If \(A=\{1,2\}\), Find \(A\times A\) and Then Form \(A\times A\times A\)
Question
If \[ A=\{1,2\}, \] first find \[ A\times A \] and then form \[ A\times A\times A. \]
Solution
Given:
\[ A=\{1,2\} \]
Step 1 : Find \(A\times A\)
The Cartesian product \(A\times A\) is the set of all ordered pairs \((a,b)\) where
\[ a,b \in A. \]
Therefore,
\[ A\times A= \{ (1,1), (1,2), (2,1), (2,2) \} \]
Step 2 : Find \(A\times A\times A\)
Now combine each ordered pair of \(A\times A\) with elements of \(A\).
Thus,
\[ A\times A\times A= \{ \]
\[ (1,1,1), (1,1,2), \]
\[ (1,2,1), (1,2,2), \]
\[ (2,1,1), (2,1,2), \]
\[ (2,2,1), (2,2,2) \} \]
Therefore,
\[ \boxed{ A\times A= \{ (1,1), (1,2), (2,1), (2,2) \} } \]
and
\[ \boxed{ A\times A\times A= \{ (1,1,1), (1,1,2), (1,2,1), (1,2,2), \]
\[ (2,1,1), (2,1,2), (2,2,1), (2,2,2) \} } \]