Equivalence Relations Reflexivity

Question

Consider the relation on $\bf{R}$ defined by $n \simeq m$ if $(n-m)\in \bf{R}$

To say this is reflexive, I can say: Let $n\in \bf{R}$ and since $n-n = 0$ and $0 \in \bf{R}$ Then $n \simeq n$.

Answer 1

Yes indeed. If zero is an element of $\bf R$, then $\forall n\in\mathbf R: (n-n)\in\mathbf R$, which means that the relation $\simeq$ is reflexive.

( This is obviously true if by $\bf R$ you mean $\Bbb R$, the set of real numbers.   It might not hold for some other set named $\bf R$. )

Now, is $\simeq$ symmetric and transitive?