Let $\Omega = \{1,2,3,4\}$ and consider $\mathcal{F} = \sigma(\{1\}, \{3\}).$ We define the functions $X, Y$ and $Z$ on $\Omega$, given by
$$X(s) = \begin{cases}
1 & \text{if } s \text{ is even}, \\
0 & \text{otherwise},
\end{cases}$$
$$Y(s) = s^2 - 6s + 8,$$ and $$Z(s) = s^2 - 4s + 3.$$
(a) Find the atoms of $\mathcal{F}$.
(b) Determine which of $X, Y$ and $Z$ are $\mathcal{F}$-measurable or not. (Justify your answers.)
(c) Reminder: For a function $X$, we denote $\sigma(X)$ the smallest $\sigma$-algebra which makes $X$ measurable.
Determine $\sigma(X), \sigma(Y), \sigma(Z)$.