If \(A=\{1,2\}\) and \(B=\{1,3\}\), Find \(A\times B\) and \(B\times A\)
Question
If \[ A=\{1,2\} \] and \[ B=\{1,3\}, \] find \[ A\times B \] and \[ B\times A. \]
Solution
The Cartesian product \(A\times B\) is the set of all ordered pairs \((a,b)\) where \[ a \in A \] and \[ b \in B. \]
Given:
\[ A=\{1,2\} \]
\[ B=\{1,3\} \]
Now form all ordered pairs for \(A\times B\):
\[ (1,1),\ (1,3),\ (2,1),\ (2,3) \]
Therefore,
\[ \boxed{A\times B=\{(1,1),\ (1,3),\ (2,1),\ (2,3)\}} \]
Now form all ordered pairs for \(B\times A\):
\[ (1,1),\ (1,2),\ (3,1),\ (3,2) \]
Therefore,
\[ \boxed{B\times A=\{(1,1),\ (1,2),\ (3,1),\ (3,2)\}} \]