DEF-TUP. Tuple.
The ordered pair / 2-tuple / couple of $a$ and $b$, written as $(a,b)$, is the set
\[\{\{a\},\{a,b\}\}.\]The 3-tuple / triple of $a$, $b$ and $c$ is
\[(a,b,c) = ((a,b),c).\]4-tuples / quadruples, 5-tuples / quintuples, etc. are defined analogously.
REM-TUP-AR.
The arity of a tuple is not an intrinsic property. For example, $(a,b,c)$, the 3-tuple of $a$, $b$ and $c$, is the 2-tuple of $(a,b)$ and $c$. Technically, we can not say $(a,b)$ is a 2-tuple, but only a 2-tuple of $a$ and $b$.
PROP-TUP.
For any $a_1$, $\cdots$, $a_n$ and $b_1$, $\cdots$, $b_n$ ($n\geq 2$),
\[(a_1,\cdots,a_n) = (b_1,\cdots,b_n) \enspace\Leftrightarrow\enspace a_1=b_1\land\cdots\land a_n=b_n.\]