Providing counterexamples for a false statement Let $A, B, C$ be three sets so that the family $\{A, B, C\}$ is disjoint. Then at least one of $A \cap B, B \cap C, A \cap C$ has no element.
Since they are disjoint sets, shouldnt the sets have no common intersection, which would make this sentence true?

If a family of sets, here $\{A, B, C\}, $ are disjoint, we know only that $A\cap B \cap C = \varnothing$
This is different than saying that a family of sets is pairwise disjoint, in which $A\cap B =\varnothing, B\cap C = \varnothing, A\cap C = \varnothing.$
Claim: Let $A, B, C$ be three sets so that the family $\{A, B, C\}$ is disjoint, i.e. $A\cap B\cap C = \varnothing)$.
 Then at least one of $A \cap B, B \cap C, A \cap C$ has no element.
In this case, a counterexample to the claim you post is $$A = \{1, 2\}, B = \{2, 3\}, C = \{1, 3\}$$
