If A and B Have 3 Common Elements, n(A)=5 and n(B)=4, Find \(n(A\times B)\) and \(n\{(A\times B)\cap(B\times A)\}\)
Question
If \(A\) and \(B\) are two sets having 3 elements in common,
\[ n(A)=5 \]
and
\[ n(B)=4, \]
find:
\[ n(A\times B) \]
and
\[ n\{(A\times B)\cap(B\times A)\}. \]
Solution
1. Find \(n(A\times B)\)
We know that:
\[ n(A\times B)=n(A)\times n(B) \]
Substituting the given values:
\[ n(A\times B)=5\times4 \]
\[ \boxed{n(A\times B)=20} \]
2. Find \(n\{(A\times B)\cap(B\times A)\}\)
The ordered pairs common to both \(A\times B\) and \(B\times A\) are formed using elements common to both sets.
Since \(A\) and \(B\) have 3 common elements,
\[ n(A\cap B)=3 \]
Therefore,
\[ (A\times B)\cap(B\times A)=(A\cap B)\times(A\cap B) \]
Hence,
\[ n\{(A\times B)\cap(B\times A)\} =3\times3 \]
\[ \boxed{n\{(A\times B)\cap(B\times A)\}=9} \]