Let R be the set of real numbers:

Statement 1 : A = {(x, y) ∈ R × R : y - x is an integer} is an equivalence relation on R.

Statement 2 : B = {(x, y) ∈ R × R : x = αy for some rational number α} is an equivalence relation of R

Statement 1 is true, Statement 2 is false

Since x - x = 0 ∈ Z, (x, x) ∈ A

⇒ A is reflexive

$(x, y) ∈ A ⇒ x - y ∈ z ⇒ y - x ∈ z$

$⇒(y,x)∈ A⇒A$ is symmetric

$(x, y) ∈ A, (y, z) ∈ A ⇒ x - y ∈ z, y - x ∈ z$

$⇒x-z∈ z⇒(x, z)∈ A$

⇒ A is equivalence relation

$(0, 1) ∈ B ∵ 0 = (0)(1), 0 ∈ ..... δ$

But (1, 0) ∉ B ⇒ B is not symmetric

⇒ B is not equivalence.