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March 7th, 2013, 10:53 PM  #1 
Member Joined: Aug 2012 Posts: 30 Thanks: 0  equivalence relations
Suppose R is an equivalence relation on a set A, with four equivalence classes. How many different equivalence relations S on A are there for which R?S? Thanking you in anticipation 
March 9th, 2013, 06:13 PM  #2 
Senior Member Joined: Aug 2012 Posts: 2,333 Thanks: 724  Re: equivalence relations
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March 11th, 2013, 08:32 PM  #3 
Member Joined: Aug 2012 Posts: 30 Thanks: 0  Re: equivalence relations
The answer that appears in my book is 15. How is that possible? 
March 13th, 2013, 03:29 PM  #4  
Senior Member Joined: Aug 2012 Posts: 2,333 Thanks: 724  Re: equivalence relations Quote:
See http://en.wikipedia.org/wiki/Partition_ ... partitions I counted by hand the number of equivalence relations on a 4element set by enumerating all the possible partitions. * 1element partitions: {1,2,3,4} One of those. * 2element partitions:  {{1,2}, {3,4}}  {{1,3}, {2,4}}  {{1,4}, {2,3}}  {1}, {2,3,4}  {2}, {1,3,4}  {3}, {1,2,4}  {4}, {1,2,3} Seven of those. * 3element partitions:  {{1,2}, {3}, {4}}  {{1,3}, {2}, {4}}  {{1,4}, {2}, {3}}  {{2,3}, {1}, {4}}  {{2,4}, {1}, {1}}  {{3,4}, {1}, {2}} Six of those. * 4element partitions:  {{1}, {2}, {3}, {4}} One of those. Total number of partitions, hence total number of equivalence relations: 1 + 7 + 6 + 1 = 15. The only 4equivclass relation is {{1}, {2}, {3}, {4}}. Which ones is it a subset of? Well, since an equivalence relation must be reflexive, the ordered pairs (1,1), (2,2), (3,3), and (4,4) must be in every possible equivalence relation. So the 4equivclass relation is a subset of every other equivalence relation. So in this case the answer to your original problem is 15. Clearly this number depends on A. I was way too lazy to work this out for any other sets, since the combinatorics rapidly become tedious. Hmmm ... if A has more than 4 elements, perhaps there's some reason that only 15 of the equivalence relations can contain the 4equivclass one. I'll have to think about this some more.  

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