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

The smallest equivalence relation is the identity relation

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

Then two ordered pairs of two distinct elements can be added to give three more equivalence relations 

$R_2=\left\{\right(a,a),(b,b),(c,c),(a,b),(b,a\left)\right\}$ Similarly $R_3$and $R_4.$

 Finally the largest equivalence relation  i.e., the universal relation 

${\mathrm{R}}_5=\left\{\right(\mathrm{a},\mathrm{a}),(\mathrm{b},\mathrm{b}),(\mathrm{c},\mathrm{c}),(\mathrm{a},\mathrm{b}),(\mathrm{b},\mathrm{a}),(\mathrm{a},\mathrm{c}),(\mathrm{c},\mathrm{a}),(\mathrm{b},\mathrm{c}),(\mathrm{c},\mathrm{b}\left)\right\}$