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Cartesian Product!

We know that given two non - void sets, say \(A\) and \(B\), their \(\textbf{Cartesian Product}\) is defined as \(A\,\times\,B\,=\,\{(a,b)\,:\,a\,\in\,A\,,\,b\,\in\,B\}\). For eg. If \(A\,=\,\{1,3\}\) and \(B\,=\,\{2,4,9\}\), then their Cartesian product is defined as \(\{(1,2),(1,4),(1,9),(3,2),(3,4),(3,9)\}\).

I was wondering what if the set \(A\) itself consists of ordered pairs, like what if \(A\,=\,\{(1,5),(1,9),(2,5)\}\) and say \(B\,=\,\{3,7\}\), then how do we define the \(A\,\times B\)?

One thing that I've observed is that ,in case, if the set that consists of ordered pairs ( n-tuples in general) can be expressed as Cartesian product of simple sets (sets that consists of numbers only), then the overall cartesian product can be found. What I mean is

Say, \(A\,=\,\{(3,4),(3,7),(2,4),(2,7)\}\) and \(B\,=\,\{2,5\}\) and \(A\times B\) is to be found, then one can proceed as follows :

\(A\times B\,=\,\{(3,4),(3,7),(2,4),(2,7)\}\,\times\,\{2,5\}\,\,=\,\{3,2\}\,\times\,\{4,7\}\,\times\,\{2,5\}\,\,=\,\{(2,3,4),(2,3,7),(2,2,4),(2,2,7),(5,3,4),(5,3,7),(5,2,4),(5,2,7)\}\).

Note by Aditya Sky
7 months, 3 weeks ago

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Do the same thing. \( A \times B = \{ ( a, b) | a \in A, b \in B \} \). It doesn't matter what the sets \(A\) and \(B\) are.

E.g. you can determine \( \{ \text{ elephant, zinc} \} \times \{ \frac{1}{2} , ( 1, 2) \} \).

Calvin Lin Staff - 7 months, 2 weeks ago

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Aditya Sky - 7 months, 2 weeks ago

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