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Let A be a finite set of size n. The number of elements in the power set of $A\times A$ is:

1. $2^{2^n}$
2. $2^{n^2}$
3. $(2^n)^2$
4. $(2^2)^n$
5. None of the above
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Cardinality of $A\times A = n^2$
Cardinality of power  set of $A\times A = 2^{n^2}$
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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}}=2n

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$2^{n^{2}}$

0
This is same as the total number of relations on Set A

$|A| = n$  ,  $|p(A)| = 2^n$

$| A×A| =n^2$

$| p(A×A)|$ =$2^{n^{2}}$

edited