Let S be the set of real numbers
Statement 1: $A=\left\{(x,y)(S\times S,y-x\mathrm{is}\mathrm{an}\mathrm{integer})\right\}$   is equivalence relation on S
Statement 2: $B=\{(x,y)\in (S\times S,x=\alpha y\mathrm{for}\mathrm{some}\mathrm{rational}\mathrm{numbers}\alpha )\}$  is an equivalence relation on S
 

Put  $y=x,\Rightarrow x-x=0$  is an integer. S is reflexive
Put  $x=2,y=3y-x=3-2=1=\mathrm{int}eger$
 $x-y=2-3=-1isan\mathrm{int}egerSissymmetric$
Put  $x=1,y=2,z=3x-y=-1,y-z=-1,z-x=2$
$x-y$ is integer,y  - z is an integer, $z-x$ is also integer
$\therefore$ S is transitive
Hence S is an equivalence relation 
S1 is true
For S2: put y = x $\Rightarrow x=\alpha x,\alpha =1isnot$ rational number   
$\therefore$ S is not reflexive
$S_2$  is not equivalence.  $S_{2}isfalse$