Total number of equivalence relations defined in the set S={a,b,c}  is:

The smallest equivalence relation is the identify relation R1={(a,a),(b,b)(c,c)} Then two ordered pairs of two distinct elements can be added to give three more equivalence relations R2={(a,a),(b,b)(c,c)(a,b),(b,a)} Similarly R3  and R4 .finally the largest equivalence relation  i.e., the universal relationR5={(a,a),(b,b)(c,c)(a,b),(b,a),(a,c),(c,a)(b,c)(c,b)}