Question

If $n\left( A \right)$ denotes the number of elements in set A and if $n\left( A \right) = 4$,$n\left( B \right) = 5$ and $n\left( {A \cap B} \right) = 3$, then $n\left( {A \times B} \right) \cap n\left( {B \times A} \right) = $
A) $8$
B) $9$
C) $10$
D) $11$

Answer 1

Hint: In order to find the value of $n\left( {A \times B} \right) \cap n\left( {B \times A} \right)$ expand the equation using the distributive property, then using the commutative law, solve and substitute the values needed and get the results. There is no need to use $n\left( A \right) = 4$,$n\left( B \right) = 5$ in solving the question.

Formula used:
Distributive Property: $A\left( {B + C} \right) = AB + AC$.
Commutative Law: \[a + b = b + a\]

Complete step by step answer:
We are given the values $n\left( A \right) = 4$,$n\left( B \right) = 5$ and $n\left( {A \cap B} \right) = 3$.
We need to find the value of $n\left( {A \times B} \right) \cap n\left( {B \times A} \right)$.
From Distributive property, we can expand the equation as:
\[ \Rightarrow n\left( {A \times B} \right) \cap n\left( {B \times A} \right) = n\left( {A \cap B} \right) \times n\left( {A \cap A} \right) \times n\left( {B \cap A} \right) \times n\left( {B \cap B} \right)\] …..(1)
Since, there are two values such as \[n\left( {A \cap A} \right)\] and \[n\left( {B \cap B} \right)\] which means A is intersected to A and will give the result as 1, because there will be all same elements.
Therefore, \[n\left( {A \cap A} \right) = 1\] and \[n\left( {B \cap B} \right) = 1\]
Substituting the values \[n\left( {A \cap A} \right) = 1\] and \[n\left( {B \cap B} \right) = 1\] in the equation 1, we get:
\[ \Rightarrow n\left( {A \times B} \right) \cap n\left( {B \times A} \right) = n\left( {A \cap B} \right) \times 1 \times n\left( {B \cap A} \right) \times 1\]
\[ \Rightarrow n\left( {A \times B} \right) \cap n\left( {B \times A} \right) = n\left( {A \cap B} \right) \times n\left( {B \cap A} \right)\] ………..(2)
From the Commutative property, we know that \[a + b = b + a\], using this property, we can write as:
\[n\left( {A \cap B} \right) = n\left( {B \cap A} \right)\]
Since, we were given $n\left( {A \cap B} \right) = 3$, that implies:
\[n\left( {A \cap B} \right) = n\left( {B \cap A} \right) = 3\]
Substituting these values in the equation 2, we get:
\[ \Rightarrow n\left( {A \times B} \right) \cap n\left( {B \times A} \right) = 3 \times 3\]
\[ \Rightarrow n\left( {A \times B} \right) \cap n\left( {B \times A} \right) = 9\]
Therefore, the value of \[n\left( {A \times B} \right) \cap n\left( {B \times A} \right)\] is equal to \[9\].

Hence, Option (B) is correct.

Note:
Since, there is no use of $n\left( A \right) = 4$ and $n\left( B \right) = 5$ while solving the above equation, so do not get confused, and do not substitute their values in the middle if not needed.
The letter n outside the brackets of the sets like n(A) and n(B) represents the number of elements in set A or number of elements in set B.