Let $X$ be a finite set. A family $\mathcal{F}$ of subsets of $X$ is said to be upward closed if the following holds for all sets $A, B \subseteq X$ :
$$ A \in \mathcal{F} \text { and } A \subseteq B \Rightarrow B \in \mathcal{F}.$$
For families $\mathcal{F}$ and $\mathcal{G}$ of subsets of $X$, let
$$ \mathcal{F} \sqcup \mathcal{G}=\{A \cup B: A \in \mathcal{F} \text { and } B \in \mathcal{G}\}.$$
Suppose $\mathcal{F}$ and $\mathcal{G}$ are upward closed families. Then which of the following is true?
- $\mathcal{F} \sqcup \mathcal{G}=\mathcal{F} \cap \mathcal{G}$
- $\mathcal{F} \sqcup \mathcal{G}=\mathcal{F} \cup \mathcal{G}$
- $\mathcal{F} \sqcup \mathcal{G}=\mathcal{F} \backslash \mathcal{G}$
- $\mathcal{F} \sqcup \mathcal{G}=\mathcal{G} \backslash \mathcal{F}$
- None of the above
