Since order doesn't matter when you have nothing or just one thing, are they the same, respectively?
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3What exactly is the "empty tuple"? – Angina Seng Sep 05 '19 at 05:58
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This depends on how you're defining your tuple. While in most reasonable set-theoretic definitions, the empty tuple is the empty set, $(x) = {x}$ may or may not be true. – eyeballfrog Sep 05 '19 at 06:03
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See the post The empty tuple or 0-tuple : its definition and properties – Mauro ALLEGRANZA Sep 05 '19 at 06:44
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In axiomatic set theory (depending on your flavor you choose), you might choose to define tuples like this so that you can use tuples before you get around to formally defining the natural numbers. See The "Empty Tuple" or "0-Tuple": Its Definition and Properties for more details.
Still, even if this is how you define 0-tuples and 1-tuples behind the curtains, I would be very reluctant to use that identification any more than absolutely necessary. The reason we define tuples is because we intend for them to be their own thing apart from sets. Tuples have terms, while sets have members.
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Well, a function $f:\{1,\ldots,n\}\rightarrow A$ can be represented as a sequence or $n$-tuple $f=(f(1),\ldots,f(n))$ or $f=(f_1,\ldots,f_n)$.
In this way, the ''empty tuple'' corresponds to the $0$-tuple or the empty function $f:\emptyset\rightarrow A$. It is by definition well-defined.
A ''one-tuple'' corresponds to the function $f:\{1\}\rightarrow A$ given by $f=(f_1)$. Functions are relations, here $f=\{(1,f(1))\}$, and relations are sets.
Hope it helps.
Wuestenfux
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