Question about ordered pairs While reading through a book on discrete mathematics, it never occured to me to think of a 1-tuple. That is, an ordered singleton (or ordered unit set). Now, do such sets exist? Using the Kuratowski Definition, the ordered singleton (a) is equal to {{a}}. But as so is the ordered pair (a,a) equal to {{a}}. So is (a,a) = (a)? Could someone point out where my logic is flawed, please. Thank you in advance.

The Kuratowski definition is used for ordered pairs, so $(a)$ is not defined as $\{\{a\}\}$.
$1$-tuples are not defined, since they carry the same amount of information as a single element. The point of defining tuples was to introduce an order into the set. When the set consists of a single element, such orders are useless.
Finally, it is true that $(a,a)=\{\{a\}\}$, which is yet another reason why one wouldn't want to define $1$-tuple the way you mentioned.
