Operation of Sets

<h1 align="center" style="color: #003399; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 23px; font-weight: normal; line-height: 30px; margin: 9px 0px 20px;"> Introduction to Sets</h1> <div class="center80" style="background-color: cornsilk; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 10px 40px; padding: 5px 8px;"> <div style="margin-bottom: 8px; margin-top: 10px;"> Forget everything you know about numbers.</div> <div style="margin-bottom: 8px; margin-top: 10px;"> In fact, forget you even know what a number is.</div> <div style="margin-bottom: 8px; margin-top: 10px;"> This is where mathematics starts.</div> <div style="margin-bottom: 8px; margin-top: 10px;"> Instead of math with numbers, we will now think about math with "things".</div> </div> <h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;"> Definition</h2> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> What is a set? Well, simply put, it's <b>a collection</b>.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> First we specify a common property among "things" (this word will be defined later) and then we gather up all the "things" that have this common property.</div> <div style="float: left; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 5px; margin-right: 10px;"> <img height="160" src="http://www.mathsisfun.com/sets/images/set-clothes.jpg" width="200" /></div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> For example, the items you wear: shoes, socks, hat, shirt, pants, and so on.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> I'm sure you could come up with at least a hundred.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> This is known as a <b>set</b>.</div> <table border="0" style="color: black; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; width: 100%px;"><tbody> <tr><td style="line-height: 22px;"><div align="right" style="line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> Or another example is <b>types of fingers</b>.</div> <div align="right" style="line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> This set includes index, middle, ring, and pinky.</div> </td><td style="line-height: 22px;"></td><td style="line-height: 22px;"><img height="96" src="http://www.mathsisfun.com/sets/images/set-fingers.jpg" width="120" /></td></tr> </tbody></table> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> So it is just things grouped together with a certain property in common.</div> <h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;"> Notation</h2> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> There is a fairly simple notation for sets. We simply list each element, separated by a comma, and then put some curly brackets around the whole thing.</div> <div align="center" style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> <img alt="Set" height="127" src="http://www.mathsisfun.com/sets/images/set.gif" width="381" /></div> <div class="def" style="background-color: honeydew; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; border: 1px solid rgb(255, 192, 112); font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 0px 40px; padding: 0px 18px;"> <div align="center" style="margin-bottom: 25px; margin-top: 16px;"> The curly brackets <span class="large" style="color: #a06000; font-size: 19px;">{ }</span> are sometimes called "set brackets" or "braces".</div> </div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> This is the notation for the two previous examples:</div> <div class="indent50px" style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-left: 50px; margin-top: 16px;"> <b>{socks, shoes, watches, shirts, ...}</b><br /> <b>{index, middle, ring, pinky}</b></div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> Notice how the first example has the "..." (three dots together).</div> <div class="def" style="background-color: honeydew; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; border: 1px solid rgb(255, 192, 112); font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 0px 40px; padding: 0px 18px;"> <div align="center" style="margin-bottom: 25px; margin-top: 16px;"> The three dots <span class="huge" style="font-size: 35px; font-weight: bold;">...</span> are called an ellipsis, and mean "continue on".</div> </div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> So that means the first example continues on ... for infinity.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> <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></div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> So:</div> <ul style="display: table; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; list-style-position: outside; margin: -5px 0px 0px 20px;"> <li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">The first set <b>{socks, shoes, watches, shirts, ...}</b> we call an <b>infinite set</b>,</li> <li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">the second set <b>{index, middle, ring, pinky}</b> we call a <b>finite set</b>.</li> </ul> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> But sometimes the "..." can be used in the middle to save writing long lists:</div> <div class="example" style="background-color: #e0f0ff; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; box-shadow: rgb(208, 208, 255) 5px 5px 1px; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 20px 0px 30px 50px; overflow: hidden; padding: 5px 10px 1px 15px;"> <div style="margin-bottom: 25px; margin-top: 16px;"> Example: the set of letters:</div> <div align="center" style="margin-bottom: 25px; margin-top: 16px;"> <b>{a, b, c, ..., x, y, z}</b></div> <div align="center" style="margin-bottom: 25px; margin-top: 16px;"> In this case it is a <b>finite set</b> (there are only 26 letters, right?)</div> </div> <h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;"> Numerical Sets</h2> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> 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?</div> <div class="indent50px" style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-left: 50px;"> Set of even numbers: {..., -4, -2, 0, 2, 4, ...}<br /> Set of odd numbers: {..., -3, -1, 1, 3, ...}<br /> Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}<br /> Positive multiples of 3 that are less than 10: {3, 6, 9}</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> And the list goes on. We can come up with all different types of sets.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> There can also be sets of numbers that have no common property, they are just <b>defined</b> that way. For example:</div> <div class="indent50px" style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-left: 50px;"> {2, 3, 6, 828, 3839, 8827}<br /> {4, 5, 6, 10, 21}<br /> {2, 949, 48282, 42882959, 119484203}</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> Are all sets that I just randomly banged on my keyboard to produce.</div> <h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;"> Why are Sets Important?</h2> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> 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.</div> <div class="indent50px" style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-left: 50px; margin-top: 16px;"> <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>.</div> <h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;"> Universal Set</h2> <table border="0" style="color: black; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px;"><tbody> <tr><td style="line-height: 22px;"><img height="74" src="http://www.mathsisfun.com/sets/images/star.jpg" width="99" /></td><td style="line-height: 22px;"></td><td style="line-height: 22px;">At the start we used the word "things" in quotes. 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></td></tr> <tr><td style="line-height: 22px;"><img height="80" src="http://www.mathsisfun.com/sets/images/various-integers.gif" width="100" /></td><td style="line-height: 22px;"></td><td style="line-height: 22px;">Then our sets included integers. The universal set for that would be all the integers. In fact, when doing Number Theory, this is almost always what the universal set is, as Number Theory is simply the study of integers.</td></tr> <tr><td style="line-height: 22px;"><img height="80" src="http://www.mathsisfun.com/sets/images/various-reals.gif" width="100" /></td><td style="line-height: 22px;"></td><td style="line-height: 22px;">However in Calculus (also known as real analysis), the universal set is almost always the real numbers. And in complex analysis, you guessed it, the universal set is the complex numbers.</td></tr> </tbody></table> <h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;"> Some More Notation</h2> <div class="simple" style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; overflow: auto;"> <table align="center" border="0" style="background-color: #cceeff; background-position: initial initial; background-repeat: initial initial; border-collapse: collapse; width: 93%px;"><tbody> <tr><td style="border: 2px solid white; font-size: 15px; line-height: 22px; padding: 5px;"><img height="80" src="http://www.mathsisfun.com/sets/images/aa.gif" width="60" /></td><td style="border: 2px solid white; font-size: 15px; line-height: 22px; padding: 5px;">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 /> <br /> So for example, A is a set, and a is an element in A. Same with B and b, and C and c.</td></tr> </tbody></table> </div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> 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><span style="font-family: Times New Roman, Times, serif;">π</span></b> years in math jail for dividing by 0), but this notation is pretty nice and easy to follow, so why not?</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> Also, when we say an element <b>a</b> is in a set <b>A</b>, we use the symbol <img height="12" src="http://www.mathsisfun.com/sets/images/element-symbol.png" width="12" /> to show it.<br /> And if something is not in a set use <img height="14" src="http://www.mathsisfun.com/sets/images/not-element-symbol.png" width="12" />.</div> <div class="example" style="background-color: #e0f0ff; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; box-shadow: rgb(208, 208, 255) 5px 5px 1px; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 20px 0px 30px 50px; overflow: hidden; padding: 5px 10px 1px 15px;"> <div style="margin-bottom: 25px; margin-top: 16px;"> Example: Set <b>A</b> is {1,2,3}. We can see that <b>1 <img height="12" src="http://www.mathsisfun.com/sets/images/element-symbol.png" width="12" /> A</b>, but <b>5 <img height="14" src="http://www.mathsisfun.com/sets/images/not-element-symbol.png" width="12" /> A</b></div> </div> <h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;"> Equality</h2> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> 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!</div> <div class="example" style="background-color: #e0f0ff; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; box-shadow: rgb(208, 208, 255) 5px 5px 1px; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 20px 0px 30px 50px; overflow: hidden; padding: 5px 10px 1px 15px;"> <div style="margin-bottom: 25px; margin-top: 16px;"> Example: Are A and B equal where:</div> <ul style="display: table; list-style-position: outside; margin: -5px 0px 0px 20px;"> <li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">A is the set whose members are the first four positive whole numbers</li> <li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">B = {4, 2, 1, 3}</li> </ul> <div style="margin-bottom: 25px; margin-top: 16px;"> 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></div> </div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> And the equals sign (=) is used to show equality, so we write:</div> <div align="center" class="large" style="color: #a06000; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 19px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> A = B</div> <h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;"> Subsets</h2> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> When we define a set, if we take pieces of that set, we can form what is called a <b>subset</b>.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> So for example, we have the set {1, 2, 3, 4, 5}. A <b>subset</b> of this is {1, 2, 3}. Another subset is {3, 4} or even another, {1}. However, {1, 6} is not a subset, since it contains an element (6) which is not in the parent set. In general:</div> <div align="center" class="larger" style="color: #000088; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 17px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> A is a subset of B if and only if every element of A is in B.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> So let's use this definition in some examples.</div> <div class="example" style="background-color: #e0f0ff; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; box-shadow: rgb(208, 208, 255) 5px 5px 1px; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 20px 0px 30px 50px; overflow: hidden; padding: 5px 10px 1px 15px;"> <h3 style="color: #6600cc; font-size: 18px; font-weight: normal; line-height: 27px; margin: 5px 0px 10px;"> Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?</h3> <div style="margin-bottom: 25px; margin-top: 16px;"> 1 is in A, and 1 is in B as well. So far so good.</div> <div style="margin-bottom: 25px; margin-top: 16px;"> 3 is in A and 3 is also in B.</div> <div style="margin-bottom: 25px; margin-top: 16px;"> 4 is in A, and 4 is in B.</div> <div style="margin-bottom: 25px; margin-top: 16px;"> That's all the elements of A, and every single one is in B, so we're done.</div> <div align="center" class="larger" style="color: #000088; font-size: 17px; margin-bottom: 25px; margin-top: 16px;"> Yes, A is a subset of B</div> <div style="margin-bottom: 25px; margin-top: 16px;"> 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.</div> </div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> Let's try a harder example.</div> <div class="example" style="background-color: #e0f0ff; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; box-shadow: rgb(208, 208, 255) 5px 5px 1px; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 20px 0px 30px 50px; overflow: hidden; padding: 5px 10px 1px 15px;"> <h3 style="color: #6600cc; font-size: 18px; font-weight: normal; line-height: 27px; margin: 5px 0px 10px;"> Example: Let A be all multiples of 4 and B be all multiples of 2. Is A a subset of B? And is B a subset of A?</h3> <div style="margin-bottom: 25px; margin-top: 16px;"> 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.</div> <div style="margin-bottom: 25px; margin-top: 16px;"> The sets are:</div> <ul style="display: table; list-style-position: outside; margin: -5px 0px 0px 20px;"> <li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">A = {..., -8, -4, 0, 4, 8, ...}</li> <li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">B = {..., -8, -6, -4, -2, 0, 2, 4, 6, 8, ...}</li> </ul> <div style="margin-bottom: 25px; margin-top: 16px;"> By pairing off members of the two sets, we can see that every member of A is also a member of B, but every member of B is not a member of A:</div> <div style="margin-bottom: 25px; margin-top: 16px;"> <br /></div> <div align="center" style="margin-bottom: 25px; margin-top: 16px;"> <img alt="pairing off A and B" height="91" src="http://www.mathsisfun.com/sets/images/sets-example2.gif" width="357" /></div> <div style="margin-bottom: 25px; margin-top: 16px;"> So:</div> <div align="center" class="larger" style="color: #000088; font-size: 17px; margin-bottom: 25px; margin-top: 16px;"> A is a subset of B, but B is not a subset of A</div> </div> <h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;"> Proper Subsets</h2> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> If we look at the defintion of subsets and let our mind wander a bit, we come to a weird conclusion.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> Let <b>A</b> be a set. Is every element<b> </b>in <b>A</b> an element in <b>A</b>? (Yes, I wrote that correctly.)</div> <div align="center" style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> Well, umm, <i>yes of course</i>, right?</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> So doesn't that mean that <b><i>A is a subset of A</i></b>?</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> This doesn't seem very <i>proper</i>, does it? We want our subsets to be <i>proper</i>. So we introduce (what else but) <b>proper subsets</b>.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> A is a <b>proper</b> subset of B if and only if every element in A is also in B, and there exists <b>at least one element</b> in B that is <b>not</b> in A.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> This little piece at the end is only there to make sure that A is not a proper subset of itself. Otherwise, a proper subset is exactly the same as a normal subset.</div> <div class="example" style="background-color: #e0f0ff; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; box-shadow: rgb(208, 208, 255) 5px 5px 1px; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 20px 0px 30px 50px; overflow: hidden; padding: 5px 10px 1px 15px;"> <h3 style="color: #6600cc; font-size: 18px; font-weight: normal; line-height: 27px; margin: 5px 0px 10px;"> Example:</h3> <div style="margin-bottom: 25px; margin-top: 16px;"> {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}.</div> </div> <div class="example" style="background-color: #e0f0ff; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; box-shadow: rgb(208, 208, 255) 5px 5px 1px; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 20px 0px 30px 50px; overflow: hidden; padding: 5px 10px 1px 15px;"> <h3 style="color: #6600cc; font-size: 18px; font-weight: normal; line-height: 27px; margin: 5px 0px 10px;"> Example:</h3> <div style="margin-bottom: 25px; margin-top: 16px;"> {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.</div> </div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> Notice that if A is a proper subset of B, then it is also a subset of B.</div> <h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;"> Even More Notation</h2> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> When we say that A is a subset of B, we write A <img height="16" src="http://www.mathsisfun.com/sets/images/subset-symbol.png" width="16" /> B.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> Or we can say that A is not a subset of B by A <img height="16" src="http://www.mathsisfun.com/sets/images/not-subset-symbol.png" width="16" /> B ("A is not a subset of B")</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> When we talk about proper subsets, we take out the line underneath and so it becomes A <img height="12" src="http://www.mathsisfun.com/sets/images/proper-subset-symbol.png" width="15" /> B or if we want to say the opposite, A <img height="13" src="http://www.mathsisfun.com/sets/images/not-proper-subset-symbol.png" width="16" /> B.</div> <h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;"> Empty (or Null) Set</h2> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> This is probably the weirdest thing about sets.</div> <div style="float: left; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 10px;"> <img height="180" src="http://www.mathsisfun.com/sets/images/keys-on-guitar.jpg" width="180" /></div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> As an example, think of the set of piano keys on a guitar.</div> <div align="center" style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> <i>"But wait!" you say, "There are no piano keys on a guitar!"</i></div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> And right you are. It is a set with <b>no elements</b>.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> This is known as the <b>Empty Set</b> (or Null Set).There aren't any elements in it. Not one. Zero.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> It is represented by <img height="13" src="http://www.mathsisfun.com/sets/images/null-set.png" width="16" /></div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> Or by <b>{}</b> (a set with no elements)</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> Some other examples of the empty set are <b>the set of countries south of the south pole</b>.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> So what's so weird about the empty set? Well, that part comes next.</div> <h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;"> Empty Set and Subsets</h2> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> 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></div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> 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?</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> It takes an introduction to logic to understand this, but this statement is one that is "vacuously" or "trivially" true.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> 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.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> So the answer to the posed question is a resounding <b>yes</b>.</div> <div align="center" class="larger" style="color: #000088; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 17px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> The empty set is a subset of every set, including the empty set itself.</div> <h2 style="color: #993300; font-family: qarmic, Verdana, sans-serif; font-size: 20px; font-weight: normal; line-height: 23px; margin: 60px 0px 12px;"> Order</h2> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> No, not the order of the elements. In sets it <b>does not matter what order the elements are in</b>.</div> <div class="example" style="background-color: #e0f0ff; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; box-shadow: rgb(208, 208, 255) 5px 5px 1px; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 20px 0px 30px 50px; overflow: hidden; padding: 5px 10px 1px 15px;"> <div style="margin-bottom: 25px; margin-top: 16px;"> Example: {1,2,3,4) is the same set as {3,1,4,2}</div> </div> <div class="center80" style="background-color: cornsilk; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 10px 40px; padding: 5px 8px;"> <div align="center" style="margin-bottom: 8px; margin-top: 10px;"> When we say "order" in sets we mean <b>the size of the set</b>.</div> </div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> Just as there are finite and infinite sets, each has finite and infinite order.</div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> For finite sets, we represent the order by a number, the number of elements.</div> <div class="example" style="background-color: #e0f0ff; border-bottom-left-radius: 10px; border-bottom-right-radius: 10px; border-top-left-radius: 10px; border-top-right-radius: 10px; box-shadow: rgb(208, 208, 255) 5px 5px 1px; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin: 20px 0px 30px 50px; overflow: hidden; padding: 5px 10px 1px 15px;"> <div style="margin-bottom: 25px; margin-top: 16px;"> Example, {10, 20, 30, 40} has an order of 4.</div> </div> <div style="font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px; margin-bottom: 25px; margin-top: 16px;"> 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.</div>

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Operation of Sets

Introduction to Sets Forget everything you know about numbers. In fact, forget you even know what a number is. This is where mathematics starts. Instead of math with numbers, we will now think about …

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Iba't ibang Kahulugan ng Wika

<div class="MsoNormal"> <span style="background: white; font-family: 'Comic Sans MS'; font-size: 12pt; line-height: 115%;"><br /></span></div> <div class="MsoNormal"> <span style="background: white; font-family: 'Comic Sans MS'; font-size: 12pt; line-height: 115%;">"Ang wika ang pangunahin at pinakamabusising anyo ng gawaing pansagisag ng tao."</span><span style="font-family: 'Comic Sans MS'; font-size: 12pt; line-height: 115%;"><br style="text-align: -webkit-center;" /> <span style="background: white; mso-bidi-font-style: italic; mso-bidi-font-weight: bold;">- Archibald Hill</span><br style="text-align: -webkit-center;" /> <span style="background: white; mso-bidi-font-style: italic; mso-bidi-font-weight: bold;"><br /><br style="text-align: -webkit-center;" /> "Itinuturing ang wika bilang saplot ng kaisipan; gayunman, mas angkop marahil sabihing ang wika ay ang saplot-kalamnan, ang mismong katawan ng kaisipan."</span><br style="text-align: -webkit-center;" /> <span style="background: white; mso-bidi-font-style: italic; mso-bidi-font-weight: bold;">- Thomas Carlyle</span><br style="text-align: -webkit-center;" /> <br style="text-align: -webkit-center;" /> <br style="text-align: -webkit-center;" /> <span style="background: white; mso-bidi-font-style: italic; mso-bidi-font-weight: bold;">"Ang wika ay isang kalipunan ng mga salita at ang pamamaraan ng pagsasama-sama ng mga ito para magkaunawaan o makapagkomyunikeyt ang isang grupo ng mga tao."</span><br style="text-align: -webkit-center;" /> <span style="background: white; mso-bidi-font-style: italic; mso-bidi-font-weight: bold;">- Pamela Constantino at Galileo Zafra<span class="apple-converted-space"> <o:p></o:p></span></span></span></div> <div class="MsoNormal"> <span style="font-family: 'Comic Sans MS'; font-size: 12pt; line-height: 115%;"><span style="background: white; mso-bidi-font-style: italic; mso-bidi-font-weight: bold;"><span class="apple-converted-space"><br /></span></span></span></div> <div class="MsoNormal"> <br /></div> <div class="MsoNormal"> <span style="background: white; font-family: 'Comic Sans MS'; font-size: 12pt; line-height: 115%;">"Ang wika ay isang sistematik na balangkas na may binibigkas na tunog na pinipili at isinasaayos sa paraang arbitrari upang magamit ng mga taong may iisang kultura."<span class="apple-converted-space"> </span></span><span style="font-family: 'Comic Sans MS'; font-size: 12pt; line-height: 115%;"><br /> <span style="background: white;">- FE Dance<span class="apple-converted-space"> </span><o:p></o:p></span></span></div> <div class="MsoNormal"> <span style="font-family: 'Comic Sans MS'; font-size: 12pt; line-height: 115%;"><span style="background: white;"><span class="apple-converted-space"><br /></span></span></span></div> <div class="MsoNormal"> <span style="font-family: 'Comic Sans MS'; font-size: 12pt; line-height: 115%;"><span style="background: white;"><span class="apple-converted-space"><br /></span></span></span></div> <div class="MsoNormal"> </div> <div class="MsoNormal"> <span style="background: white; font-family: 'Comic Sans MS'; font-size: 12pt; line-height: 115%;">"Ang wika ay isang kalipunan ng mga salita at ang pamamaraan ng pagsasama-sama ng mga ito para magkaunawaan o makapagkomyunikeyt ang isang grupo ng mga tao."<span class="apple-converted-space"> </span></span><span style="font-family: 'Comic Sans MS'; font-size: 12pt; line-height: 115%;"><br /> <span style="background: white;">- Valentine<span class="apple-converted-space"> </span><o:p></o:p></span></span></div> <div class="MsoNormal"> <span style="font-family: 'Comic Sans MS'; font-size: 12pt; line-height: 115%;"><span style="background: white;"><span class="apple-converted-space"><br /></span></span></span></div> <div class="MsoNormal"> <br /></div>

Iba't ibang Kahulugan ng Wika

"Ang wika ang pangunahin at pinakamabusising anyo ng gawaing pansagisag ng tao." - Archibald Hill "Itinuturing ang wika bilang saplot ng kaisipan; gayunman, mas angkop marahil sabihing ang wika ay an…

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Some Definations College Algebra

<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <br /></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">ALGEBRAIC EXPRESSION</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is an </span><a href="http://en.wikipedia.org/wiki/Expression_(mathematics)" title="Expression (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">expression</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> built up from </span><a href="http://en.wikipedia.org/wiki/Constant_(mathematics)" title="Constant (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">constants</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, </span><a href="http://en.wikipedia.org/wiki/Variable_(mathematics)" title="Variable (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">variables</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, and the </span><a href="http://en.wikipedia.org/wiki/Algebraic_operation" title="Algebraic operation"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">algebraic operations</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> (</span><a href="http://en.wikipedia.org/wiki/Addition" title="Addition"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">addition</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, </span><a href="http://en.wikipedia.org/wiki/Subtraction" title="Subtraction"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">subtraction</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, </span><a href="http://en.wikipedia.org/wiki/Multiplication" title="Multiplication"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">multiplication</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, </span><a href="http://en.wikipedia.org/wiki/Division_(mathematics)" title="Division (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">division</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> and</span><a href="http://en.wikipedia.org/wiki/Exponentiation" title="Exponentiation"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">exponentiation</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> by an exponent that is a </span><a href="http://en.wikipedia.org/wiki/Rational_number" title="Rational number"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">rational number</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">).<o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">POLYNOMIAL</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is an </span><a href="http://en.wikipedia.org/wiki/Mathematical_expression" title="Mathematical expression"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">expression</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> consisting of </span><a href="http://en.wikipedia.org/wiki/Variable_(mathematics)" title="Variable (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">variables</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> (or <a href="http://en.wikipedia.org/wiki/Indeterminate_(variable)" title="Indeterminate (variable)"><span style="color: black; mso-themecolor: text1;">indeterminates</span></a>) and </span><a href="http://en.wikipedia.org/wiki/Coefficient" title="Coefficient"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">coefficients</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, that involves only the operations of</span><a href="http://en.wikipedia.org/wiki/Addition" title="Addition"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">addition</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, </span><a href="http://en.wikipedia.org/wiki/Subtraction" title="Subtraction"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">subtraction</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, </span><a href="http://en.wikipedia.org/wiki/Multiplication" title="Multiplication"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">multiplication</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, and non-negative </span><a href="http://en.wikipedia.org/wiki/Integer" title="Integer"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">integer</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> </span><a href="http://en.wikipedia.org/wiki/Exponent" title="Exponent"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">exponents</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">. An example of a polynomial of a single indeterminate (or variable), x, isx2 − 4x + 7, which is a </span><a href="http://en.wikipedia.org/wiki/Quadratic_function" title="Quadratic function"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">quadratic polynomial</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">.<o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">A MONOMIAL</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is a product of powers of </span><a href="http://en.wikipedia.org/wiki/Variable_(mathematics)" title="Variable (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">variables</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> with </span><a href="http://en.wikipedia.org/wiki/Nonnegative_integer" title="Nonnegative integer"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">nonnegative integer</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> exponents, or, in other words, a product of variables, possibly with repetitions.</span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <br /></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">A BINOMIAL</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is a </span><a href="http://en.wikipedia.org/wiki/Polynomial" title="Polynomial"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">polynomial</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> which is the sum of two terms, which are </span><a href="http://en.wikipedia.org/wiki/Monomial" title="Monomial"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">monomials</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">.</span><a href="http://en.wikipedia.org/wiki/Binomial#cite_note-1"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">[1]</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> It is the simplest kind of polynomial after the </span><a href="http://en.wikipedia.org/wiki/Monomial" title="Monomial"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">monomials</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">.<o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">A TRINOMIAL</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is a </span><a href="http://en.wikipedia.org/wiki/Polynomial" title="Polynomial"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">polynomial</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> consisting of three terms or </span><a href="http://en.wikipedia.org/wiki/Monomial" title="Monomial"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">monomials</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">.<o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">A MULTINOMIAL</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is an algebraic expression having more than one term. For example: 9x3 + 2x2 + 5<br /> <b>IN ALGEBRA</b> a term is either a single number or variable, or numbers and variables multiplied together. Terms are separated by + or – signs<o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">A VARIABLE</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is a symbol that denotes a mathematical object, which could be a </span><a href="http://en.wikipedia.org/wiki/Number" title="Number"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">number</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, a </span><a href="http://en.wikipedia.org/wiki/Vector_(mathematics)" title="Vector (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">vector</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, a </span><a href="http://en.wikipedia.org/wiki/Matrix_(mathematics)" title="Matrix (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">matrix</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, or even a function. In this case, the original property of "variability" of a variable is not kept (except, sometimes, for informal explanations).<o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <a href="http://www.math.com/school/subject2/lessons/S2U2L4GL.html"><b><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">LIKE TERMS</span></b></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> are terms that contain the same variables raised to the same power. Only the </span><a href="http://www.math.com/school/subject2/lessons/S2U2L4GL.html"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">coefficients</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> of like terms are different.<o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">UNLIKE TERMS</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> : The terms having the same variable with different exponents or different variable with same exponents are calledUnlike terms .<o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">EXPONENTS</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 53. The "exponent", being 3 in this example, stands for however many times the value is being multiplied. The thing that's being multiplied, being5 in this example, is called the "base".<o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">DEFINITION OF NUMERICAL COEFFICIENT -</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">The constant multiplicative factors attached to the variables in an expression are known as Numerical Coefficient.<b><o:p></o:p></b></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">FACTORS</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> are numbers you can multiply together to get another number:<o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">THE PRODUCT RULE</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is a formal rule for differentiating problems where one function is multiplied by another.</span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <br /></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">THE QUOTIENT RULE</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is a formal rule for differentiating problems where one function is divided by another.<o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> <b>THE POWER RULE</b> is one of the most important </span><a href="http://en.wikipedia.org/wiki/Differentiation_rules" title="Differentiation rules"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">differentiation rules</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">. Since differentiation is linear, </span><a href="http://en.wikipedia.org/wiki/Polynomial" title="Polynomial"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">polynomials</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> can be differentiated using this rule.<o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">SYNTHETIC DIVISION</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is a shorthand, or shortcut, method of </span><a href="http://www.purplemath.com/modules/polydiv.htm"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">polynomial division</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> in the special case of dividing by a linear factor -- and it only works in this case. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials. More about this later.<o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <br /></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <br /></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <br /></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <br /></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <br /></div> <h3 style="background: white; margin-bottom: 2.25pt; margin-left: 0in; margin-right: 0in; margin-top: 11.25pt;"> <span style="font-family: Arial, sans-serif; font-size: 17pt; font-weight: normal; line-height: 115%;">Exponents rules and properties<o:p></o:p></span></h3> <table border="1" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="background: white; border-collapse: collapse; border: none; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 3.75pt 3.75pt 3.75pt 3.75pt; mso-yfti-tbllook: 1184; width: 620px;"> <tbody> <tr> <td style="background: #E0E0E0; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div align="center" class="MsoNormal" style="margin-bottom: 3.75pt; text-align: center;"> <span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">Rule name<o:p></o:p></span></div> </td> <td style="background: #E0E0E0; border-left: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div align="center" class="MsoNormal" style="margin-bottom: 3.75pt; text-align: center;"> <span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">Rule<o:p></o:p></span></div> </td> <td style="background: #E0E0E0; border-left: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div align="center" class="MsoNormal" style="margin-bottom: 3.75pt; text-align: center;"> <span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">Example<o:p></o:p></span></div> </td> </tr> <tr> <td rowspan="2" style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><a href="http://www.rapidtables.com/math/number/exponent.htm#product"><span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; line-height: 115%; mso-themecolor: text1;">Product rules</span></a><span style="font-size: 10.5pt; line-height: 115%;"><o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">a<span class="apple-converted-space"></span>ⁿ</span></i><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">·<span class="apple-converted-space"> </span><i>a<span class="apple-converted-space"></span>^m</i><span class="apple-converted-space"> </span>=<span class="apple-converted-space"> </span><i>a<span class="apple-converted-space"></span>^n+m</i><o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">2³<span class="apple-converted-space"> </span>· 2⁴<span class="apple-converted-space"> </span>= 2³⁺⁴<span class="apple-converted-space"> </span>= 128<o:p></o:p></span></div> </td> </tr> <tr> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">a<span class="apple-converted-space"></span>ⁿ</span></i><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">·<span class="apple-converted-space"> </span><i>b<span class="apple-converted-space"></span>ⁿ</i><span class="apple-converted-space"> </span>= (<i>a<span class="apple-converted-space"> </span></i>·<span class="apple-converted-space"><i> </i></span><i>b</i>)<span class="apple-converted-space"><i></i></span><i>ⁿ</i><o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">3²<span class="apple-converted-space"> </span>· 4²<span class="apple-converted-space"> </span>= (3·4)²<span class="apple-converted-space"> </span>= 144<o:p></o:p></span></div> </td> </tr> <tr> <td rowspan="2" style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><a href="http://www.rapidtables.com/math/number/exponent.htm#quotient"><span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; line-height: 115%; mso-themecolor: text1;">Quotient rules</span></a><span style="font-size: 10.5pt; line-height: 115%;"><o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">a<span class="apple-converted-space"></span>ⁿ</span></i><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">/<span class="apple-converted-space"> </span><i>a<span class="apple-converted-space"></span>^m</i><span class="apple-converted-space"> </span>=<span class="apple-converted-space"> </span><i>a<span class="apple-converted-space"></span>ⁿ</i>^-<i>m</i><o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">2⁵<span class="apple-converted-space"> </span>/ 2³<span class="apple-converted-space"> </span>= 2^5-3<span class="apple-converted-space"> </span>= 4<o:p></o:p></span></div> </td> </tr> <tr> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">a<span class="apple-converted-space"></span>ⁿ</span></i><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">/<span class="apple-converted-space"> </span><i>b<span class="apple-converted-space"></span>ⁿ</i><span class="apple-converted-space"> </span>= (<i>a<span class="apple-converted-space"> </span></i>/<span class="apple-converted-space"><i> </i></span><i>b</i>)<span class="apple-converted-space"><i></i></span><i>ⁿ</i><o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">4³<span class="apple-converted-space"> </span>/ 2³<span class="apple-converted-space"> </span>= (4/2)³<span class="apple-converted-space"> </span>= 8<o:p></o:p></span></div> </td> </tr> <tr> <td rowspan="4" style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><a href="http://www.rapidtables.com/math/number/exponent.htm#power"><span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; line-height: 115%; mso-themecolor: text1;">Power rules</span></a><span style="font-size: 10.5pt; line-height: 115%;"><o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">(<i>bⁿ</i>)<i>^m</i><span class="apple-converted-space"> </span>=<span class="apple-converted-space"> </span><i>b^n·m</i><o:p></o:p></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">(2³)²<span class="apple-converted-space"> </span>= 2^3·2<span class="apple-converted-space"> </span>= 64<o:p></o:p></span></div> </td> </tr> <tr> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><i>_{<span style="font-size: 17pt; line-height: 115%;">b</span>}</i><i><span style="font-size: 13.5pt; line-height: 115%;">n^m</span></i><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span>_{<span style="font-size: 17pt; line-height: 115%;">= b</span>}<span style="font-size: 17pt; line-height: 115%;">(</span><i><span style="font-size: 13.5pt; line-height: 115%;">n^m</span></i><span style="font-size: 17pt; line-height: 115%;">)<o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;">_{<span style="font-size: 17pt; line-height: 115%;">2</span>}<span style="font-size: 13.5pt; line-height: 115%;">3²</span><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span>_{<span style="font-size: 17pt; line-height: 115%;">= 2</span>}<span style="font-size: 17pt; line-height: 115%;">(</span><span style="font-size: 13.5pt; line-height: 115%;">3²</span><span style="font-size: 17pt; line-height: 115%;">)_{= 512}<o:p></o:p></span></span></div> </td> </tr> <tr> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><i>^{<span style="font-size: 17pt; line-height: 115%;">m</span>}</i><span style="font-size: 17pt; line-height: 115%;">√(<i>bⁿ</i>) =<span class="apple-converted-space"> </span><i>b</i><span class="apple-converted-space"> </span><i>ⁿ</i>^/<i>m</i><o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;">^{<span style="font-size: 17pt; line-height: 115%;">2</span>}<span style="font-size: 17pt; line-height: 115%;">√(2⁶) = 2^6/2<span class="apple-converted-space"> </span>= 8<o:p></o:p></span></span></div> </td> </tr> <tr> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">b</span></i>^{<span style="font-size: 17pt; line-height: 115%;">1/<i>n</i></span>}<span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">=<span class="apple-converted-space"> </span><i>ⁿ</i>√<i>b</i><o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">8^1/3<span class="apple-converted-space"> </span>=<span class="apple-converted-space"> </span>³√8<span class="apple-converted-space"> </span>= 2<o:p></o:p></span></div> </td> </tr> <tr> <td style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><a href="http://www.rapidtables.com/math/number/exponent.htm#negative"><span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; line-height: 115%; mso-themecolor: text1;">Negative exponents</span></a><span style="font-size: 10.5pt; line-height: 115%;"><o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">b^-n</span></i><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">= 1 /<span class="apple-converted-space"> </span><i>bⁿ</i><o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">2^-3<span class="apple-converted-space"> </span>= 1/2³<span class="apple-converted-space"> </span>= 0.125<o:p></o:p></span></div> </td> </tr> <tr> <td rowspan="2" style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">Zero rules<o:p></o:p></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">b</span></i>^{<span style="font-size: 17pt; line-height: 115%;">0</span>}<span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">= 1<o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">5⁰<span class="apple-converted-space"> </span>= 1<o:p></o:p></span></div> </td> </tr> <tr> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">0<i>ⁿ</i><span class="apple-converted-space"> </span>= 0 , for<span class="apple-converted-space"> </span><i>n</i>>0<o:p></o:p></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">0⁵<span class="apple-converted-space"> </span>= 0<o:p></o:p></span></div> </td> </tr> <tr> <td rowspan="2" style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">One rules<o:p></o:p></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">b</span></i>^{<span style="font-size: 17pt; line-height: 115%;">1</span>}<span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">=<span class="apple-converted-space"> </span><i>b</i><o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">5¹<span class="apple-converted-space"> </span>= 5<o:p></o:p></span></div> </td> </tr> <tr> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">1<i>ⁿ</i><span class="apple-converted-space"> </span>= 1<o:p></o:p></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">1⁵<span class="apple-converted-space"> </span>= 1<o:p></o:p></span></div> </td> </tr> <tr> <td style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">Minus one rule<o:p></o:p></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit;"><span style="font-size: 17pt; line-height: 115%;"><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"/> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"/> <v:f eqn="sum @0 1 0"/> <v:f eqn="sum 0 0 @1"/> <v:f eqn="prod @2 1 2"/> <v:f eqn="prod @3 21600 pixelWidth"/> <v:f eqn="prod @3 21600 pixelHeight"/> <v:f eqn="sum @0 0 1"/> <v:f eqn="prod @6 1 2"/> <v:f eqn="prod @7 21600 pixelWidth"/> <v:f eqn="sum @8 21600 0"/> <v:f eqn="prod @7 21600 pixelHeight"/> <v:f eqn="sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/> <o:lock v:ext="edit" aspectratio="t"/> </v:shapetype><v:shape id="Picture_x0020_1" o:spid="_x0000_i1025" type="#_x0000_t75" alt="http://www.rapidtables.com/math/number/exponent/minus_1_rule.gif" style='width:157.5pt;height:39.75pt;visibility:visible;mso-wrap-style:square'> <v:imagedata src="file:///C:\Users\STATIO~1\AppData\Local\Temp\msohtmlclip1\01\clip_image001.gif" o:title="minus_1_rule"/> </v:shape><![endif]--><!--[if !vml]--><img alt="http://www.rapidtables.com/math/number/exponent/minus_1_rule.gif" border="0" height="53" src="file:///C:/Users/STATIO~1/AppData/Local/Temp/msohtmlclip1/01/clip_image001.gif" v:shapes="Picture_x0020_1" width="210" /><!--[endif]--></span><span style="font-size: 17pt; line-height: 115%;"><o:p></o:p></span></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">(-1)⁵<span class="apple-converted-space"> </span>= -1<o:p></o:p></span></div> </td> </tr> <tr> <td style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">Derivative rule<o:p></o:p></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">(<i>xⁿ</i>)<i>'<span class="apple-converted-space"> </span></i>=<span class="apple-converted-space"> </span><i>n</i>·<i>x<span class="apple-converted-space"></span>ⁿ</i>^-1<o:p></o:p></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">(<i>x</i>³)<i>'<span class="apple-converted-space"> </span></i>= 3·<i>x</i>^3-1<o:p></o:p></span></div> </td> </tr> <tr> <td style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">Integral rule<o:p></o:p></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">∫<span class="apple-converted-space"><i> </i></span><i>xⁿdx</i><span class="apple-converted-space"> </span>=<span class="apple-converted-space"> </span><i>xⁿ</i>⁺¹/(<i>n</i>+1)+<i>C</i><o:p></o:p></span></div> </td> <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;"> <span style="font-family: inherit; font-size: 17pt; line-height: 115%;">∫<span class="apple-converted-space"><i> </i></span><i>x</i>²<i>dx</i><span class="apple-converted-space"> </span>=<span class="apple-converted-space"> </span><i>x</i>²⁺¹/(2+1)+<i>C</i><o:p></o:p></span></div> </td> </tr> </tbody></table> <span style="font-family: inherit;"><br /></span> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;"> <br /></div>

Some Definations College Algebra

ALGEBRAIC EXPRESSION is an expression built up from constants, variables, and the algebraic operations (addition, subtraction, multiplication, division andexponentiation by an exponent that is a rati…

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