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 :

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

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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 - 4 years, 3 months ago

OK...

Aditya Sky - 4 years, 3 months ago

Infact, this is the method followed to generate n-ary relations.......

Aaghaz Mahajan - 2 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$