If $∩F$ and $∩G$ are disjoint, then for some $A ∈ F$ and $B ∈ G$, $A$ and $B$ are disjoint.

Question

If $∩F$ and $∩G$ are disjoint, then for some $A ∈ F$ and $B ∈ G$, $A$ and $B$ are disjoint. Give a proof or counter-example.

I'm like 99 percent the theorem is true, can't seem to prove it. Have tried a few methods, closest I have got is trying to prove the contra-positive. Any help with this problem would be extremely appreciated, thanks in advance!

Answer 1

Counterexample : $A = \{1,2,3\}$, $C = \{3,4,5\}$, $B = \{3,6,7\}$, $D = \{1,5,6\}$ with $F = \{A,C\}$ and $G = \{B,D\}$.

In particular, $3 \in A \cap B$, $3 \in C \cap B$, $5 \in C \cap D$ , $1 \in A \cap D$. However, $\cap F = \{3\}$ and $\cap G = \{6\}$ are disjoint.

Answer 2

Think small: $\mathcal{F}=\bigl\{\{1,2\}\bigr\}$, $\mathcal{G}=\bigl\{\{1\},\{2\}\bigr\}$.

If you don't like that the sets have empty intersection, use $\mathcal{G}=\bigl\{\{1,3\},\{2,3\}\bigr\}$