Let R be the real line. Consider the following subsets of the plane R×R $S=\{(x, y): y=x+1 \text { and } 0<x<2\} \\ T=\{(x, y): x-y \text { is an integer }\}$

Question

Let $R$ be the real line. Consider the following subsets of the plane $R \times R$

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

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

Answer 1

SOLUTION : $S$ is not reflexive so not equivalence as $x \neq x+1$

 $(x, y) \in T \Rightarrow x-y$ is an integer

(i) $x-x$ is an integer $\Rightarrow$ reflexive

(ii) $x-y=$ integer $\Rightarrow y-x=$ integer $\therefore T$ is symmetric

(iii) $x-y=m, y-z=n$

$\Rightarrow x-y+y-z=m+n$

$x-z=m+n \quad \Rightarrow$ Transitive

so $T$ is equivalence relation