Number of equivalence relations = Number of possible partitions

Partition into one set : 1 Way

Partition into two sets : (4,1) = 5 Ways ; (3,2) = 5C2 Ways

Partition into three sets : (3,1,1) = 5C3 Ways ; (2,2,1) = 5C1 * (4C2 / 2!) Ways

Partition into four sets : (2,1,1,1) = 5C2 Ways

Partition into five sets : 1 Way

Total = 52

Method 2 :

The number of equivalence relations on a set of 5 elements = bell number

so 52

https://en.wikipedia.org/wiki/Bell_number