Let $R$ be the real line. Consider the following subsets of the plane $R imes R$ : $S=\{(x, y): y=

Question

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

Professor PC Thomas Classes · Answer

(a) : To be an equivalence relation the relation must be all reflexive, symmetric and transitive.\ $T=\{(x, y): x-y \in Z\}$ is reflexive $-\operatorname{tor}(x, x)$ $\in Z$ i.e., $x-x=0 \in Z$\ symmetric - for $(x, y) \in Z \Rightarrow x-y \in Z$\ $\Rightarrow \quad y-x \in Z$ i.e. $(y, x) \in Z$\ transitive - for $(x, y) \in Z$ and $(y, w) \in Z$\ $\Rightarrow \quad x-y \in Z$ and $y-w \in Z$,\ giving $x-w \in Z$ i.e. $(x, w) \in Z$.\ $	herefore \quad T$ is an equivalence relation on $R$.\ $S=\{(x, y): y=x+1,0<x<2\}$ is not reflexive for $(x, x) \in S$ would imply\ $x=x+1 \Rightarrow 0=1$ (impossible).\ Thus $S$ is not an equivalence relation.\