If \(A=\{1,2,3\}\) and \(B=\{2,4\}\), Find \(A\times B\), \(B\times A\), \(A\times A\times A\), \(B\times B\) and \((A\times B)\cap(B\times A)\)
Question
If \[ A=\{1,2,3\} \] and \[ B=\{2,4\}, \] find:
\[ A\times B,\quad B\times A,\quad A\times A\times A,\quad B\times B \]
and
\[ (A\times B)\cap(B\times A) \]
Solution
Given:
\[ A=\{1,2,3\} \]
\[ B=\{2,4\} \]
1. Find \(A\times B\)
The Cartesian product \(A\times B\) is:
\[ A\times B= \{ (1,2),(1,4), (2,2),(2,4), (3,2),(3,4) \} \]
2. Find \(B\times A\)
\[ B\times A= \{ (2,1),(2,2),(2,3), (4,1),(4,2),(4,3) \} \]
3. Find \(A\times A\times A\)
\[ A\times A\times A= \{ \]
\[ (1,1,1),(1,1,2),(1,1,3), \]
\[ (1,2,1),(1,2,2),(1,2,3), \]
\[ (1,3,1),(1,3,2),(1,3,3), \]
\[ (2,1,1),(2,1,2),(2,1,3), \]
\[ (2,2,1),(2,2,2),(2,2,3), \]
\[ (2,3,1),(2,3,2),(2,3,3), \]
\[ (3,1,1),(3,1,2),(3,1,3), \]
\[ (3,2,1),(3,2,2),(3,2,3), \]
\[ (3,3,1),(3,3,2),(3,3,3) \} \]
4. Find \(B\times B\)
\[ B\times B= \{ (2,2),(2,4), (4,2),(4,4) \} \]
5. Find \((A\times B)\cap(B\times A)\)
Common ordered pairs in both sets are:
\[ (2,2) \]
Therefore,
\[ \boxed{(A\times B)\cap(B\times A)=\{(2,2)\}} \]