Question

Let $A, B$ and $C$ be finite sets such that $A \cap B \cap C=\phi$ and each one of the sets $A \Delta B, B \Delta C$ and $C \Delta A$ has $100$ elements. The number of elements in $A \cup B \cup C$ is

• A. 250
• B. 200
• C. 150
• D. 300

Answer 1

Answer: (C)

Let $n(X)$ denote the number of elements in $X$.
Then,
$n ( A \cup B \cup C )= n ( A )+ n ( B )+ n ( C )- n ( A \cap B )$
$- n ( B \cap C )- n ( C \cap A )+ n ( A \cap B \cap C )$
$=\sum n ( A )-\sum n ( A \cap B ) \quad(\because A \cap B \cap C =\phi)$
Now,
$A \Delta B=(A-B) \cup(B-A)=(A \cup B)-(A \cap B)$
Therefore
$n(A \Delta B)=n(A \cup B)-n(A \cap B)=n(A)+n(B)-2 n(A \cap B)$
and
$300=\sum n ( A \Delta B )=\sum[ n ( A )+ n ( B )-2 n ( A \cap B )]$
$=2\left[\sum n ( A )-\sum n ( A \cap B )\right]$
Therefore
$n ( A \cup B \cup C )=\sum n ( A )-\sum n ( A \cap B )=300 / 2=150$