How to show the equivalence between two definitions of completion of sigma algebra

Question

Given a Probability space $(\Omega,\mathcal{A},P)$ , I knew the definiton of the completion of a sub-sigma algebra $\mathcal{F}$ to be $\overline{\mathcal{F}}=\sigma(\mathcal{F}\cup \mathcal{N})$ where $\mathcal{N}=\{A\subset X : \exists B\in\mathcal{A} , P(B)=0\}$ .

Now I read in this that $\overline{\mathcal{F}}=\{A\subset\Omega : \exists B\in\mathcal{F},N\subset\Omega \,\text{such that}\, A\Delta B\subset N , P(N)=0\}$ .

For one side, I have done the following :-

Let $B\cup N\in \mathcal{F}\cup{\mathcal{N}}$ . Then $P((B\cup N)\Delta B)=0$ and hence $\mathcal{F}\cup\mathcal{N}\subset \overline{\mathcal{F}}$ which gives that $\sigma(\mathcal{F}\cup\mathcal{N})\subset \overline{\mathcal{F}}$.

I am really having trouble in proceeding with the proof of the other direction . Can anyone help me with this question?

Answer 1

$$A = (B \cup (A \setminus B) ) \setminus (B \setminus A)$$ and $A \setminus B \subset A \Delta B \in {\cal N}$, $B \setminus A \subset A \Delta B \in {\cal N}$.