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s1_1 - CHAPTER 1 Introduction to Sets and Functions 1...

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CHAPTER 1 Introduction to Sets and Functions 1. Introduction to Sets 1.1. Basic Terminology. We begin with a refresher in the basics of set theory. Our treatment will be an informal one rather than taking an axiomatic approach at this time. Later in the semester we will revisit sets with a more formal approach. A set is a collection or group of objects or elements or members . (Cantor 1895) A set is said to contain its elements. In each situation or context, there must be an underlying universal set U , either specifically stated or understood. Notation: If x is a member or element of the set S , we write x S . If x is not an element of S we write x 6∈ S . 1.2. Notation for Describing a Set. Example 1.2.1 . List the elements between braces: S = { a, b, c, d } = { b, c, a, d, d } Specify by attributes: S = { x | x 5 or x < 0 } , where the universe is the set of real numbers. Use brace notation with ellipses: S = { . . . , - 3 , - 2 , - 1 } , the set of negative integers. Discussion 9
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1. INTRODUCTION TO SETS 10 Sets can be written in a variety of ways. One can, of course, simply list the elements if there are only a few. Another way is to use set builder notation, which specifies the sets using a predicate to indicate the attributes of the elements of the set. For example, the set of even integers is { x | x = 2 n, n Z } or { . . . , - 2 , 0 , 2 , 4 , 6 , . . . } . The first set could be read as “the set of all x’s such that x is twice an integer.” The symbol | stands for “such that.” A colon is often used for “such that” as well, so the set of even integers could also be written { x : x = 2 n, n Z } . 1.3. Common Universal Sets. The following notation will be used throughout these notes. R = the real numbers N = the natural numbers = { 0 , 1 , 2 , 3 , . . . } Z = the integers = { . . . , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , . . . } Z + = the positive integers = { 1 , 2 , 3 , . . . } Discussion
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