Properties Of The Mod Relation

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Problem 61: (Properties Of The Mod Relation)

Let $A=\mathbb{Z}$. Let $n\in \mathbb{N}$. Given $a,b\in \mathbb{Z}$, let's say that $a$ and $b$ are related by $\mathrm{R}$ if and only if $n$ divides the difference $a-b$. So using symbols we can write $a\mathrm{R}b$ if and only if $a\cong b \pmod{n}$.

1. Is $\mathrm{R}$ reflexive? Remember to justify your answer, here and below.
2. Is $\mathrm{R}$ symmetric?
3. Is $\mathrm{R}$ transitive?
4. Is $\mathrm{R}$ antisymmetric?

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