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

$T=\{(x,y):x-y∈Z$

As $0 ∈ zT$ is a reflexive relation

If $x - y ∈ z ⇒ y - x ∈ z$

∴ T is symmetrical also

If $x - y = z_1$ and $y-x=z_2$

Then $x-z=(x-y)+(y-z)=z_1+z_2∈ z$

∴ T is also transitive.

Hence T is an equivalence relation

Clearly $x≠x+1⇒(x,x)∈S$

∴ S is not reflexive