Say set is A= {1,2,3}=n
Subset of set A ={phi, {1} , {2} , {3} , {1,2} , {2,3}, {1,3} , {1,2,3}}=2^{n}
Now, $A\times A=\left \{ 1,2,3 \right \}\times \left \{ 1,2,3 \right \}$
$=\left \{ \left \{ 1,1 \right \},\left \{ 1,2 \right \} \left \{ 1,3 \right \}\left \{ 2,1 \right \}\left \{ 2,2 \right \}\left \{ 2,3 \right \}\left \{ 3,1 \right \}\left \{ 3,2 \right \}\left \{ 3,3 \right \}\right \}$
So, number of subsets will be$2^{n^{2}}$