Use a Venn diagram or some other method to obtain a formula for
$n ( A \cup B \cup C )$ in terms of $n ( A )$ , $n ( B )$ , $n ( C )$ , $n ( A \cap B )$ , $n ( A \cap C )$ , $n ( B \cap C )$ , and $n ( A \cap B \cap C )$ .

A) $n ( A \cup B \cup C ) = n ( A ) + n ( B ) + n ( C ) - n ( A \cap B ) - n ( A \cap C ) - n ( B \cap C ) + n ( A \cap B \cap C )$

B) $n ( A \cup B \cup C ) = n ( A ) + n ( B ) + n ( C ) - n ( A \cap B ) - n ( A \cap C ) - n ( B \cap C ) - n ( A \cap B \cap C )$

C) $n ( A \cup B \cup C ) = n ( A ) + n ( B ) + n ( C ) - n ( A \cap B \cap C )$

D) $n ( A \cup B \cup C ) = n ( A ) + n ( B ) + n ( C ) + n ( A \cap B ) + n ( A \cap C ) + n ( B \cap C ) - n ( A \cap B \cap C )$

E) $n ( A \cup B \cup C ) = n ( A ) + n ( B ) + n ( C ) - n ( A \cap B ) - n ( A \cap C ) - n ( B \cap C )$

Question

Use a Venn diagram or some other method to obtain a formula for
 $n ( A \cup B \cup C )$ in terms of $n ( A )$ , $n ( B )$ , $n ( C )$ , $n ( A \cap B )$ , $n ( A \cap C )$ , $n ( B \cap C )$ , and $n ( A \cap B \cap C )$ .

A) $n ( A \cup B \cup C ) = n ( A ) + n ( B ) + n ( C ) - n ( A \cap B ) - n ( A \cap C ) - n ( B \cap C ) + n ( A \cap B \cap C )$

B) $n ( A \cup B \cup C ) = n ( A ) + n ( B ) + n ( C ) - n ( A \cap B ) - n ( A \cap C ) - n ( B \cap C ) - n ( A \cap B \cap C )$

C) $n ( A \cup B \cup C ) = n ( A ) + n ( B ) + n ( C ) - n ( A \cap B \cap C )$

D) $n ( A \cup B \cup C ) = n ( A ) + n ( B ) + n ( C ) + n ( A \cap B ) + n ( A \cap C ) + n ( B \cap C ) - n ( A \cap B \cap C )$

E) $n ( A \cup B \cup C ) = n ( A ) + n ( B ) + n ( C ) - n ( A \cap B ) - n ( A \cap C ) - n ( B \cap C )$

Quizplus · Accepted Answer

The formula for $n(A \cup B \cup C)$ is derived using the principle of inclusion-exclusion to avoid overcounting elements that are in more than one set. It adds the sizes of the individual sets, subtracts the sizes of each pairwise intersection to correct for double counting, and then adds back the size of the triple intersection to correct for the elements that were subtracted three times.

Use a Venn Diagram or Some Other Method to Obtain n(A∪B∪C)n ( A \cup B \cup C )n(A∪B∪C)

Use a Venn Diagram or Some Other Method to Obtain $n ( A \cup B \cup C )$