(a) : "lo be an equivalence relation the relation must be all reflexive, symmetric and transitive.\\ $T = \{(x,y):x - y \in Z\}$ is\\ reflexive - for $(x,x) \in Z$ i.e., $x - x = 0 \in Z$\\ symmetric - for $(x,y) \in Z$\\ $\Rightarrow x - y \in Z$ $\Rightarrow \ y - x \in Z$ i.e. $(y,x) \in Z$\\ transitive - for $(x,y) \in Z$ and $(y,w) \in Z$\\ $\Rightarrow \ x - y \in Z$ and $y - w \in Z$, giving $x - w \in Z$ i.e. $(x,w) \in Z$.\\ $\therefore\ T$ 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.