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

The smallest equivalence relation is the identify relation

_{$R_1=\{\left(a,a\right),\left(b,b\right)\left(c,c\right)\}$}

Then two ordered pairs of two distinct elements can be added to give three more equivalence relations _{$R_2=\{\left(a,a\right),\left(b,b\right)\left(c,c\right)\left(a,b\right),\left(b,a\right)\}$}

Similarly _{$R_3$}  and _{$R_4$} .finally the largest equivalence relation  i.e., the universal relation

_{$R_5=\{\left(a,a\right),\left(b,b\right)\left(c,c\right)\left(a,b\right),\left(b,a\right),\left(a,c\right),\left(c,a\right)\left(b,c\right)\left(c,b\right)\}$}