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Unformatted text preview: Applications of Equivalence Relations Margaret M. Fleck 26 April 2010 This lecture finishes the topic of equivalence relations (section 8.5 of Rosen). We show how to show that an operation on equivalence classes is well-defined. And we see more applications of equivalence relations and equivalence classes. 1 Announcements Quiz on Wednesday! If you missed a quiz (and it wasn’t excused) or have a truly horrible quiz grade, there will be makeup quiz (for 80% of its face value) on the last day of classes. Remember to check your final exam schedule for conflicts. Conflicts must be reported to the instructors no later than the last day of classes. Notice that issues near the end of term are regulated by a fairly strict set of rules, which instructors don’t have much ability to modify. In particular, if you get sick towards the end of the term, it’s vital to involve the emergency dean because only they can authorize extensions past the end of the term. 2 Proving that an operation is well-defined [In lecture, I actually did multiplication of fractions rather than addition. All the ideas are the same but a few of the equations differ.] 1 We’ve used equivalence classes to define new types of objects, such as rational numbers. To do anything with these new data types, we need to define operations on them, such as addition and multiplication. Given that we know how to do arithmetic on integers, we can define addition of fractions by x y + p q = xq + py yq We’d like to apply this definition to our new rational numbers, by simply throwing equivalence class brackets around the inputs and outputs: [ x y ] + [ p q ] = [ xq + py yq ] To work right, the output of this definition should not depend on which representative we pick for the input equivalence classes. E.g. [ 2 3 ]+[ 10 2 ] should be equal to (say) [- 4- 6 ] + [ 5 1 ]. In math jargon, we say that we want to make sure the operation is well-defined on the equivalence classes....
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- Spring '10
- Equivalence relation, Rational number, equivalence classes