Consider a family $\mathcal{F}$ of subsets of $\{1, 2, \dots , n\}$ such that for any two distinct sets $A$ and $B$ in $\mathcal{F}$ we have: $A \subset B$ or $ B \subset A$ or $A \cap B = \emptyset$. Which of the following statements is TRUE? (Hint: what does the Venn diagram of this family look like?)

$\mid \mathcal{F} \mid \leq 2n$ and there exists a family $\mathcal{F}$ such that $\mid \mathcal{F} \mid =2n$

$\mid \mathcal{F} \mid \leq n^2$ and there exists a family $\mathcal{F}$ such that $\mid \mathcal{F} \mid =n^2$

$\mid \mathcal{F} \mid \leq 2n^2$ and there exists a family $\mathcal{F}$ such that $\mid \mathcal{F} \mid =2n^2$

$\mid \mathcal{F} \mid \leq 2^{n-1}$ and there exists a family $\mathcal{F}$ such that $\mid \mathcal{F} \mid =2^{n-1}$