Translate the following formulas into the official language of set theory

Question

I would appreciate some help for the following exercise!

Translate the following formulas into the official language of set theory:

1. $b=S(a)$, where $S(a)$ is generally defined as $S(a):=a\cup\{a\}$.
2. $n\in\omega$.
3. $x=\omega$.
4. $y=S(\omega)$.
5. $x=S(S(\omega))$.

My solution:

1. $\forall a \forall b: t\in b \longleftrightarrow((t \in a) \lor (t\in \{a\}))$.
2. $n\in\omega$ is already in set theory language.

In my understanding I have to rephrase the formulas in a way that the only non logical symbol is $\in$. Can someone explain if there is meant something different and maybe give me an example how this works?