lect_11 - Sets I Margaret M. Fleck 12 February 2010 This...

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Unformatted text preview: Sets I Margaret M. Fleck 12 February 2010 This lecture presents sets and basic operations on sets (Rosen sections 2.1 and 2.2). 1 Announcements The quiz solutions are on the web. I expect the quizzes themselves to be handed back in discussion sections next week, though the grades may be available online somewhat sooner. 2 Sets Im sure youve all seen the basic ideas of sets and, indeed, weve been using some of them all term. Its time to discuss sets systematically, showing you a useful range of constructions, notation, and special cases. A few operations (e.g. power sets and Cartesian products) are probably unfamiliar to many of you. And well 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 1 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. Weve already seen the notation for this: x A means that x is a member of the set A . Theres three basic ways to define a set: describe its contents in mathematical English, e.g. the integers be- tween 3 and 7, inclusive. list all its members, e.g. { 3 , 4 , 5 , 6 , 7 } use so-called 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. Heres 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...
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lect_11 - Sets I Margaret M. Fleck 12 February 2010 This...

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