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+12 votes
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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
asked in Set Theory & Algebra by Veteran (59.6k points)
edited by | 939 views

2 Answers

+21 votes
Best answer
Cardinality of $A\times A = n^2$
Cardinality of power  set of $A\times A = 2^{n^2}$
answered by Veteran (55.5k points)
selected by
+4

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
+6 votes

$|A| = n$  ,  $ |p(A)| = 2^n$

$| A×A| =n^2$

$| p(A×A)|$ =$2^{n^{2}}$

Option (B) is the correct answer.

answered by Loyal (7k points)
edited by
Answer:

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