# Finding a Cartesian Product. Let A_{1}=\{x,y\},\ A_{2}=\{1,2,3\}, and A_{3}

Finding a Cartesian Product.
Let $$\displaystyle{A}_{{{1}}}={\left\lbrace{x},{y}\right\rbrace},\ {A}_{{{2}}}={\left\lbrace{1},{2},{3}\right\rbrace},$$ and $$\displaystyle{A}_{{{3}}}={\left\lbrace{a},{b}\right\rbrace}.$$
a) Find $$\displaystyle{A}_{{{1}}}\times{A}_{{{2}}}.$$
b) Find $$\displaystyle{A}_{{{1}}}\times{A}_{{{2}}}\times{A}_{{{3}}}.$$

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Szeteib

Step 1
It is given that, the set is
$$\displaystyle{A}_{{{1}}}={\left\lbrace{x},{y}\right\rbrace},\ {A}_{{{2}}}={\left\lbrace{1},{2},{3}\right\rbrace}$$ and $$\displaystyle{A}_{{{3}}}={\left\lbrace{a},{b}\right\rbrace},$$
Step 2
a) Obtain the $$\displaystyle{A}_{{{1}}}\times{A}_{{{2}}}$$ as follows:
$$\displaystyle{A}_{{{1}}}\times{A}_{{{2}}}={\left\lbrace{x},{y}\right\rbrace}\times{\left\lbrace{1},{2},{3}\right\rbrace}$$
$$\displaystyle={\left\lbrace{\left({x},{1}\right)},{\left({x},{2}\right)},{\left({x},{3}\right)},{\left({y},{1}\right)},{\left({y},{2}\right)},{\left({y},{3}\right)}\right\rbrace}$$
Step 3
b) Obtain the $$\displaystyle{A}_{{{1}}}\times{A}_{{{2}}}\times{A}_{{{3}}}$$ as follows:
$$\displaystyle{A}_{{{1}}}\times{A}_{{{2}}}\times{A}_{{{3}}}={\left({A}_{{{1}}}\times{A}_{{{2}}}\right)}\times{\left\lbrace{a},{b}\right\rbrace}$$
$$\displaystyle={\left\lbrace{\left({x},{1}\right)},{\left({x},{2}\right)},{\left({x},{3}\right)},{\left({y},{1}\right)},{\left({y},{2}\right)},{\left({y},{3}\right)}\right\rbrace}\times{\left\lbrace{a},{b}\right\rbrace}$$
$$=\left\{\begin{array}{c} (x,1,a),(x,2,a),(x,3,a),(y,1,a),(y,2,a),(y,3,a)\\ (x,1,b),(x,2,b),(x,3,b),(y,1,b),(y,2,b),(y,3,b)\end{array}\right\}$$