Question

Find\({{\bf{A}}^{\bf{3}}}\)if

(a) \({\bf{A = }}\left\{ {\bf{a}} \right\}\)

(b) \({\bf{A = }}\left\{ {{\bf{0,a}}} \right\}\)

Short answer

(a) \({A^3} = \left\{ {\left( {a,a,a} \right)} \right\}\)

(b) \({A^2} = \left\{ {\left( {0,0,0} \right),\left( {a,0,0} \right),(0,a,0),(0,0,a),(0,a,a),(a,a,a),(a,a,0),(a,0,a)} \right\}\)

Step-by-step solution

Cartesian Product

The cartesian product of A and B denoted by\({\bf{A \times B}}\), is the set of all ordered pairs (a,b).

Therefore, \(A \times B = \left\{ {\left( {a,b} \right)\left| {a \in A\Lambda b \in B} \right.} \right\}\)

To determine \({{\bf{A}}^{\bf{3}}}\)(a)

Here,\(A = \left\{ a \right\}\)

Therefore,

\(\begin{aligned}{}{A^3} = A \times A \times A\\ = \left\{ {\left( {a,a,a} \right)} \right\}\end{aligned}\)

To determine \({{\bf{A}}^{\bf{3}}}\)(b)

Here,\(A = \left\{ {0,a} \right\}\)

Therefore,

\(\begin{aligned}{}{A^2} = A \times A \times A\\ = \left\{ {\left( {0,0,0} \right),\left( {a,0,0} \right),(0,a,0),(0,0,a),(0,a,a),(a,a,a),(a,a,0),(a,0,a)} \right\}\end{aligned}\)