{"id":23127,"date":"2023-03-03T05:24:48","date_gmt":"2023-03-03T05:24:48","guid":{"rendered":"https:\/\/www.booksofall.com\/cn\/?post_type=product&p=23127"},"modified":"2023-03-03T05:24:49","modified_gmt":"2023-03-03T05:24:49","slug":"precalculus-version-3-corrected-edition","status":"publish","type":"product","link":"https:\/\/www.booksofall.com\/cn\/precalculus-version-3-corrected-edition\/","title":{"rendered":"Precalculus Version 3, Corrected Edition"},"content":{"rendered":"

Another version of Precalculus: Precalculus Prerequisites a.k.a. \u2018Chapter 0\u2019<\/a><\/p>\n

Chapter 1 – Relations and Functions<\/h2>\n

1.1 Sets of Real Numbers and the Cartesian Coordinate Plane<\/h3>\n

1.1.1 Sets of Numbers<\/p>\n

While the authors would like nothing more than to delve quickly and deeply into the sheer excitement that is Precalculus, experience1 has taught us that a brief refresher on some basic notions is welcome, if not completely necessary, at this stage. To that end, we present a brief summary of \u2018set theory\u2019 and some of the associated vocabulary and notations we use in the text. Like all good Math books, we begin with a definition.<\/p>\n

Definition 1.1.<\/strong> A set is a well-defined collection of objects which are called the \u2018elements\u2019 of the set. Here, \u2018well-defined\u2019 means that it is possible to determine if something belongs to the collection or not, without prejudice.<\/em><\/p>\n

For example, the collection of letters that make up the word \u201csmolko\u201d is well-defined and is a set, but the collection of the worst math teachers in the world is not well-defined, and so is not a set.2 In general, there are three ways to describe sets. They are<\/p>\n

Ways to Describe Sets <\/strong><\/p>\n

    \n
  1. The Verbal Method<\/a>: Use a sentence to define a set.<\/li>\n
  2. The Roster Method<\/a>: Begin with a left brace \u2018{\u2019, list each element of the set only once and then end with a right brace \u2018}\u2019.<\/li>\n
  3. The Set-Builder Method<\/a>: A combination of the verbal and roster methods using a \u201cdummy variable\u201d such as x.<\/li>\n<\/ol>\n

    For example, let S be the set described verbally as the set of letters that make up the word \u201csmolko\u201d. A roster description of S would be {s,m, o, l, k}. Note that we listed \u2018o\u2019 only once, even though it appears twice in \u201csmolko.\u201d Also, the order of the elements doesn\u2019t matter, so {k, l,m, o, s} is also a roster description of S. A set-builder description of S is:<\/p>\n

    {x |x is a letter in the word \u201csmolko\u201d.}<\/p>\n

    The way to read this is: \u2018The set of elements x such that x is a letter in the word \u201csmolko.\u201d\u2019 In each of the above cases, we may use the familiar equals sign \u2018=\u2019 and write S = {s,m, o, l, k} or S = {x |x is a letter in the word \u201csmolko\u201d.}. Clearly m is in S and q is not in S. We express these sentiments mathematically by writing m \u2208 S and q \/\u2208 S. Throughout your mathematical upbringing, you have encountered several famous sets of numbers. They are listed below.<\/p>\n

    Sets of Numbers <\/strong><\/p>\n

      \n
    1. The Empty Set<\/a>: \u2205 = {} = {x |x 6= x}. This is the set with no elements. Like the number \u20180,\u2019 it plays a vital role in mathematics.<\/li>\n
    2. The Natural Numbers<\/a>: N = {1, 2, 3, . . .} The periods of ellipsis here indicate that the natural numbers contain 1, 2, 3, \u2018and so forth\u2019.<\/li>\n
    3. The Whole Numbers<\/a>: W = {0, 1, 2, . . .}<\/li>\n
    4. The Integers: Z = {. . . ,\u22123,\u22122,\u22121, 0, 1, 2, 3, . . .}<\/li>\n
    5. The Rational Numbers<\/a>: Q = { a b | a \u2208 Z and b \u2208 Z } . Rational numbers are the ratios of integers (provided the denominator is not zero!) It turns out that another way to describe the rational numbers is:
      \nQ = {x |x possesses a repeating or terminating decimal representation.}<\/li>\n
    6. The Real Numbers<\/a>: R = {x |x possesses a decimal representation.}<\/li>\n
    7. The Irrational Numbers<\/a>: P = {x |x is a non-rational real number.} Said another way, an irrational number is a decimal which neither repeats nor terminates.<\/li>\n
    8. The Complex Number<\/a>s: C = {a+ bi | a,b \u2208 R and i = \u221a \u22121} Despite their importance, the complex numbers play only a minor role in the text.<\/li>\n<\/ol>\n

      It is important to note that every natural number is a whole number, which, in turn, is an integer. Each integer is a rational number (take b = 1 in the above definition for Q) and the rational numbers are all real numbers, since they possess decimal representations.3 If we take b = 0 in the above definition of<\/span> C<\/span>, we see that every real number is a complex number. In this sense, the sets <\/span>N<\/span>,<\/span> W<\/span>,<\/span> Z<\/span>,<\/span> Q<\/span>,<\/span> R<\/span>, and<\/span> C<\/span> are \u2018nested\u2019 like<\/span> Matryoshka dolls<\/span><\/a>.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"