Question

If two sets A and B are having 43 elements in common, then find the number of elements common to each of the sets \[A \times B\] and \[B \times A\].
A. \[{43^2}\]
B. \[{2^{43}}\]
C. \[{43^{43}}\]
D. \[{2^{86}}\]

Answer 1

Hint: We have to calculate the number of elements \[n\left( {A \times B} \right) \cap n\left( {B \times A} \right)\]. To calculate the \[n\left( {A \times B} \right) \cap n\left( {B \times A} \right)\], we will apply the formula \[n\left( {A \times B} \right) \cap n\left( {B \times A} \right) = n\left( {A \cap B} \right) \times n\left( {B \cap A} \right)\].

Formula Used:
\[n\left( {A \times B} \right) \cap n\left( {B \times A} \right) = n\left( {A \cap B} \right) \times n\left( {B \cap A} \right)\]

Complete step by step solution:
Given that two sets A and B are having 43 elements in common.
Thus
\[n\left( {A \times B} \right) = 43\] and \[n\left( {B \times A} \right) = 43\].
We have to calculate \[n\left( {A \times B} \right) \cap n\left( {B \times A} \right)\]
Now applying the formula \[n\left( {A \times B} \right) \cap n\left( {B \times A} \right) = n\left( {A \cap B} \right) \times n\left( {B \cap A} \right)\]
 \[n\left( {A \times B} \right) \cap n\left( {B \times A} \right) = n\left( {A \cap B} \right) \times n\left( {B \cap A} \right)\]
Now putting \[n\left( {A \times B} \right) = 43\] and \[n\left( {B \times A} \right) = 43\]
\[n\left( {A \times B} \right) \cap n\left( {B \times A} \right) = 43 \times 43\]
\[ \Rightarrow n\left( {A \times B} \right) \cap n\left( {B \times A} \right) = {43^2}\]

Hence option A is the correct option.

Note: Sometimes students often do a mistake to multiply \[43 \times 43\]. They do \[43 \times 43 = {43^{43}}\] which incorrect. If we multiply \[43 \times 43 \times \cdots 43\,{\rm{times}}\], then the result will be \[{43^{43}}\]. The correct answer is \[43 \times 43 = {43^2}\].