</p>
|
||
<p style="float:left; margin: 10px;"><img src="images/keys-on-guitar.jpg" alt="keys on guitar" height="180" width="180"></p>
|
||
<p>As an example, think of the set of piano keys on a guitar.</p>
|
||
<p class="center"><i>"But wait!" you say, "There are no piano keys on a guitar!" </i></p>
|
||
<p>And right you are. It is a set with <b>no elements</b>.</p>
|
||
<p>This is known as the <b>Empty Set</b> (or Null Set).There aren't any elements in it. Not one. Zero.</p>
|
||
<p>It is represented by <img src="images/null-set.png" alt="null set" height="13" width="16"></p>
|
||
<p>Or by <b>{}</b> (a set with no elements)</p>
|
||
<p>Some other examples of the empty set are <b>the set of countries south of the south pole</b>.</p>
|
||
<p>So what's so weird about the empty set? Well, that part comes next.</p>
|
||
<h2>Empty Set and Subsets</h2>
|
||
<p>So let's go back to our definition of subsets. We have a set A. We won't define it any more than that, it could be any set. <i><b>Is the empty set a subset of A?</b></i></p>
|
||
<p>Going back to our definition of subsets, <b>if every element in the empty set is also in A, then the empty set is a subset of A</b>. But what if we have <b>no</b> elements?</p>
|
||
<p>It takes an introduction to logic to understand this, but this statement is one that is "vacuously" or "trivially" true.</p>
|
||
<p>A good way to think about it is: <i>we can't find any elements in the empty set that aren't in A</i>, so it must be that all elements in the empty set are in A.</p>
|
||
<p>So the answer to the posed question is a resounding <b>yes</b>.</p>
|
||
<div class="def">
|
||
<p class="center">The empty set is a subset of every set, including the empty set itself.</p>
|
||
</div>
|
||
<h2>Order</h2>
|
||
<p>No, not the order of the elements. In sets it <b>does not matter what order the elements are in</b>.</p>
|
||
<div class="example">
|
||
<p>Example: {1,2,3,4} is the same set as {3,1,4,2}</p>
|
||
</div>
|
||
<div class="def">
|
||
<p class="center">When we say <b>order</b> in sets we mean <b>the size of the set</b>.</p>
|
||
<p class="center">Another (better) name for this is <b>cardinality</b>.</p>
|
||
</div>
|
||
<p>A finite set has finite order (or cardinality). An infinite set has infinite order (or cardinality).</p>
|
||
<p>For finite sets the order (or cardinality) is the <b>number of elements</b>.</p>
|
||
<div class="example">
|
||
<p>Example: {10, 20, 30, 40} has an order of 4.</p>
|
||
</div>
|
||
<p>For infinite sets, all we can say is that the order is infinite. Oddly enough, we can say with sets that some infinities are larger than others, but this is a more advanced topic in sets.</p>
|
||
<h2>Arg! Not more notation!</h2>
|
||
<p>Nah, just kidding. No more notation.</p>
|
||
<table width="100%" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td align="right"> by
|
||
|
||
|
||
|
||
|
||
<div class="large">Ricky Shadrach</div> and
|
||
|
||
|
||
|
||
|
||
<div class="large">Rod Pierce</div></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<div class="questions">
|
||
|
||
<script>getQ(366, 367, 368, 1051, 1052, 1053, 9070, 2424, 2425, 2426);</script>
|
||
|
||
</div>
|
||
<div class="activity"> <a href="../activity/subsets.html">Activity: Subsets</a></div>
|
||
|
||
<div class="related">
|
||
<a href="venn-diagrams.html">Venn Diagrams</a>
|
||
<a href="set-calculator.html">Set Calculator</a>
|
||
<a href="groups-introduction.html">Introduction to Groups</a>
|
||
<a href="index.html">Sets Index</a>
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