### An alternate set-theoretic representation of n-tuples

This is just a brief note regarding the alternate set-theoretic representations of $n$-tuples.

I had read in a text on set theory that the $2$-tuple $(a,b)$ could be represented as a set in the form $\{\{a,b\},\{a\}\}$. I tried hard to think of other ways of representing such a tuple, but could not think of any then.

Today, I again went back to thinking about the issue for some reason, and came up with $\{\{a,b\},\{\{a\},b\}\}$.

What is the main issue we’re dealing with here? In a set, the order in which the elements are written does not matter. For example, $\{a,b,c\}=\{a,c,b\}=\{b,c,a\}$, etc. Hence, denoting the order in which elements appear in the tuple is tricky. The essential observation is that the reader should be able to recognize which element comes at which position in the tuple by comparing the number of parentheses it has in one element in the set as compared to the number of parentheses it has in the base element of the set (the one with the lowest number of parentheses for all elements).

For example, consider the set

$\{\{a,b,c\},\{\{a\},b,c\},\{a,\{\{b\}\},c\},\{a,b,\{\{\{c\}\}\}\}\}$.

Here the “base” element can be considered to be $\{a,b,c\}$ as it has the minimum number of parentheses for all elements. $\{\{a\},b,c\}$ has one additional set of parenthesis containing $a$. Hence, $a$ occupies the first position in the tuple. $b$ has two sets of extra parentheses containing it in $\{a,\{\{b\}\},c\}$. Hence $b$ occupies the second position in the tuple; and so on.

I can represent an $n$-tuple $(a,b,c,\dots,n)$ in the form $\{\{a,b,c,\dots,n\},\{\{a\},b,c,\dots,n\},\{a,\{\{b\}\},c,d,\dots,n\},\dots\}$.

EDIT: A representation that takes much lesser effort to write is $\{\{a, b, c, \dots, n\}, \{a\}, \{\{b\}\}, \{\{\{c\}\}\}, \dots, \}$.