{"id":23122,"date":"2023-03-03T03:55:41","date_gmt":"2023-03-03T03:55:41","guid":{"rendered":"https:\/\/www.booksofall.com\/id\/?post_type=product&p=23122"},"modified":"2023-03-03T05:25:30","modified_gmt":"2023-03-03T05:25:30","slug":"precalculus-prerequisites-a-k-a-chapter-0","status":"publish","type":"product","link":"https:\/\/www.booksofall.com\/id\/precalculus-prerequisites-a-k-a-chapter-0\/","title":{"rendered":"Precalculus Prerequisites a.k.a. \u2018Chapter 0\u2019"},"content":{"rendered":"

Another version of Precalculus: Precalculus Version 3, Corrected Edition<\/a><\/p>\n

0.1 Basic Set Theory and Interval Notation<\/h2>\n

0.1.1 Some Basic Set Theory Notions<\/h3>\n

Like all good Math books, we begin with a definition.<\/p>\n

Definition 0.1.<\/strong>\u00a0A\u00a0set\u00a0is 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

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\u00a0not\u00a0well-defined and therefore is\u00a0not\u00a0a set.1 In general, there are three ways to describe sets and those methods are listed below.<\/p>\n

Ways to Describe Sets\u00a0<\/strong><\/p>\n

    \n
  1. The Verbal Method<\/a>: Use a sentence to define the 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\u00a0roster\u00a0description of S is {s, m, o, l , k}. Note that we listed \u2018o\u2019 only once, even though it appears twice in the word \u201csmolko\u201d. Also, the order of the elements doesn\u2019t matter, so {k , l , m, o, s} is also a roster description of S. Moving right along, a\u00a0set-builder\u00a0description of S is: {x | x is a letter in the word \u201csmolko\u201d}. 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}. Notice that m is in S but many other letters, such as q, are not in S. We express these ideas of set inclusion and exclusion mathematically using the symbols m 2 S (read \u2018m is in S\u2019) and q \/2 S (read \u2018q is not in S\u2019). More precisely, we have the following.<\/p>\n

    Definition 0.2.<\/strong>\u00a0Let A be a set.<\/em><\/p>\n