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lect_12 - Sets II Margaret M Fleck 15 February 2010 This...

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Sets II Margaret M. Fleck 15 February 2010 This lecture shows how to prove facts about set equality and set inclusion. (Rosen section 2.2). 1 Announcements First midterm is a week from Wednesday (i.e. February 24th) 7-9pm in 141 Wohlers. Please tell me ASAP about any conflicts or special needs. For conflicts, please include a copy of your class schedule. Quizzes will be handed out in sections today and tomorrow. Grades should be on compass this afternoon. Check out the Exams web page for information on how to interpret your score. 2 Recap Last class, we saw a bunch of set theory notation and operations on sets. Much of this was (more or less) familiar. The least familiar operations were Powerset: P ( A ) = { all subsets of A } Cartesian product: A × B = { ( x,y ) | x A and y B } 1
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Remember that ( x,y ) is an ordered pair containing two objects, which is very different from the two-object set { x,y } . In ( x,y ), the order of the two elements matters and you can have duplicates e.g. (2 , 2). If you add the same thing to a set twice, you only get one copy in the set, e.g. { 2 , 2 } = { 2 } . 3 Set identities Rosen lists a large number of identities showing when two sequences of set operations yield the same output sets. For example: DeMorgan’s Law: A B = A B I won’t go through these in detail because they are largely identical to the identities you saw for logical operations, if you make the following corre- spondences: • ∪ is like • ∩ is like A is like ¬ P • ∅ (the empty set) is like F U (the universal set) is like T The two systems aren’t exactly the same. E.g. set theory doesn’t use a close analog of the operator. But they are very similar. You can use these set theory identities to prove new identites via a chain of equalities. This is exactly like what you did with logical identities and there’s no point in walking you through the same exercise twice. I’m also going to skip showing you set membership tables, which are in Rosen, because those are just like truth tables and I’m confident you’re already on top of that idea.
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