13. Five mark exercise problems I

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 $A = \{a, c, e, f, h\}$, $B = \{c, d, e, f\}$ and $C = \{a, b, c, f\}$.

 

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.