Question:
Let $A=\{x: x=6 n \in N$ ) and $B=\{x: x=9 n, n \in N\}$, find $A \cap B$.
Solution:
$A=\{x: x=6 n \forall n \in N)$
As $x=6 n$ hence for $n=1,2,3,4,5,6 \ldots x=6,12,18,24,30,36 \ldots$
Hence $A=\{6,12,18,24,30,36 \ldots\}$
$B=\{x: x=9 n \forall n \in N)$
As $x=9 n$ hence for $n=1,2,3,4 \ldots x=9,18,27,36 \ldots$
Hence $B=\{9,18,27,36 \ldots\}$
$A \cap B$ means common elements to both sets
The common elements are $18,36,54, \ldots$
Hence $A \cap B=\{18,36,54, \ldots\}$
All the elements are multiple of 18
Hence $A \cap B=\{x: x=18 n \forall n \in N\}$