14. Five mark exercise problems II

Verify \(n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)\) for the sets \(A = \{1, 3, 5\}\), \(B = \{2, 3, 5, 6\}\) and \(C = \{1, 5, 6, 7\}\).

 

Proof:

 

\(n(A \cup B \cup C) =\)  ---- (\(1\))

 

\(n(A) =\)

 

\(n(B) =\)

 

\(n(C) =\)

 

\(n(A \cap B) =\)

 

\(n(B \cap C) =\)

 

\(n(A \cap C) =\)

 

\(n(A \cap B \cap C) =\)

 

\(n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C) =\)  ---- (\(2\))

 

From equations (\(1\)) and (\(2\)), we have: \(n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)\).

 

Hence, we proved.