Equivalence Relations on SetsDate: 2/3/96 at 18:30:14 From: Anonymous Subject: Equivalence Relations on sets Please tell me how many equivalence relations can be defined on the set S={a,b,c}. I came up with 13. They are listed below. Your input and/or verification would be appreciated. Relation, R = aa;bb;cc;ab;ac;ba;bc;ca;cb (I know this is not proper notation, but to save time I'll list them this way) 1. {a,a} 2. {a,a};{b,b} 3. {a,a};{b,b};{c,c} 4. aa;cc 5. bb 6. cc 7. aa,bb,ab,ba 8. aa,cc,ac,ca 9. bb,cc,bc,cb 10. aa,bb,cc,ab,ba 11. aa,bb,cc,ac,ca 12. aa,bb,cc,bc,cb 13. aa,bb,cc,ab,ba,ac,ca,bc,cb This is what I have. Thanks for your help. Date: 6/27/96 at 12:17:39 From: Doctor Ceeks Subject: Re: Equivalence Relations on sets The number of distinct equivalence relations on S is the same as the number of partitions of S via the notion of "equivalence class". For the three element set, there are 5 different equivalence relations: 1. x is equivalent to y if and only if x = y (for x,y in S). 2. a is equivalent to b, but not c. 3,4. cyclic permutations of #2. 5. a, b, and c are all equivalent to each other. -Doctor Ceeks, The Math Forum |
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