MHB Cartesian Product of Non-Real Sets

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The Cartesian product of non-real sets can still be defined, as demonstrated with sets A and B containing tuples from different dimensions. When combining A, a subset of R², and B, a subset of R³, the resulting product is indeed in R⁵, represented as tuples. Although the elements are not identical, they are isomorphic, meaning they share the same structure. The distinction between tuples and their dimensions is often overlooked, leading to potential confusion. Clear definitions in mathematical texts are essential for understanding these concepts accurately.
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Suppose we have the sets $A=\left\{2,3\right\}$ and $B=\left\{5\right\}$, then $A$ X $B$ is defined as $\left\{(x,y)|x \in A, y\in B\right\}=\left\{(2,5), (3,5)\right\}$. But what happens when $A$ contains elements that are not in $\Bbb{R}$?

Example:
$A=\left\{(2,3),(3,4)\right\}\subset \Bbb{R}^2$ and $B=\left\{(3,2,5)\right\}\subset \Bbb{R}^3$, then following the same definition as above, we have $A$ X $B=\left\{((2,3),(3,2,5)), ((3,4),(3,2,5))\right\}$, but my book tells me that $A$ X $B$ should have elements in $\Bbb{R}^5$. Did I make a mistake?
 
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Hi Rido12,

You haven't made a mistake. Your elements belong to $\Bbb R^5$: $((2,3), (3,2,5)) = (2,3,3,2,5)$ and $((3,4),(3,2,5)) = (3,4,3,2,5)$.
 
Hey Rido! (Happy)

Strictly speaking they are not the same, but they are isomorphic (literally meaning same shape with symbol ≅).

Note that:
$$\mathbb R^2 \times \mathbb R^3 = (\mathbb R \times \mathbb R) \times (\mathbb R \times \mathbb R \times \mathbb R)
≅ \mathbb R \times \mathbb R \times \mathbb R \times \mathbb R \times \mathbb R
= \mathbb R^5
$$
Anyway, it boils down to the same thing and the distinction is often not made.
 
I like Serena said:
Hey Rido! (Happy)

Strictly speaking they are not the same, but they are isomorphic (literally meaning same shape with symbol ≅).

Note that:
$$\mathbb R^2 \times \mathbb R^3 = (\mathbb R \times \mathbb R) \times (\mathbb R \times \mathbb R \times \mathbb R)
≅ \mathbb R \times \mathbb R \times \mathbb R \times \mathbb R \times \mathbb R
= \mathbb R^5
$$
Anyway, it boils down to the same thing and the distinction is often not made.

Thanks for making this note. I assumed that in the book, a tuple made of up an $m$-tuple and an $n$-tuple is an $(m + n)$-tuple. However, that needs to be made explicitly clear in the text, for otherwise we consider them as different objects.
 
Hey ILS and Euge,

Thanks for clarifying my confusion! That is interesting to know. :D
 
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