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
Answer & explanation
Correct answer: option 1
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.