Question
If $$A = \left\{ {4n + 2|n{\text{ is a natural number}}} \right\}$$ and $$B = \left\{ {3n|n{\text{ is a natural number}}} \right\},$$ then what is $$\left( {A \cap B} \right)$$ equal to ?
A.
$$\left\{ {12{n^2} + 6n|n{\text{ is a natural number}}} \right\}$$
B.
$$\left\{ {24n - 12|n{\text{ is a natural number}}} \right\}$$
C.
$$\left\{ {60n + 30|n{\text{ is a natural number}}} \right\}$$
D.
$$\left\{ {12n - 6|n{\text{ is a natural number}}} \right\}$$
Answer :
$$\left\{ {12n - 6|n{\text{ is a natural number}}} \right\}$$
Solution :
$$\eqalign{
& {\text{Let }}A = \left\{ {4n + 2:n\, \in N} \right\}\,{\text{and }}B = \left\{ {3n:n\, \in \,N} \right\} \cr
& \Rightarrow \,A = \left\{ {6,\,10,\,14,\,18,\,22,\,26,\,30,\,34,\,38,\,42,.....} \right\} \cr
& {\text{and }}B = \left\{ {3,\,6,\,9,\,12,\,15,\,18,\,21,\,24,\,27,\,30,.....} \right\} \cr
& \therefore \,A \cap B = \left\{ {6,\,18,\,30,\,42,.....} \right\} \cr
& = 6 + 12n - 12 \cr
& = 12n - 6 \cr
& {\text{Hence, }}A \cap B = \left\{ {12n - 6:n{\text{ is a natural number}}} \right\} \cr} $$