Equivalence RelationsDate: 10/02/98 at 00:24:35 From: shawn Subject: Equivalence relations Let X = {people in the world} and let R be a relation on the set X such that (x,y) is in R if either x and y live in a state that starts with the same letter or x and y live outside of the United States. For example, two people would be related if one lived in Michigan and the other lived in Massachusetts and two other people would be related if one lived in England and the other lived in Paris. (a) Show that this relation is an equivalence relation (b) List and describe the equivalence classes that would describe this relation. The problem is that I can not figure out the set or how this would be written in symbols. Any help would be appreciated. Thank you. Date: 10/02/98 at 17:36:55 From: Doctor Anthony Subject: Re: Equivalence relations There are three criteria for an equivalence relation: (1) Reflexive a R a (2) Symmetric a R b -> b R a (3) Transitive a R b and b R c -> a R c Clearly if you take the relation as defined above, it satisfies these three criteria. I'll do the first one as an example. To show that a R a, you need to show that (a,a) is in R. Can you say that one person lives in a state that starts with the same letter as the state that the person lives in or lives outside of the United States? Yes. Now you need to show that if (a,b) is in R, then (b,a) is in R, and if (a,b),(b,c) are in R, (a,c) is in R. An equivalence class of the element a is the set of all elements equivalent to a under the relation R. One equivalence class is everyone not living in the United States. The other equivalence classes are: Individuals living in states with a name starting with the letter A, the letter B, the letter C, and so on. Note that equivalence classes will partition the set X. No individual can be in two equivalence classes. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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