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<title>Set-Builder Notation</title>
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<meta name="description" content="Learn how to describe a set by saying what properties its members have." />
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Set-Builder Notation</h1>
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<p align="center"><i>How to describe a set by saying what properties its members have.</i></p>
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<div class="def">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/various-integers.svg" alt="set of various integers" /></p>
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<p>A <a href="sets-introduction.html">Set</a> is <b>a collection of things </b>(usually numbers).</p>
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<p>Example: <b>{5, 7, 11}</b> is a set.</p>
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</div>
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<p>But we can also "build" a set by describing what is in it.</p>
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<p>Here is a simple example of set-builder notation:</p>
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<p align="center"><img src="images/set-builder-notation-a.svg" alt="Set Builder Notation" /></p>
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<p align="center" class="larger">It says <b><i>"the set of all x's, such that x is greater than 0"</i></b>.</p>
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<p align="center">In other words <b>any value greater than 0</b></p>
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<p>Notes:</p>
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<div class="bigul">
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<ul>
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<li>The "x" is just a place-holder, it could be anything, such as <b>{ q | q > 0 }</b></li>
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<li>Some people use "<b>:</b>" instead of "<b>|</b>", so they write <b>{ x : x > 0 }</b></li>
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</ul>
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</div>
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<h2>Type of Number</h2>
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<p>It is also normal to show what <a href="number-types.html">type of number</a> <b>x</b> is, like this:</p>
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<p align="center"><img src="images/set-builder-notation.svg" alt="Set Builder Notation" /></p>
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<ul>
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<li>The <img src="../images/symbols/member-of-sm.gif" alt="member of" width="14" height="19" style="vertical-align:middle;" /> means "a member of" (or simply "in")</li>
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<li>The <img src="../images/symbols/set-r.svg" alt="reals" height="16" style="vertical-align:middle;" /> is the special symbol for <a href="../numbers/real-numbers.html">Real Numbers</a>. </li>
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</ul>
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<p>So it says:</p>
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<p align="center" class="larger"> <i>"the set of all x's that are <b>a member of the Real Numbers</b>, <br />
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such that x is greater than or equal to 3"</i></p>
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<p align="center">In other words <b>"all Real Numbers from 3 upwards"</b></p>
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<p>There are other ways we could have shown that:</p>
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<div class="center80">
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<p>On the <a href="../number-line.html">Number Line</a> it looks like: <img src="images/interval-3-inf.gif" alt="[3, infinity)" width="131" height="52" style="vertical-align:middle;"/></p>
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<p>In <a href="intervals.html">Interval notation</a> it looks like:<span class="large"> [3, +∞)</span> </p>
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</div>
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<h2>Number Types</h2>
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<p>We saw <img src="../images/symbols/set-r.svg" alt="reals" height="16" style="vertical-align:middle;" /> (the special symbol for Real Numbers). Here are the <a href="number-types.html">common number types</a>:</p>
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<table border="0" align="center">
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<tr align="center" valign="top">
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<td width="80"><img src="../images/symbols/set-n.svg" alt="Natural Numbers" /></td>
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<td width="80"><img src="../images/symbols/set-z.svg" alt="Integers" /></td>
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<td width="80"><img src="../images/symbols/set-q.svg" alt="Rational Numbers" /></td>
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<td width="80"><img src="../images/symbols/set-r.svg" alt="Real Numbers" /></td>
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<td width="80"><img src="../images/symbols/set-i.svg" alt="Imaginary Numbers" /></td>
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<td width="80"><img src="../images/symbols/set-c.svg" alt="Complex Numbers" /></td>
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</tr>
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<tr align="center">
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<td width="80"><a href="../whole-numbers.html">Natural Numbers</a></td>
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<td width="80"><a href="../whole-numbers.html">Integers</a></td>
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<td width="80"><a href="../rational-numbers.html">Rational Numbers</a></td>
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<td width="80"><a href="../numbers/real-numbers.html">Real Numbers</a></td>
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<td width="80"><a href="../numbers/imaginary-numbers.html">Imaginary Numbers</a></td>
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<td width="80"><a href="../numbers/complex-numbers.html">Complex Numbers</a></td>
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</tr>
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</table>
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<p> </p>
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<div class="example">
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<p>Example: <span class="larger">{ k <img src="../images/symbols/member-of-sm.gif" alt="member of" width="14" height="19" style="vertical-align:middle;" /> <img src="../images/symbols/set-z.svg" alt="reals" height="16" style="vertical-align:middle;" /> | k > 5 }</span></p>
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<p><i>"the set of all k's that are a member of the Integers,
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such that k is greater than 5"</i></p>
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<p>In other words <b>all integers greater than 5</b>.</p>
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<p>This could also be written {6, 7, 8, ... } , so:</p>
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<p align="center"> <span class="larger">{ k <img src="../images/symbols/member-of-sm.gif" alt="member of" width="14" height="19" style="vertical-align:middle;" /> <img src="../images/symbols/set-z.svg" alt="reals" height="16" style="vertical-align:middle;" /> | k > 5 } = {6, 7, 8, ... }</span></p>
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</div>
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<h2>Why Use It?</h2>
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<p>When we have a simple set like the <b>integers from 2 to 6</b> we can write:</p>
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<p align="center" class="larger">{2, 3, 4, 5, 6}</p>
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<p>But how do we list the Real Numbers in the same interval?</p>
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<p align="center" class="larger">{2, 2.1, 2.01, 2.001, 2.0001, ... <b>???</b></p>
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<p>So instead we say<b> how to build the list</b>:</p>
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<p align="center"><span class="larger">{ x <img src="../images/symbols/member-of-sm.gif" alt="member of" width="14" height="19" style="vertical-align:middle;" /> <img src="../images/symbols/set-r.svg" alt="reals" height="16" style="vertical-align:middle;" /> | x ≥ 2 and x ≤ 6 }</span><br />
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<i>Start with all Real Numbers, then limit them between 2 and 6 inclusive.</i></p>
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<p>We can also use set builder notation to do other things, like this:</p>
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<p align="center"><span class="larger">{ x <img src="../images/symbols/member-of-sm.gif" alt="member of" width="14" height="19" style="vertical-align:middle;" /> <img src="../images/symbols/set-r.svg" alt="reals" height="16" style="vertical-align:middle;" /> | x = x<sup>2</sup> } = {0, 1}</span><br />
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<i>All Real Numbers such that x = x<sup>2</sup><br />
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<b>0 and 1</b> are the only cases where x = x<sup>2</sup></i></p>
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<h2>Another Example:</h2>
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<div class="example">
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<h3>Example: x ≤ 2 or x > 3</h3>
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<p>Set-Builder Notation looks like this:</p>
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<p align="center"><span class="larger">{ x <img src="../images/symbols/member-of-sm.gif" alt="member of" width="14" height="19" style="vertical-align:middle;" /> <img src="../images/symbols/set-r.svg" alt="reals" height="16" style="vertical-align:middle;" /> | x ≤ 2 or x >3 }</span></p>
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<p>On the Number Line it looks like:</p>
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<p align="center"><img src="images/interval-to-2-3-up.gif" alt="two intervals" width="240" height="50" /></p>
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<p>Using Interval notation it looks like:</p>
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<p align="center" class="large"> (−∞, 2] U (3, +∞)</p>
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<p align="center">We used a "U" to mean Union (the <a href="venn-diagrams.html">joining together of two sets</a>).</p>
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</div>
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<p> </p>
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<h2>Defining a Domain</h2>
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<p>Set Builder Notation is very useful for defining <a href="domain-range-codomain.html">domains</a>.</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/range-domain-graph.svg" alt="domain and range graph" /></p>
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<p>In its simplest form the domain is <b>the set of all the values</b> that go into a function.</p>
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<p>The <a href="function.html">function</a> must work for all values we give it, so it is <b>up to us</b> to make sure we get the domain correct!</p>
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<div style="clear:both"></div>
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<div class="example">
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<h3>Example: The domain of 1/x </h3>
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<p>1/x is <b>undefined</b> at x=0 (because 1/0 is <a href="../numbers/dividing-by-zero.html">dividing by zero</a>).</p>
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<p>So we must exclude x=0 from the Domain:</p>
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<p align="center" class="larger">The Domain of 1/x is all the Real Numbers, except 0</p>
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<p>We can write this as</p>
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<p align="center" class="larger">Dom(1/x) = {x <img src="../images/symbols/member-of-sm.gif" alt="member of" width="14" height="19" style="vertical-align:middle;" /> <img src="../images/symbols/set-r.svg" alt="reals" height="16" style="vertical-align:middle;" /> | x ≠ 0}</p>
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</div>
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<div class="example">
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<h3>Example: The domain of g(x)=1/(x−1) </h3>
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<p>1/(x−1) is <b>undefined</b> at x=1, so we must exclude x=1 from the Domain:</p>
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<p align="center" class="larger">The Domain of 1/(x−1) is all the Real Numbers, except 1</p>
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<p>Using set-builder notation it is written:</p>
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<p align="center"><span class="larger">Dom( g(x) ) = { x<img src="../images/symbols/member-of-sm.gif" alt="member of" width="14" height="19" style="vertical-align:middle;" /><img src="../images/symbols/set-r.svg" alt="reals" height="16" style="vertical-align:middle;" /> | x ≠ 1}</span></p>
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</div>
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<div class="example">
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<h3>Example: The domain of √x </h3>
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<p>Is all the Real Numbers from 0 onwards, because we can't take the square root of a negative number (unless we use <a href="../numbers/imaginary-numbers.html">Imaginary Numbers</a>, which we aren't).</p>
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<p>We can write this as</p>
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<p align="center" class="larger">Dom(√x) = {x <img src="../images/symbols/member-of-sm.gif" alt="member of" width="14" height="19" style="vertical-align:middle;" /> <img src="../images/symbols/set-r.svg" alt="reals" height="16" style="vertical-align:middle;" /> | x ≥ 0}</p>
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</div>
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<div class="example">
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<h3>Example The domain of f(x) = x/(x<sup>2</sup> − 1) </h3>
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<p>To avoid dividing by zero we need: x<sup>2</sup> - 1 ≠ 0 </p>
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<p align="center">Factor: <b>x<sup>2</sup> - 1 = (x−1)(x+1)</b></p>
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<p align="center">(x−1)(x+1) = 0 when <b>x = 1</b> or <b>x = −1</b>, which we want to avoid!</p>
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<p>So:</p>
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<p align="center" class="larger">Dom( f(x) ) = {x <img src="../images/symbols/member-of-sm.gif" alt="member of" width="14" height="19" style="vertical-align:middle;" /> <img src="../images/symbols/set-r.svg" alt="reals" height="16" style="vertical-align:middle;" /> | x ≠ 1, x ≠ −1}</p>
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</div>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(7211, 551, 552, 1187, 1188, 7212, 553, 1189, 1190, 554);</script>
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</div>
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<div class="related">
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<a href="intervals.html">Intervals</a>
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<a href="index.html">Sets Index</a>
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<a href="../algebra/index.html">Algebra Index</a> </div>
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Copyright © 2018 MathsIsFun.com
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