Let  $R_1 and R_2$  be two relations defined as follows:
 $R_1=\{(a,b)\in R^2:a^2+b^2\in Q\}and$
 $R_2=\{(a,b)\in R^2:a^2+b^2\notin Q\}$
Where Q is the set of all rational numbers. Then 

Neither  $R_1 nor{R}_2$ is equivalence relation.

$R_1$  is  equivalence relation but  $R_2$ is not equivalence relation.

$R_2$  is  equivalence relation but $R_1$  is not equivalence relation.

Both  $R_1,R_2$ are equivalence relations.