If \(A=\{-1,1\}\), Find \(A\times A\times A\)
Question
If \[ A=\{-1,1\}, \] find \[ A\times A\times A. \]
Solution
Given:
\[ A=\{-1,1\} \]
The Cartesian product \[ A\times A\times A \] is the set of all ordered triples \((a,b,c)\) where \[ a,b,c \in A. \]
Thus, all possible ordered triples are:
\[ A\times A\times A= \{ \]
\[ (-1,-1,-1), (-1,-1,1), \]
\[ (-1,1,-1), (-1,1,1), \]
\[ (1,-1,-1), (1,-1,1), \]
\[ (1,1,-1), (1,1,1) \} \]
Therefore,
\[ \boxed{ A\times A\times A= \{ (-1,-1,-1), (-1,-1,1), (-1,1,-1), (-1,1,1), \]
\[ (1,-1,-1), (1,-1,1), (1,1,-1), (1,1,1) \} } \]