Show that $A∩B∩C= ∅$ is only true when $A∩B = ∅, A∩C = ∅$ or $B∩C = ∅$ or show a counterexample.
Solution 1:
(Just another counterexample.)
Solution 2:
Hint: consider $A=\{1,2\}$, $B=\{2,3\}$ and $C=\{1,3\}$.
When you have a statement of the form
If $A\cap B\cap C=\emptyset$, then $A\cap B=\emptyset$, $A\cap C=\emptyset$ and $B\cap C=\emptyset$
you can't prove it true by example. You can prove it false by showing a counterexample.
Solution 3:
$$A=\{1,2\}, B=\{2,3,4,5\}, C=\{1,5\}$$ then:
A∩B∩C= ∅
A∩B={2}
A∩C= {1}
B∩C= {5}