the notation a≡b (mod n) to mean that a and b are both represented by
the same element of Zn (under addition (mod n)). Show that a≡b (mod n) if and only if n divide (b-a)
Assume $\displaystyle a \equiv b (mod n)$, then by definition $\displaystyle a = s \cdot n$ and $\displaystyle b = r \cdot n$ for $\displaystyle s,r \in \mathbb{N}$.
Now, you can see that $\displaystyle b-a = rn - sn = (r-s)n = t \cdot n$ for some $\displaystyle t \in \mathbb{N}$, thus $\displaystyle n|(b-a)$.
Now, for the second part: $\displaystyle n|(b-a) \Rightarrow b-a = k \cdot n \Rightarrow b = kn + a \Rightarrow b \equiv a (mod n)$.