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

vestirme4 2021-08-19 Answered
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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Expert Answer

Szeteib
Answered 2021-08-20 Author has 5586 answers

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:
\(\displaystyle{A}_{{{1}}}\times{A}_{{{2}}}\times{A}_{{{3}}}={\left({A}_{{{1}}}\times{A}_{{{2}}}\right)}\times{\left\lbrace{a},{b}\right\rbrace}\)
\(\displaystyle={\left\lbrace{\left({x},{1}\right)},{\left({x},{2}\right)},{\left({x},{3}\right)},{\left({y},{1}\right)},{\left({y},{2}\right)},{\left({y},{3}\right)}\right\rbrace}\times{\left\lbrace{a},{b}\right\rbrace}\)
\(=\left\{\begin{array}{c} (x,1,a),(x,2,a),(x,3,a),(y,1,a),(y,2,a),(y,3,a)\\ (x,1,b),(x,2,b),(x,3,b),(y,1,b),(y,2,b),(y,3,b)\end{array}\right\}\)

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