Bell formula is something complicated to remember..For smaller number of numbers , we can apply some basic combinatorics and the following fact :
No of equivalence relations = No of unordered partitions of the given set
Hence as the given set contains 3 elements , hence these can be partitioned in 3 possible ways..Lets find no of ways for each sort of unordered partition :
a) 1 partition containing all 3 elements : For this we have no of ways = 3! / 3! = 1 way
b) 2 partitions , one containing 1 and other containing 2 elements : = 3! / (2! * 1!) = 3 ways
c) 3 partitions , each containing 1 element : 3! / ((1!)3 * 3!) = 1 way
Hence total number of ways = 1 + 3 + 1 = 5 ways
Hence no of equivalence relations for given set = 5
EDIT : HOW TO FIND EQUIVALENCE RELATION GIVEN A PARTITION : ------------------------------------------------------------------------------------------------------------
To find the individual equivalence relations, we consider each partition one by one as :
For a given partition of set we consider cross product of each partition with itself in order to get the pairs of equivalence relation.
So , for example :
Consider the partition p = { {4,7} , {1} }
So equivalence relation will be constructed as indiviual cross products of p which is : {4,7} X {4,7} and {1} X {1}
So on doing {4,7} X {4,7} , we get : { (4,4) , (4,7) , (7,4) , (7,7) } and { (1,1) } from second one doing union of which we get the equivalence relation as : { (4,4) , (4,7) , (7,4) , (7,7) , (1,1) }
So equivalence relation corresponding to the partition { {4,7} , {1} } is : { (4,4) , (4,7) , (7,4) , (7,7) , (1,1) } .
Similarly we proceed for other partitions to find their corresponding equivalence relations