We are required to find sets A, B and C such that \( A \cap B \), \( B \cap C \) and \( A \cap C \) are all non-empty, but \[ A \cap B \cap C = \varnothing. \]
Consider the following sets:
\[ A = \{1, 2\}, \quad B = \{2, 3\}, \quad C = \{1, 3\}. \]
Now, let us find their pairwise intersections:
\[ A \cap B = \{2\} \neq \varnothing \]
\[ B \cap C = \{3\} \neq \varnothing \]
\[ A \cap C = \{1\} \neq \varnothing \]
Now, find the intersection of all three sets:
\[ A \cap B \cap C = \varnothing \]
Hence, the sets \[ A = \{1, 2\}, \quad B = \{2, 3\}, \quad C = \{1, 3\} \] satisfy the given conditions.