1 - LECTURE 1 SETS SUBSETS AND SET OPERATIONS(1.1-1.4 There...

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LECTURE 1: SETS, SUBSETS, AND SET OPERATIONS ( § 1.1-1.4) There is surely a piece of divinity in us, something that was before the elements, and owes no homage unto the sun. Sir Thomas Browne (1605 - 1682) 1. What is a set? A set is a collection of elements , where we don’t allow repetition and we ignore order. Example 1. Here are some examples of sets. 1. Let S = { 1 , 2 , 3 } = { 1 , 3 , 2 } = { 3 , 2 , 1 } . This is a set whose elements are 1, 2, and 3. We write 1 S , 2 S , and 3 S . Notice that 4 / S . There are many ways to write down the same set. 2. Some sets get special names. N = { 1 , 2 , 3 , . . . } = the natural numbers Z = { . . . , - 2 , - 1 , 0 , 1 , 2 , . . . } = the integers 3. Some sets are described by properties: T = { 1 , 2 , 3 , 4 , 5 } = { x N : x 5 } . U = { 1 , 4 , 9 , 16 , . . . } = { x N : x = y 2 for some y Z } = { x 2 : x N } , 4. Here are five more important sets. R = the real numbers C = the complex numbers = { a + bi : a, b R , i 2 = - 1 } Q = the rational numbers = { a/b : a, b Z , b 6 = 0 } I = the irrational numbers = { x R : x / Q } Where does 2 live? We have 2 I , 2 R , 2 C , but 2 / Q .
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