Let R be the real line. Consider the following subsets of the plane R × R:

$S=\{(x, y): y=x+1\, and\, 0 < x <2\}$,

$T=\{(x, y): x-y\, is\, an\, integer\}$

Which one of the following is true?

T is an equivalence relation on R but S is not

For any $x ∈ (0, 2)$, we find that $x ≠ x + 1$. 

So, $(x, x) ∉ S$.

∴ S is not a reflexive relation.

Hence, S is not an equivalence relation.

Reflexivity of T: For any $x ∈ R$, we have

$x-x=0$, which is an integer.

$⇒(x, x) ∈T$

∴ T is reflexive on R

Symmetry of T: Let $(x, y) ∈ T$. Then,

$x-y$ is an integer

$⇒y-x$ is an integer

$⇒(y, x) ∈T$

T is symmetric on R.

Transitivity of T: Let $(x, y) ∈ T$ and $(y, z) ∈ T$. Then,

$x-y$ is an integer and $y-z$ is an integer

$⇒x-z$ is an integer

$⇒(x, z) ∈T$

∴ T is transitive on R.

Hence, T is an equivalence relation on R.