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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?

  1. $\mathcal{F} \sqcup \mathcal{G}=\mathcal{F} \cap \mathcal{G}$
  2. $\mathcal{F} \sqcup \mathcal{G}=\mathcal{F} \cup \mathcal{G}$
  3. $\mathcal{F} \sqcup \mathcal{G}=\mathcal{F} \backslash \mathcal{G}$
  4. $\mathcal{F} \sqcup \mathcal{G}=\mathcal{G} \backslash \mathcal{F}$
  5. None of the above
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