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Properties of Set Operation

A and B are...

Question

$A$ and $B$ are the two subsets of Universal Set $U$ such that $n (U) = 700, n (A) = 200, n (B) = 300$ and $n (A \cap B) = 100$ . Find $n (A^{'} \cap B^{'})$

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Solution

Given that:
$n (U) = 700, n (A) = 200, n (B) = 300$ and $n (A \cap B) = 100$
To find:
$n (A^{'} \cap B^{'}) = ?$
Solution:
We know that,
$n (A^{'} \cap B^{'}) = n (A \cup B)^{'} = n (U) - n (A \cup B)$ $. . . . . . . . . . . . (i)$
$n (A \cup B) = n (A) + n (B) - n (A \cap B)$
$= 200 + 300 - 100$
$= 400$
Putting value of $n (A \cup B)$ in eqn. in (i) we get,
$n (A^{'} \cap B^{'}) = n (A \cup B)^{'} = n (U) - n (A \cup B)$
$= 700 - 400$
$= 300$
$∴ n (A^{'} \cap B^{'}) = 300$

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