Equivalence Relations on Sets

Date: 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