Question

If $ n\left( \cup \right) = 60 $ , $ n\left( A \right) = 35 $ , $ n\left( B \right) = 24 $ and $ n\left( {A \cup B} \right)' = 10 $ , then $ n\left( {A \cap B} \right) $ is:
A. $ 9 $
B. $ 8 $
C. $ 6 $
D. $ 7 $

Answer 1

Hint: In order to find $ n\left( {A \cap B} \right) $ , we just need to know some of the basic formulas of sets, like $ n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) $ and $ n\left( {A \cup B} \right) = n\left( \cup \right) - n\left( {A \cup B} \right)' $ etc. We are given some of the values so just substitute them in the above formulas and get the value of $ n\left( {A \cap B} \right) $ .

Complete step by step solution:
The values we are given with are: $ n\left( \cup \right) = 60 $ , $ n\left( A \right) = 35 $ , $ n\left( B \right) = 24 $ and $ n\left( {A \cup B} \right)' = 10 $
From the basic formula of sets we know that:
 $ n\left( {A \cup B} \right) = n\left( \cup \right) - n\left( {A \cup B} \right)' $
So, just put the value of $ n\left( \cup \right) = 60 $ and $ n\left( {A \cup B} \right)' = 10 $ in the above equation and solve for $ n\left( {A \cup B} \right) $ and we get:
\[
  n\left( {A \cup B} \right) = n\left( \cup \right) - n\left( {A \cup B} \right)' \\
  n\left( {A \cup B} \right) = 60 - 10 \\
  n\left( {A \cup B} \right) = 50 \;
 \]
Hence, we got $ n\left( {A \cup B} \right) = 50 $ .
Now, to solve for $ n\left( {A \cap B} \right) $ substitute $ n\left( A \right) = 35 $ , $ n\left( B \right) = 24 $ and $ n\left( {A \cup B} \right) = 50 $ in the formula of set that is $ n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) $ and on solving it, we get:
 $
  n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) \\
  50 = 35 + 24 - n\left( {A \cap B} \right) \;
  $
Add both the sides with $ n\left( {A \cap B} \right) $ and subtract $ 50 $ , and we get:
 $
  50 = 35 + 24 - n\left( {A \cap B} \right) \\
  50 - 50 + n\left( {A \cap B} \right) = 35 + 24 - n\left( {A \cap B} \right) + n\left( {A \cap B} \right) - 50 \\
  n\left( {A \cap B} \right) = 59 - 50 \\
  n\left( {A \cap B} \right) = 9 \;
  $
And, hence we got that $ n\left( {A \cap B} \right) $ is equal to $ 9 $ .
Therefore, the correct option is: Option A, i.e $ 9 $ .
So, the correct answer is “Option A”.

Note: Some of the formulas used in sets are:
 $ n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) $
 $ n\left( {A \cup B} \right) = n\left( \cup \right) - n\left( {A \cup B} \right)' $
 $ n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) $ , if $ A $ and $ B $ are disjoint sets.
 $ n\left( {A - B} \right) = n\left( A \right) - n\left( {A \cap B} \right) $
\[n\left( {A \cup B \cup C} \right) = n\left( A \right) + n\left( B \right) + n\left( C \right) - n\left( {A \cap B} \right) - n\left( {B \cap C} \right) - n\left( {A \cap C} \right) + n\left( {A \cap B \cap C} \right)\]