CHAPTER 1- Basic Ideas - CHAPTER 1 Basic Ideas In the end...

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CHAPTER 1 Basic Ideas In the end, all mathematics can be boiled down to logic and set theory. Because of this, any careful presentation of fundamental mathematical ideas is inevitably couched in the language of logic and sets. This chapter defines enough of that language to allow the presentation of basic real analysis. Much of it will be familiar to you, but look at it anyway to make sure you understand the notation. 1. Sets Set theory is a large and complicated subject in its own right. There is no time in this course to touch on any but the simplest parts of it. Instead, we’ll just look at a few topics from what is often called “naive set theory.” We begin with a few definitions. A set is a collection of objects called elements. Usually, sets are denoted by the capital letters A, B, · · · , Z . A set can consist of any type and number of elements. Even other sets can be elements of a set. The sets dealt with here usually have real numbers as their elements. If a is an element of the set A , we write a A . If a is not an element of the set A , we write a / A . If all the elements of A are also elements of B , then A is a subset of B . In this case, we write A B or B A . In particular, notice that whenever A is a set, then A A . Two sets A and B are equal , if they have the same elements. In this case we write A = B . It is easy to see that A = B i ff A B and B A . Establishing that both of these containments are true is the most common way to show that two sets are equal. If A B and A ̸ = B , then A is a proper subset of B . In cases when this is important, it is written A B instead of just A B . There are several ways to describe a set. A set can be described in words such as “ P is the set of all presidents of the United States.” This is cumbersome for complicated sets. All the elements of the set could be listed in curly braces as S = { 2 , 0 , a } . If the set has many elements, this is impractical, or impossible. More common in mathematics is set builder notation. Some examples are P = { p : p is a president of the United states } = { Washington, Adams, Je ff erson, · · · , Clinton, Bush, Obama } and A = { n : n is a prime number } = { 2 , 3 , 5 , 7 , 11 , · · · } . 1-1
1-2 CHAPTER 1. BASIC IDEAS In general, the set builder notation defines a set in the form { formula for a typical element : objects to plug into the formula } . A more complicated example is the set of perfect squares: S = { n 2 : n is an integer } = { 0 , 1 , 4 , 9 , · · · } . The existence of several sets will be assumed. The simplest of these is the empty set , which is the set with no elements. It is denoted as . The natural numbers is the set N = { 1 , 2 , 3 , · · · } consisting of the positive integers. The set Z = {· · · , 2 , 1 , 0 , 1 , 2 , · · · } is the set of all integers. ω = { n Z : n 0 } = { 0 , 1 , 2 , · · · } is the nonnegative integers. Clearly, ∅ ⊂ A , for any set A and ∅ ⊂ N ω Z .

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