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?

It's always good to be careful with definitions.
A standard definition for "disjoint family" is:

A family of sets $\{A_i\}_{i\in I}$ is disjoint if $$\bigcap_{i\in I} A_i=\emptyset$$


So, following the standard definition, your statement says:

If $A\cap B\cap C=\emptyset$, then at least one of $A\cap B, B\cap C$ or $A\cap C$ has no element.

This statement is not true, and a simple counterexample can show it is not.
