Question

Let $A,B$ and $C$ be finite sets such that $A\cap B\cap C=\varphi$ 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,
$n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)$
$-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C)$
$=\sum n(A)-\sum n(A\cap B)(\because A\cap B\cap C=\varphi )$
Now,
$A\Delta B=(A-B)\cup (B-A)=(A\cup B)-(A\cap B)$
Therefore
$n(A\Delta B)=n(A\cup B)-n(A\cap B)=n(A)+n(B)-2n(A\cap B)$
and
$300=\sum n(A\Delta B)=\sum [n(A)+n(B)-2n(A\cap B)]$
$=2\left[\sum n(A)-\sum n(A\cap B)\right]$
Therefore
$n(A\cup B\cup C)=\sum n(A)-\sum n(A\cap B)=300/2=150$