Definition. A non empty family $\mathcal{M}\subseteq\mathcal{P}(X)$ is called monotone class on $X$ if for every monotone sequence $\{E_n\}\subseteq \mathcal{M}$ it holds that $\lim E_n \in \mathcal{M}$.
The family $\mathcal{A}$ is an algebra, and $\mathcal{M}_0(\mathcal{A})$ is the monotone class generated by $\mathcal{A}$, $F\in\mathcal{A}$.
We consider the following three families of sets: $$\tilde{\mathcal{A}}=\{E\in\mathcal{M}_0(\mathcal{A})\;|\;\complement E\in\mathcal{M}_0(\mathcal{A})\},$$
$$\hat{\mathcal{A}_F}=\{E\in\mathcal{M}_0(\mathcal{A})\;|\;E\cup F\in\mathcal{M}_0(\mathcal{A})\},$$
$$\hat{\mathcal{A}}=\{F\in\mathcal{M}_0(\mathcal{A})\;|\; E\cup F\in\mathcal{M}_0(\mathcal{A})\quad\forall E\in\mathcal{M}_0(\mathcal{A})\}.$$
Question. I must show that these three families are monotone classes. Could someone provide me a proof of this fact?
Thanks!