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Unformatted text preview: Sets Margaret M. Fleck 8 September 2011 These notes cover set notation, operations on sets, measuring the size of sets, and proving claims involving sets. They also discuss vacuous truth. 1 Sets So far, we’ve been assuming only a basic understanding of sets. It’s time to discuss sets systematically, including a useful range of constructions, opera tions, notation, and special cases. And we’ll see how to do proofs of claims involving sets. Definition: A set is an unordered collection of objects. For example, the natural numbers are a set. So are the integers between 3 and 7 (inclusive). So are all the planets in this solar system or all the programs written by students in CS 225 in the last three years. The objects in a set can be anything you want. The items in the set are called its elements or members. We’ve already seen the notation for this: x ∈ A means that x is a member of the set A . There’s three basic ways to define a set: • describe its contents in mathematical English, e.g. “the integers be tween 3 and 7, inclusive.” 1 • list all its members, e.g. { 3 , 4 , 5 , 6 , 7 } • use socalled set builder notation, e.g. { x ∈ Z  3 ≤ x ≤ 7 } Set builder notation has two parts separated with a vertical bar (or, by some writers, a colon). The first part names a variable (in this case x ) that ranges over all objects in the set. The second part one or more constraints that these objects must satisfy, e.g. 3 ≤ x ≤ 7. The type of the variable (integer in our example) can be specified either before or after the vertical bar. The separator (  or :) is often read “such that.” Here’s an example of a set containing an infinite number of objects • “multiples of 7” • { ... − 14 , − 7 , , 7 , 14 , 21 , 18 ,... } • { x ∈ Z  x is a multiple of 7 } We couldn’t list all the elements, so we had to use “ ... ”. This is only a good idea if the pattern will be clear to your reader. If you aren’t sure, use one of the other methods. If we wanted to use shorthand for “multiple of”, it might be confusing to have  used for two different purposes. So it would probably be best to use the colon variant of set builder notation: { x ∈ Z : 7  x } 2 Things to be careful about A set is an unordered collection. So { 1 , 2 , 3 } and { 2 , 3 , 1 } are two descriptions of the same set. Each element occurs only once in a set. Or, alternatively, it doesn’t matter how many times you write it. So { 1 , 2 , 3 } and { 1 , 2 , 3 , 2 } also describe the same set. 2 We’ve seen ordered pairs and triples of numbers, such as (3 , 4) and (4 , 5 , 2). The general term for an ordered sequence of k numbers is a ktuple. 1 Tuples are very different from sets, in that the order of values in a tuple matters and duplicate elements don’t magically collapse. So (1 , 2 , 2 , 3) negationslash = (1 , 2 , 3) and (1 , 2 , 2 , 3) negationslash = (2 , 2 , 1 , 3). Therefore, make sure to enclose the elements of a set in curly brackets and carefully distinguish curly brackets (set) from...
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 Spring '08
 Erickson
 Set Theory, Sets, Empty set

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