Question
Consider the following statements.
I. If $${A_n}$$ is the set of first $$n$$ prime numbers then $$\mathop U\limits_{n = 2}^{10} \,{A_n}$$ is equal to $$\left\{ {2,\,3,\,5,\,7,\,11,\,13,\,17,\,19,\,23,\,29} \right\}$$
II. if $$A$$ and $$B$$ are two sets such that $$n\left( {A \cup B} \right) = 50,\,\,n\left( A \right) = 28,\,\,n\left( B \right) = 32,$$ then $$n\left( {A \cap B} \right) = 10$$
Which of these is correct ?
A.
Only I is true
B.
Only II is true
C.
Both are true
D.
Both are false
Answer :
Both are true
Solution :
$$\eqalign{
& {\text{I}}{\text{.}}\,\,\,\mathop U\limits_{n = 2}^{10} \,{A_n}{\text{ is the set of first 10 prime numbers}} \cr
& = \left\{ {2,\,3,\,5,\,7,\,11,\,13,\,17,\,19,\,23,\,29} \right\} \cr
& {\text{II}}{\text{.}}\,\,n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) \cr
& \Rightarrow 50 = 28 + 32 - n\left( {A \cap B} \right) \cr
& \Rightarrow n\left( {A \cap B} \right) = 60 - 50 = 10 \cr} $$