27 votes 27 votes Let A be a finite set of size n. The number of elements in the power set of $A\times A$ is: $2^{2^n}$ $2^{n^2}$ $\left(2^n\right)^2$ $\left(2^2\right)^n$ None of the above Set Theory & Algebra gate1993 set-theory&algebra easy set-theory + – Kathleen asked Sep 29, 2014 recategorized Apr 22, 2021 by Lakshman Bhaiya Kathleen 6.8k views answer comment Share Follow See 1 comment See all 1 1 comment reply Deepak Poonia commented Apr 23, 2022 i edited by Deepak Poonia Apr 23, 2022 reply Follow Share Asking for number of elements in $A \times A$ is same as asking for number of relations possible over set $A.$Every relation on set $A$ is subset of $A \times A.$ Every subset of $A \times A$ is a relation over set $A$.Power set of $A \times A$ contains subsets of $A \times A$. So, Every element in power set of $A \times A$ is a relation on set $A.$Hence, the cardinality of $P(A \times A)$ is same as number of relations possible on set $A.$If $|A| = n, $ then $|P(A)| = 2^n, |A \times A| = n^2$, So, $|P(A \times A)| = 2^{n^2} = 2^{(n^2)}$Detailed Video Solution 2 votes 2 votes Please log in or register to add a comment.
Best answer 41 votes 41 votes Cardinality of $A\times A = n^2$ Cardinality of power set of $A\times A = 2^{n^2}$ Correct Answer: $B$ Digvijay Pandey answered Apr 26, 2015 edited Apr 24, 2019 by Naveen Kumar 3 Digvijay Pandey comment Share Follow See all 2 Comments See all 2 2 Comments reply srestha commented Jun 11, 2017 reply Follow Share 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$2^{n^{2}}$ 12 votes 12 votes Ayush Upadhyaya commented Jun 30, 2018 reply Follow Share This is same as the total number of relations on Set A 2 votes 2 votes Please log in or register to add a comment.
14 votes 14 votes $|A| = n$ , $ |p(A)| = 2^n$ $| A×A| =n^2$ $| p(A×A)|$ =$2^{n^{2}}$ Option (B) is the correct answer. Warrior answered Jul 30, 2017 edited Jun 8, 2018 by Milicevic3306 Warrior comment Share Follow See all 0 reply Please log in or register to add a comment.
0 votes 0 votes . akshay_123 answered Sep 2, 2023 akshay_123 comment Share Follow See all 0 reply Please log in or register to add a comment.